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D. W. TAYLOR, E.D., D. Sc., L.L.D. 










Entered at Stationers' Hall. London 

Stanhope fl>res 




THE intention of this work is to treat in a consistent and con- 
nected manner, for the use of students, the theory of resistance and 
propulsion of vessels and to give methods,, rules and formulae which 
may be applied in practice by those who have to deal with such 
matters. The contents are based largely upon model experiments, 
such as were initiated in England nearly half a century ago by Mr. 
William Froude and are now generally recognized as our most effec- 
tive means of investigation in the field of resistance and propulsion. 
At the same time care has been taken to point out the limitations 
of the model experiment method and the regions where it ceases 
to be a reliable guide. 

During the years that the author has directed the work of the 
U. S. Experimental Model Basin many results obtained there have 
been published in the Transactions of the Society of Naval Archi- 
tects and Marine Engineers and elsewhere, so, naturally, the 
experiments at the U. S. Model Basin have been made large use of 
wherever applicable. It will be found, however, that they are in 
substantial agreement with the many published results of the 
work of other experimental establishments of this kind. 

Although the coefficients and constants for practical application 
are mainly derived from the author's experience at the Model 
Basin and elsewhere, and are necessarily general in their nature, 
endeavor has been made wherever possible to develop formulae 
and methods in such a manner that naval architects and engineers 
using the book may, if they wish, adopt their own constants derived 
from their special experience. 

For instance, by the methods given it will be found possible to 
estimate closely the effective horse-power of a vessel having the 
form of what I have called the Standard Series, but it will also be 
found possible, by the same methods, to determine with fair accu- 



racy the variation of resistance with changes of dimensions, etc., 
of vessels upon almost any lines for which a naval architect may 
have reliable data, and which, on account of satisfactory past 
results, or for other reasons, he may wish to use. 

The science of Naval Architecture is not yet developed to a point 
where our knowledge of resistance and propulsion is complete. 
While the author naturally hopes that this volume will at least 
partially bridge some of the gaps hitherto existing, much work 
remains to be done, and in a number of places attention is called 
to the need of further investigation of various questions. While 
we know something, for instance, in a qualitative way of the effect 
of shallow water upon resistance, information which would enable 
us to solve satisfactorily many problems arising in this connection 
is lacking, and apparently can be obtained only by much experi- 
mental investigation. When dealing with questions of wake and 
thrust deduction we are not yet upon firm ground, and it is to be 
hoped that the excellent work recently done by Luke in this con- 
nection will soon be supplemented by even more extensive investi- 


WASHINGTON, D.C., July, 1910. 


Preliminary and General 









































Trials and Their Analysis 




The Powering of Ships 





i. Stream Lines 

1. Assumptions Made. The consideration of stream lines or 
lines of flow will be restricted mainly to the case of the motion of 
liquid past a solid. It is sufficient for present purposes to define 
a liquid as a fluid which is incompressible, or virtually so, such as 

The difficulties in the way of adequate mathematical determi- 
nation of the motion of liquids past solids such as ships have 
hitherto been found insuperable. The mathematics of the motion 
of liquids is complicated; even the simple cases which can be dealt 
with mathematically require assumptions which are far from actual 
conditions in practice. Thus, when considering the motion of 
solids through a liquid, or what is the same thing mathematically, 
the motion of a liquid past solids, it is assumed that the liquid 
is " perfect " or has no viscosity and that the solid is frictionless, 
that is to say, that the liquid can act upon the solid only by pres- 
sure which must at each point be normal to the surface. In most 
cases that are dealt with mathematically, it is further assumed 
that the fluid or liquid extends to an infinite distance from the 

2. Steady Motion Formula. We cannot deal satisfactorily with 
problems of resistance by mathematical analysis, but in spite of 
the somewhat artificial assumptions involved, the results of mathe- 
matical analysis applied to a perfect liquid are of interest and value 
as they indicate tendencies and have large qualitative bearing upon 
the phenomena of the motion of water past ships. 


One mathematical conclusion in this connection is particularly 
valuable. It is known as the steady motion formula and is as 

t+*+ z = h . 

W 2g 

In the above formula, p denotes pressure of the liquid per unit 
area, w denotes weight per unit volume, v denotes velocity of 
flow in units of length per second, g acceleration due to gravity 
in units of length per second, z denotes height above a fixed level 
and h is a constant for each stream line, being called the head. 
It is usually convenient to express p in pounds per square foot, W 
in pounds per cubic foot, v and g in feet per second, z and h in feet. 

The above formula applies to the steady motion of an infinite 
mass of perfect liquid. For such liquid the value of h is constant 
for all particles passing a point fixed in the liquid. These particles 
form a continuous line called a stream line, and in steady motion, no 
matter how many twists and turns the stream line takes, the above 
formula applies to its pressure, velocity and elevation at every point. 
It will be observed that contrary to what might at first be thought, 
the greater the velocity at a point of the stream line the less the 
pressure, and vice versa. That is to say, if a stream of perfect 
liquid flows in a frictionless pipe of gently varying section, the 
pressure increases as the size of the pipe increases and decreases 
as the size of the pipe decreases. This is demonstrable in the case 
of flow through pipes, although it is necessary to have the changes 
of section very gradual in order to obtain the smooth continuous 
motion to which alone the steady motion formula is applicable. 

3. Application of Steady Motion Formula to Ships. The 
steady motion formula applies to the motion of a liquid, including 
motion past a solid at rest. In the case of ships, we are interested 
in the motion of a solid through a liquid at rest. The two cases 
are, however, as already stated, mathematically interchangeable. 
Suppose we have a ship moving uniformly through still water 
which extends indefinitely ahead and astern. If we suppose both 
ship and water given the same velocity, equal and opposite to- the 
velocity of the ship in the still water we have the ship at rest and 
the water flowing past it. The mutual reactions between ship 


and water are identical whether we have the ship moving through 
still water or the water flowing past the fixed ship. To the latter 
case, however, the steady motion formula applies if we neglect 
friction and the mathematical treatment is much easier. 

If the ship is in a restricted channel so shallow and narrow that 
the area of the midship section of the ship is an appreciable fraction 
of the area of the channel section, the steady motion formula 
teaches us that with the water flowing past the fixed ship there 
will be abreast the central portion of the ship where the channel 
area is diminished an appreciable increase in velocity of flow and 
reduction of pressure. 

The surface being free, reduction of pressure would result in 
depression of surface. Passing to the case of the ship moving 
through the channel we would infer that the water is flowing aft 
abreast the central portion of the ship and that there is a depression 
in this vicinity. 

This, as a matter of fact, occurs in all cases, but in open water the 
motions are not so pronounced, and it is seldom possible to detect 
them by the eye. In a constricted channel, however, it is generally 
easy to detect the depression abreast the ship since it extends to 
the banks. If these are sloping the depression shows more plainly 
than it does against vertical or steep banks. 

There might be quoted many other illustrations of the validity 
of the steady motion formula taken from phenomena of experience. 
There is no doubt of its general validity within certain limits as 
regards motion of water around solids, but in considering any par- 
ticular case it should not be applied regardless of its limitations. 

4. Failure of Steady Motion Formula. The steady motion 
formula assumes frictionless motion. Water is not frictionless, 
but its friction is not sufficiently great in the majority of cases to 
seriously affect steady motion directly. 

The main failure of the steady motion formula as regards prac- 
tical cases is in connection with the transformation of pressure into 
velocity and vice versa. Neglecting variations of level the steady 

motion formula is *- -\ = a constant. By the formula the 

W 2g 

greater the velocity the less the pressure, and if the velocity be 


made sufficiently great the pressure must become negative. Now, 
negative pressure would be a tension, and liquids are physically in- 
capable of standing a tension. Hence, when the case is such that the 
steady motion formula would give a tension the motion that would 
be given by the steady motion formula becomes impossible and 
the formula fails. In practice, in such a case, instead of steady mo- 
tion we have eddying, disturbed motion. In fact, in actual liquids, 
when the motion is such as to cause a reduction of pressure, eddy- 
ing generally makes its appearance some time before the pressure 
becomes zero. But for moderate variations of pressure we find for 
actual liquids pressure transformed into velocity according to the 
steady motion formula with great accuracy. The transformation 
of velocity into pressure, however, according to the steady motion 
formula, without loss of energy, is not common in practice. For 
instance, experiments at the United States Model Basin have 
shown that air will pass through converging conical pipes with 
practically no loss of head except that due to friction of the pipe 
surface. But when passing through diverging cones, even when the 
taper is but one-half inch of diameter per foot of length, there is 
material loss of head beyond that due to friction. It appears 
reasonable to suppose that the difficulties found in converting 
velocity of actual fluids into pressure without loss of energy are 
connected with the friction of the actual fluids, both their internal 
friction or viscosity and their friction against the pipes or vessels 
containing them. 

To sum up, we appear warranted in concluding that in flowing 
water pressure will be transformed into velocity according to the 
steady motion formula with little or no loss of energy in most 
cases, provided the pressure is not reduced to the neighborhood 
of zero, and that velocity will be transformed into pressure but with 
a loss of energy dependent upon the conditions. 

It is evident that if the total head or average pressure is great, 
given variations of pressure and velocity can take place with closer 
approximation to the steady motion formula than if the total head 
be small. 

5. Sink and Source Motion. The mathematics of fluid motion 
or hydrodynamics being somewhat complicated will not be gone 


into here, but results will be given in a few of the simplest cases 
which are of interest and have practical bearing. Suppose we 
have liquid filling the space between two frictionless planes which 
are very close together. The motion will be everywhere parallel 
to the planes, and hence will be uniplanar or in two dimensions 
only. Suppose now that liquid is being continually introduced 
between the planes at some point. It will spread radially at an 
equal rate in every direction. The point of introduction of the 
liquid is called a "source." Fig. i indicates the motion, S being 
the source. If liquid were being abstracted at S the motion at 
every point would be directly opposite that shown in Fig. i and 5 
would be what is called a "sink." The sink and source motion is 
not physically possible because the steady motion formula applies, 
and for velocity and pressure finite at a distance from 5 the velocity 
at 5 would be infinite. But it will be seen presently that the mathe- 
matical concept of sinks and sources has a bearing upon possible 
motions. Suppose that instead of a single source or sink we have 
in Fig. 2 a source at A and a sink of equal strength at B. Liquid 
is being withdrawn at B at the same rate at which it is being intro- 
duced at A and in time every particle introduced at A must find 
its way out at B. The motion being steady the paths followed are 
stream lines. These paths are arcs of circles. A number of these 
circular arcs are indicated in Fig. 2. They are so chosen that the 
" flow " or quantity of fluid passing between each pair of circles 
is the same. Adjacent to the line connecting the sink and source the 
path is direct, the velocity great and the circles close together. As we 
leave this line the path followed from source to sink is circuitous, 
the velocity low and the spacing of the circles greater and greater. 
6. Sink and Source Motion Combined with Uniform Stream. - 
Suppose, now, that the liquid in which the source is found is not at 
rest but is flowing with constant speed from right to left. Fig. 3 
shows the result of the injection of a source into such a uniform 
stream. In this case we have a curve of demarcation DDD sepa- 
rating the liquid which comes rom the source and the other liquid. 
No liquid crosses this curve. Now, the motion being frictionless 
it makes no difference whether DDD is an imaginary line in the 
moving liquid or the boundary of a frictionless solid. Hence if in 


a uniform stream we put a frictionless solid of the shape DDD the 
motion outside of it will be the same as in Fig. 3. This motion 
will be completely possible if we could have a frictionless solid like 
DDD, since we no longer have the source with its impossible con- 
ditions as regards velocity and pressure. 

In Fig. 3 DDD extends to infinity. Suppose, now, in a uniform 
stream we put a sink and a source of equal strength as at A and B 
in Fig. 4. The direction of flow of the uniform stream is supposed 
parallel to AB. In this case the closed oval curve CCC separates 
the liquid which appears at the source and disappears at the sink 
from the liquid of the uniform stream. Hence, if a frictionless 
solid of the shape of CCC took the place of the liquid inside the 
oval the motion of the stream outside would be unchanged. 

Of course the shape and dimensions of CCC would vary with 
the relative strengths of source and sink and velocity of stream. 
Instead of one source and one sink we may distribute a number 
along the line A B enabling us to modify the shape and proportions 
of the line of demarcation CCC. The author (see Transactions of 
the Institution of Naval Architects for 1894 and 1895) nas extended 
this method to cover the case of an infinite number of infinitely 
small sources and sinks, thus enabling us to determine lines of 
demarcation or stream forms both in plane and solid motions, 
closely resembling actual ships' lines. Not only the stream forms 
but also the velocities and pressures along them can be determined, 
but the process is laborious and has not so far been given sufficient 
practical application to warrant following further here. 

The closed ovals due to a source and a sink in a uniform stream 
somewhat resemble ellipses as appears from Fig. 4. 

7. Flow in Two Dimensions in Practice. While to reduce the 
motion to one plane or two dimensions, the assumption was made 
that it took place between two frictionless parallel planes so close 
together that the space between them practically constituted a 
single plane, it should be pointed out that motion practically iden- 
tical with plane motion occurs in practice. Suppose we have a 
body of cylindrical type of infinite length moving in some direction 
perpendicular to its axis. The motion past will be identical in all 
planes perpendicular to the axis. 


The motion past an actual body of cylindrical type whose length 
though not infinite is great compared with its transverse dimensions 
will, over a great portion of the length, be practically the same as 
if the length were infinite. A propeller strut is a case in point. 
Ideal plane flow has direct practical bearing upon the motion past 
such fittings. 

8. Stream Lines past Elliptic Cylinders. One general case of 
uniplanar motion that has been solved mathematically is that of an 
elliptic cylinder moving parallel to either axis in an infinite mass of 
liquid. The circle is a special case and a plane lamina is another 
special case where one axis of the ellipse is zero. The general 
mathematical formulae expressing the motion of an elliptic cylinder 
through liquid may be referred to in Lamb's " Hydrodynamics," 
edition of 1906, Article 71. They do not give directly the stream 
lines past an elliptic cylinder but the latter can be deduced from 
them. Figs. 5 to 15 show plane stream lines or lines of flow past 
various types of elliptic cylinders. The lines in the first quadrant 
only are shown as they are symmetrical in the other three. The 
proportions of the ellipses are given, the semi-major axis being 
always taken as unity. Fig. 10 shows flow around a circular 
cylinder and Fig. 15 flow past a plane lamina of indefinite length 
and unit half breadth. The flow around a lamina is, however, 
impossible since the formula would require an infinite velocity 
around the edges, or, as indicated in Fig. 15, the stream line spac- 
ing in the immediate vicinity of the edge would become infinitely 

9. Pressure Variations around Elliptic Cylinders. Figs. 16 and 
17 give some idea of variation of pressure along the central stream 
line and around the surface of the cylinders. A particle approach- 
ing a cylinder along the axis steadily loses velocity and gains pressure 
until it comes to rest against the cylinder when its pressure is in- 
creased by the total velocity head of the undisturbed stream. The 
particle then starts around the cylinder, rapidly gaining velocity 
and losing pressure until at a point where it has moved but a short 
distance around the cylinder it has regained the velocity and re- 
turned to the pressure it had in the undisturbed stream. The 
velocity then continues to increase and the pressure falls as shown 


until the particle is abreast the center of the cylinder when the 
velocity is at a maximum and the pressure at a minimum. 

Figs. 1 6 and 17 show negative pressures but these are only 
relatively negative. For convenience the diagrams are drawn as 
if the pressure in the undisturbed stream were zero. The actual 
pressure in any case is the pressure of the figure with the pressure 
in the undisturbed stream added. Bearing in mind also that in 
each figure the unit of pressure is the pressure head due to the 
velocity of the undisturbed stream, or the velocity head of the 
stream, Figs. 16 and 17 shed a good deal of light upon the effect of 
variation of proportions. Thus, for an ellipse one-tenth as wide 
as long, the maximum reduction of pressure abreast the center is 
about one-fifth the velocity head. For the ellipse four-tenths as 
wide as long, the maximum reduction is nearly the velocity head. 
For the ellipse as wide as long (the circle), the reduction is three 
times the velocity head. For the ellipse two and one-half times as 
wide as long, the reduction is over eleven times the velocity head, 
and for the ellipse five times as wide as long, the reduction is thirty- 
five times the velocity head and about one hundred and seventy- 
five times the reduction for the ellipse one-tenth as wide as long. 

The velocity head being proportional to the square of the speed, 
the reduction in or increase of pressure at every point is propor- 
tional to the square of the speed, and hence if any of the cylinders 
were pushed to a high enough speed the reduction of pressure 
abreast the center would equal the original pressure in the undis- 
turbed stream, and hence the pressure abreast its center would 
reduce to zero resulting in eddying. But eddying would appear 
in the case of an actual cylinder long before the pressure abreast 
the center became zero. For the excess velocity amidships would 
not be fully converted into excess pressure on the rear of the cylinder 
as required for perfect stream motion, and eddying would show 
itself aft. 

10. Disturbance Abreast Cylinder Centers. It is evident from 
Figs. 1 6 and 17 that in the case of a cylinder moving through still 
water the maximum sternward velocity of the water at any point 
of the cylindrical surface is abreast the center of the cylinder. It 
is also true that for motion parallel to the axis of x the greatest 


sternward velocity for any value of y is on the axis of y. It is of 
interest to trace the variation of velocity as we pass along the axis 
of y. Fig. 18 shows sections of seven types of cylinders ranging 
from the flat plate, No. i, which is all breadth, to the circle, 
No. 4, and the ellipse five times as long as wide, No. 7. They all 
have unit half breadth on the axis of y and are supposed to move 
with velocity V parallel to the axis of x. 

Fig. 1 8 shows also curves of sternward velocity u of the water 
as we pass out from the cylinder along the axis of y expressed as 
a fraction of the speed of advance of the cylinder. It is seen that 
the long cylinder causes the minimum disturbance at the surface 
of the cylinder where y = i, but the maximum beyond y = 4. 
Fig. 1 8 shows markedly the very great variations of disturbance 
in the vicinity of the cylinder with variation of ratio of breadth to 
length. The areas of all the curves of Fig. 18 are the same, being 
equal to V X (half breadth). The dotted square in the figure 
shows this area. 

u. Tracks of Particles. While Figs. 5 to 15 show stream lines 
or flow past the cylinders, they give little idea of the paths followed 
by particles of water when a cylinder is moved through water 
initially at rest. 

Rankine gave, many years ago, the differential equation to these 
paths for the motion of a circular cylinder, and while this equation 
cannot be integrated it is possible by graphic methods to determine 
the resulting paths with ample accuracy. 

Fig. 19 shows the paths followed by a few particles at various 
distances from the axis as a cylinder of the size indicated by the 
dotted semicircles in the figure passes along the axis from an infinite 
distance to the right to an infinite distance to the left. 

A on each path shows the original position of the particle when 
the cylinder is at an infinite distance to the right. B, C, D, E 
and F on the paths of the particles show positions when the cylinder 
is at B, C, D, E, and F, on the axis as indicated. 

The paths are symmetrical, and G denotes the position of each par- 
ticle when the cylinder has passed to an infinite distance to the left. 

Fig. 19 shows the curious result that each particle is shifted 
ultimately a certain distance parallel to the direction of motion of 


the cylinder. This could not occur if the cylinder started from 
rest at a finite distance from the particle, and came to rest within 
a finite distance of the particle. For such motion the particles must 
on the average be slightly displaced in a direction opposite to the 
direction of motion of the cylinder. 

12. Stream Lines around Sphere. While there are very few 
mathematical determinations of stream lines in three dimensions 
those for the sphere are known and it is of interest to compare them 
with those for a circular cylinder shown in Fig. 10. The stream 
lines past a sphere are identical in all planes through the axis parallel 
to the direction of undisturbed flow. 

They are shown in Fig. 20, and in Fig. 21 are shown curves of 
pressure variation along the horizontal axis and around the sphere, 
also along the horizontal axis and around a circular cylinder. The 
curves show as might be expected that the sphere creates less dis- 
turbance. This is evidently because the water is free to move in 
three dimensions around the sphere, while it is restricted to plane 
motion around the cylinder. 

The increase of pressure in front of the sphere is less. There is 
a sudden rise close to the intersection of axis and sphere. At this 
point the increase of pressure is the same as in the case of the 
cylinder, being the pressure head due to the undisturbed velocity. 
Abreast the center the loss of pressure is one and one-half times that 
due to the velocity as contrasted with three times the velocity head 
in the case of the cylinder. In other words, if a sphere is advancing 
with perfect stream line action through water otherwise undisturbed 
the water abreast the center is flowing aft with one-half the velocity 
of advance of the sphere. In the case of the circular cylinder the 
water abreast its center flows aft with velocity equal to the velocity 
of advance. 

2. Trochoidal Water Waves 

i. Mathematical Waves. Ocean waves during a storm are 
generally confused rather than regular. They are not of uniform 
height or length from crest to crest, and the crests and hollows 
extend but comparatively short distances. After a storm, how- 
ever, the confused motion settles down into rather uniform and 


regular swells and the motion approaches that of mathematical 
waves. For mathematical treatment it is necessary to assume 
regularity of motion. We may define a series of mathematical 
waves as an infinite series of parallel infinitely long identically 
similar undulations advancing at uniform speed in a direction 
perpendicular to that of their crests and hollows. The constant 
distance between successive crests is called the length of the waves 
or the wave length, the distance between the level of the crest and 
the level of the hollow is called the height of the wave, and the time 
interval between the passage of successive crests by a fixed point 
is called the period of the wave. 

Mathematical waves are cases of motion in two dimensions, 
since the motion is identical in all planes perpendicular to the 
wave crests. 

2. Trochoidal Wave Theory. The most commonly accepted 
theory of regular wave motion is that called the " trochoidal 
theory." Its mathematics is too long and difficult to be gone 
into here, and I shall undertake only to give some of the formulae 
and conclusions that have been evolved by the eminent mathe- 
maticians who have worked in this field. 

Of the British mathematicians who have contributed to the 
trochoidal theory, Airy and Rankine were especially prominent 
shortly after the middle of the last century. 

By the trochoidal theory, in water of unlimited depth each 
particle describes at a uniform rate a circular orbit, making one 
complete revolution per wave period, the radii of the orbits being 
a maximum for surface particles and decreasing indefinitely with 

Referring to Fig. 22 let the wave length be denoted by L and 
let R be the radius of a circle whose circumference is L. Then 

R = - Suppose we locate this circle with its center midway 

2 7T 

between the levels of crest and hollow and take a point P on the 


radius at a distance r or from the center, H being the wave 

height. Then, if the circle rolls on the line AB the point P will 
describe a trochoid giving the outline of the wave surface. This 


trochoid shows the contour assumed by particles originally at the 
surface level. Similarly, particles originally at any level below the 
surface are found along a trochoidal surface having the same 
diameter of rolling circle but less orbit radius, the radius diminish- 
ing indefinitely with depth. 

Fig. 23 shows the trochoids at various levels, orbit diameters 
and contours of lines of particles which in undisturbed water were 
equally spaced verticals. The cycloid the limiting trochoid 
is shown, but it is not possible for sharp crested waves to appear 
in practice. They break long before they approach closely the 
limiting cycloid. 

Fig. 23 is for water of unlimited depth. In water of finite depth, 
by the trochoidal theory each particle describes an elliptical orbit 
instead of the circular orbit of deep water. Referring to Fig. 24 
let A BCD be the " rolling circle" whose perimeter, as before, is 
equal to the wave length from crest to crest. Let the ellipse 
EFGH of center the same as the center of the rolling circle be the 
orbit of the surface particles. Let OP' be the radius of a concentric 
circle of diameter the same as the major (horizontal) axis of the 
ellipse. Then, as the rolling circle moves, let the radius OP' 
revolve with it and the ellipse move horizontally with it without 
revolving. Draw vertical lines as P'N from the successive posi- 
tions of P' to meet the ellipse in points such as P. The modified 
trochoid obtained by joining all points such as P is the surface profile 
of the wave. 

The horizontal and vertical axes of the elliptical orbits are not 
independent but vary with the depth of water, the depth below 
the surface, etc. 

Thus let a and b denote the horizontal and vertical semi-axes, 
respectively, of an elliptical orbit whose center is a distance h 
below the orbit centers of the surface particles. Let oo&o denote 
the semi-axes of the surface orbit. Let d denote the depth from 
center of surface orbits to the bottom. Let R denote the radius 
of the rolling circle and w the angular velocity with which it must 
roll to have its center travel at the speed of the wave. 

Let L denote the wave length in feet, v the wave speed in feet 
per second, g the acceleration of gravity and e the base of hyper- 


bolic logarithms. Then the formulae connecting the above quan- 
tities are as follows: 

e L i 


b = 

e L e 


g L e L 

1x(d-h) -2*( 

L + e 

e L e L 

and w = m/ 

a R 


and if T denote the period in seconds 

bo g 

To pass to the case of indefinitely deep water, we put d = oo . 
Then a Q = b = r , say, and if r denote the radius of the circular 
orbit at a distance h below the surface orbits, we have 


a = b = r = r e L . 
As before, v = &R, but 

R 2 TT 

Substituting for g the value 32.16 and for TT its value, we have the 
following formulas for deep-water trochoidal waves: 

Velocity in feet per second = v = 2.26 
Velocity in knots = V = 1.34 

Period in seconds = T= 0.442 

Length in feet = .557 V 2 = 5.118 r 2 . 

The above rather complicated-looking formulae express com- 
pletely the motion under the trochoidal theory. 


3. Mechanical Possibility of Trochoidal Waves. For the 
motion to be possible it must satisfy, 

1. The condition of continuity. 

2. The condition of dynamical equilibrium. 

3. The boundary conditions. 

4. The conditions of formation. 

The mathematical investigation of the above conditions is too 
long and complicated to be given here. The results only can be 
given. . As regards continuity, it is found that the motion is possible 
in water of infinite depth, but that in water of finite depth the 
equation of continuity is not quite satisfied. 

As regards dynamical equilibrium, again we find that the motion 
is not quite possible in finite depth, the pressure at the surface being 
not quite constant, which it must be from boundary conditions* 
In infinite depth, however, the pressure as deduced from the tro- 
choidal formulae is constant along the wave profile and hence the 
motion is possible. 

The only other boundary conditions to be satisfied are those at 
the bottom, and these are satisfied by the trochoidal formulae, 
since they give at the bottom horizontal motion only (b = o) when 
the water is of finite depth and no motion at all (r = o) when the 
water is of infinite depth. 

Finally, as regards the condition of formation, it is a theorem 
of hydrodynamics that a perfect liquid, originally at rest, that has 
been acted upon by natural forces only, cannot show molecular 
rotation. The trochoidal wave motion involves a slight molecular 
rotation, and hence falls slightly short of being a possible motion 
in both finite and infinite depths. 

We conclude, then, that trochoidal wave motion falls slightly 
short of being mathematically possible; but it would require a very 
small change in the motion to render it possible. This and other 
considerations which will be pointed out later warrant the adoption 
of the trochoidal theory as a working approximation. 

4. Trochoidal Wave Profiles. The formulae already given may 
be supplemented by those representing the trochoidal contours at 
various depths. They are x = Rd a sin 6, y = h b cos 6, where 
x is measured horizontally, y is measured vertically down from the 


surface orbit centers R, a and b have the values already given and 
6 is angle rolled through by the rolling circle, being = o for an 
initial condition where the radius of the rolling circle is vertical and 
its center ;under the crest of the trochoid. Of course, in deep water 
a = b = r. 

Fig. 25 shows the wave surface profiles for three waves, each 300 
feet long and 20 feet high, but in three depths of water, namely oo , 
25 feet and 15 feet. These three profiles have the same line of 
undisturbed water level. It is seen that in each case the orbit 
center, or mid height of wave, is above the level of the undisturbed 

water. For deep-water waves the amount of this elevation is 
r 2 

- > r being the surface orbit radius. For shallow- water waves it 
2 R 

is j^- The pressure on any trochoidal subsurface for deep-water 

waves is uniform and the same as the pressure in undisturbed water 
on the corresponding layer. 

For subsurface trochoids the elevation of orbit centers is given 

by e R , where h is the distance of the orbit centers from the 

level of surface orbit centers. 

5. Energy of Trochoidal Waves. Consider now the energy of 
waves in deep water. This is partly potential, due to the fact that 
in wave motion the particles are elevated on the average above their 
still-water positions, and partly kinetic, due to the velocity with 
which the particles of water are revolving in their circular orbits. 

Let w denote the weight of one cubic foot of water. Then the 
potential energy of a mass of water one foot wide and one wave 
length long, i.e., extending from one crest to the next, is 


where r is surface orbit radius or one-half the wave height. 
Now R = - - Substituting this value we may write 

2 7T 


\ L~ 


In practice, for actual waves is a small fraction and for 


most purposes can be ignored. The kinetic energy of the mass of 
water as above is exactly the same as the potential energy, or if 
we denote it by E k , 



V 1 

While the potential and kinetic energies of a mass of water in 
wave motion remain constant, there is constant transmission of 
energy going on. 

Fig. 26 shows a number of positions of a distorted vertical or 
line of particles originally vertical in still water. During part of 
the motion, energy is being transmitted across this vertical in the 
direction in which the wave is traveling and during the rest of the 
motion it is being transmitted backward. One wave length away 
is a similar distorted vertical moving in the same way, so there is 
at no time net gain or loss of energy to a mass of water one wave 
length long. But the energy transmitted forward across a surface 
originally a vertical plane is during one wave passage greater than 

the energy transmitted backward by the quantity ( i - J - 

4 \ J-i / 

This is identical with the kinetic or potential energy of the wave, 
so that a mass of water extending over one wave length receives 
from the water behind it and communicates to the water in front 
of it during the passage of one wave a net amount of energy equal 
to its kinetic or potential energy. 

While this is the net energy transmitted the rate of transmission 
is much higher during a portion of the wave passage than the aver- 
age. Thus, if 6 is the angle in its orbit from the vertical of the 
radius r of a surface particle, the rate of transmission of energy 
through the distorted vertical terminating in the surface particle 
(see Fig. 26) is given by 

By integrating this between the limits 6 = o and 2 IT, we get the 
expression given above for the net energy transmitted. Fig. 27 


shows a curve of rate of transmission of energy for a deep-water 
wave 300 feet long and 20 feet high. Between o and 90 and 
270 and 360 there is positive transmission. Between 90 and 
270 there is negative transmission. The average rate of trans- 
mission is indicated on the figure. 

6. Superposition of Trochoidal Waves. If we superpose two 
trochoidal wave series of the same length L, and hence the same 
speed of advance, which are traveling in the same direction with 
parallel crests a distance a apart, the result is a single series of 
length L. 

If we denote by HI, HZ the wave heights of the two components 
and by H the height of the resultant series, we have 

R L 

Evidently if a = o, or the crests of the component series are im- 

mediately over one another, cos =i and H 2 =(Hi+ H^) 2 . In 


this case the wave height of the resultant series is the sum of 

the component heights. If a = irR we have cos = i and 


H z = (Hi Hz) 2 . In this case the crest of one component is 
immediately over the hollow of the other, and the height of the 
resultant series is the difference of the heights of the components. 
If in this case Hi=H z , the components extinguish each other and 
the resultant is still water. 

7. Wave Groups. A very important deduction from the tro- 
choidal theory is the theory of wave groups. If we superpose 
two trochoidal systems of equal heights, but slightly different 
lengths, we have at one point of the resultant series waves of 
double the height of either component and at another point waves 
of zero height, since at one point of the series we would have crest 
superposed on crest and at another point crest superposed on 
hollow. The resultant series in this case would consist of a number 
of groups of waves, each with a wave of maximum height in the 
middle and of heights steadily decreasing ahead and astern of the 
middle until waves of infinitesimal height or bands of practically 
still water separate the groups. It can be easily proved from the 


trochoidal theory that each group will travel as a whole at just half 
the speed appropriate to the wave length of the original compo- 
nents. The individual waves, however, travel at their natural 
speed, which is double the group speed. A wave will advance 
from the rear of a group where its height is infinitesimal and pass 
through the group, growing until it reaches a maximum at the 
center of the group and then dwindling as it goes forward until 
its height again becomes infinitesimal at the front of the group. 
One can readily start a group of circular waves by dropping a 
pebble from a bridge into a placid stream. This shows general 
features somewhat similar to the theoretical trochoidal group. 
If the reflection in the water of the side of the bridge is distinct a 
wave can be watched as, first becoming noticeable at the rear, it 
passes through the group, reaching a maximum height and dying 
down again. as it gets further and further ahead of the center of 
the group. It will be found, however, that unlike the theoretical 
trochoidal group, which has similar groups some distance ahead 
and astern of it, the circular group gets wider and wider from front 
to rear. If, for instance, at a given time it shows five appreci- 
able waves, it will be seen a little later to show six, then seven, and 
so on. 

8. Applicability of Trochoidal Theory. Having considered the 
nature of the motions and the conclusions that can be drawn from 
the trochoidal wave theory, it is time to consider its applicability 
to actual water waves. We know that actual waves cannot be 
exactly trochoidal, and we are not warranted in assuming without 
some confirmatory evidence that the trochoidal theory gives us 
waves substantially the same as actual waves. Now, as already 
pointed out, actual waves are almost never regular, so that a rather 
rough approximation, mathematically, to the ideal regular waves 
would, as a rule, resemble them more closely than do the actual 
waves. Hence, if we find that the trochoidal theory adequately 
represents the most important feature or features of wave motion 
we need not be concerned as to minor features. 

Stokes has developed a mechanically possible theory of wave 
motion where the wave profiles are sines and the speed of the wave 
is not independent of the height, but increases slightly with it. 


For waves of ordinary proportions, however, the speed is practi- 
cally the same as by the less complex trochoidal theory. 

It appears, then, that for the proportions occurring in practice 
trochoidal waves are in substantial agreement with mathematical 
waves free from their minor mechanical imperfections. 

Now, what is the basic feature of trochoidal waves? It seems 
that it may fairly be said to be the fact that the velocity of advance 
depends only upon the length from crest to crest and the depth of 
the water. We have seen that the formula for this velocity is 

2nd -2xd 

2 7T ^ H^d 2 7T 

e L + e L 

Small-scale experiments in tanks, such as those of the Weber 
Brothers, who published their results in 1825, have given results 
consistent with the trochoidal theory; but it is obviously desirable 
to compare the theory with actual full-sized waves, which it is 
very difficult to do with accuracy. 

9. Gaillard's Experimental Investigations of Trochoidal Theory. 
Major D. D. Gaillard, U. S. A., in a monograph on Wave 
Action in Relation to Engineering Structures (Professional Papers, 
No. 31, Corps of Engineers, U. S. Army), has compared reported 
speeds of advance and speeds computed by the trochoidal theory 
in eighty-five cases of ocean waves observed by various people at 
various places. Of these eighty-five reported velocities, twenty- 
three were higher than the computed velocities corresponding to 
the observed length and sixty-two were lower, the average of the 
whole number being nearly 9 per cent below the average computed 
velocity. While giving due consideration to the difficulties in the 
way of accurate observation, the agreement between these observa- 
tions and the trochoidal theory is certainly not wholly satisfactory. 

Fortunately, Major Gaillard gives a further comparison of the 
trochoidal theory with a large number of observations, taken by 
himself or under his direction, under conditions favorable to 
accuracy. These observations were made in 1901 and 1902 in the 
Duluth, Minn., ship canal and in Lake Superior near the canal. 

The canal in question is about 300 feet wide, 26 feet deep, 


where the observations were taken, and about 1000 feet long. It 
connects the harbor of Duluth with Lake Superior, and natural 
conditions are such that during and after storms, waves often 
pass squarely into its mouth and on through it. By means of 
instantaneous photography accurate profiles of waves against the 
walls, either in the canal or outside, in gently shoaling water, 
could be determined. The velocity of the waves could also be 
determined quite accurately, velocity observations being usually 
taken between stations 300 feet apart. The observations during 
two years numbered 631 in all. The wave heights varied from 2 
to 23 feet, the wave lengths from 45 to 425 feet, and the wave 
velocities from 9.1 to 33.3 feet per second. The depth of the 
water varied from 3.3 to 27 feet, though 533 of the observations 
were taken in the canal 26 feet deep. For these 533 observations 
the mean observed velocity and the mean velocity as computed 
from the shallow-water trochoidal formula agreed within less than 
one-half of one per cent. This is practically exact agreement. 
For the ninety-eight observations made outside the canal in varying 
depths the computed velocities averaged nearly 5 per cent more 
than the observed velocities. Major Gaillard states that conditions 
and facilities were such that the last series of observations could 
not be taken with the same degree of accuracy as those on waves 
inside the canal. Major Gaillard's observations appear to furnish 
conclusive evidence of the reliability of the trochoidal theory as 
regards its most important feature, the relation between length 
and speed of advance. 

It is true that Major Gaillard dealt only with shallow-water 
waves, but it is evident from what has gone before that shallow- 
water trochoidal waves are more likely to misrepresent the actual 
waves than the deep-water trochoidal waves. 

The actual wave profiles in the Duluth canal as obtained by 
photography agreed reasonably well with the profiles from the 
trochoidal formula. The differences, generally speaking, were 
greatest at about mid-height of the wave, where the failure of the 
trochoidal theory to satisfy the conditions of continuity and 
dynamical stability is most marked. Major Gaillard states that 
the elevated portion of an actual wave " is always narrower and 


the depressed portion broader and flatter than is indicated by 
theory, and this difference becomes more marked as the wave 
approaches the point of breaking." The actual wave profiles, 
however, were by no means uniform, differing from each other 
quite as much as from the trochoidal form. 

To sum up it seems fair to say that the trochoidal formulae 
represent actual waves very closely as regards speed, with a suffi- 
cient approximation as regards profile, and for practical purposes 
are much better than more complicated and difficult formulas that 
have been devised. They are themselves quite complicated and 
difficult enough. 

10. Shallow Water and Solitary Waves. The trochoidal for- 
mula for wave speed in shallow water of depth d may be written 

L - 

i gL 


For a constant length of wave v decreases as the water shoals, the 
ratio between the velocity of a wave of given length L in water 
of depth d below orbit centers and a wave of the 'same length in 
indefinitely deep water being 

Fig. 28 shows a curve of the value of this ratio plotted on It 


is seen that for depths of water greater than half the wave length 
there is practically no change of speed. 

Figs. 29 and 30 show graphically the relations between depth of 
water, length of wave and speed of wave, the speeds being ex- 
pressed in knots per hour. Fig. 30 simply reproduces on a large 
scale for clearness the lower part of Fig. 29. It is seen that as the 
depth of water becomes very small the speed tends to become 
independent of the length. So let us investigate the results of 
assuming that the wave length is very much greater than the depth 
of water. 


The formula for wave speed in shallow water is, as we have seen, 

do 2 7T i^ 2 7T 

L + I 

Now expanding we have 


f + 

2 7T 

Now when > or the ratio between depth and length, becomes very 


small all terms of the long fraction above except two can be neg- 
lected, and the fraction reduces to 


47r Z d 

= ^L 
Then 1?= 2 w- * = gd. 

L 2 7T 

In the above 6? is not the original depth of water but the depth 
to surface orbit centers, or to mid-height of the waves. This 
depth is somewhat greater than undisturbed still water depth, but 
not very much greater. 

The above result is interesting as indicating that in shallow water, 
on the trochoidal theory, there is a limit to the speed of waves no 
matter what their length. This conclusion is confirmed by ex- 
perience, and the value of the limit obtained above is in reasonable 
agreement with experiments. It is interesting to note in this 
connection that, as indicated in Fig. 25, the shoaler the water the 
more a trochoidal wave system tends to approach a series of sharp 
crests separated by long hollows that are nearly flat. That is to 
say, it tends to become a series of solitary waves, or waves of 
translation, consisting of humps or crests without hollows. Scott 


Russell, as a result of numerous experiments on the so-called solitary 
wave, or wave of translation, made in a trough, concluded that the 
velocity of this wave was equal to that of a body falling freely 
through a height equal to half the depth from the top of the wave. 
The formula above gives the velocity of the trochoidal wave ap- 
proaching the wave of translation type as that of a body falling 
through a height equal to half the depth measured from mid-height 
of the wave. The difference is not great for possible waves whose 
height is generally but a fraction of the depth. There is, however, 
testimony to indicate that Scott Russell's formula gives too great a 
velocity. Rankine gives a formula practically equivalent to Scott 
Russell's. Major Gaillard states that he has applied Rankine's for- 
mula to several hundred observations upon shallow-water waves, 
taken at North Beach, Fla., and on Lake Superior, and has found 
that it almost invariably gives results considerably in excess of the 
observed velocities. The trochoidal formula, then, with its velocity 
somewhat smaller than Scott Russell's or Rankine's, would agree 
more closely with Gaillard's observations. 

ii. Dimensions of Sea Waves. It may be well to supplement 
the mathematical theory of waves with some information regard- 
ing waves found in practice. The heights of sea waves are their 
most striking feature and the most important for seagoing people. 
From the nature of the case it is very difficult to observe with 
accuracy the heights of deep-sea waves. From observations made 
by a number of observers of various nationalities in various seas 
it seems reasonable to consider that waves 40 feet high from 
trough to crest can be generated in deep water by unusually severe 
and long continued storms. This exceptional height is liable to 
be materially surpassed by abnormal waves, the result of super- 
position. Thus Major Gaillard quotes a case where a photograph 
taken on the United States Fish Commission steamer Albatross, 
and furnished him by Commander Tanner, U. S. N., showed the 
fore yard of the ship parallel to the crest of a huge wave and a 
little below it, the photograph being taken from aft. From the 
known dimensions of the vessel and position of the camera it seems 
that this crest must have been from 55 to 60 feet above its trough. 
This wave was photographed in the North Pacific off the United 


States coast. Estimated heights as great as this are not infre- 
quently reported by captains of steamers crossing the Atlantic, 
but accurate estimates of wave heights are difficult to make. 
Probably it would be a fair statement of the case to say that very 
heavy seas with maximum wave heights of 30 feet are not unusual. 
Exceptionally heavy seas with maximum wave heights of 40 feet 
are encountered at times, and there is good evidence that abnormal 
crests 60 feet in height have been encountered. The maximum 
wave height would not be found for every wave of a heavy sea. 
The 30 and 40 foot waves would appear at intervals. Intervening 
waves would be lower. 

For the purpose of estimating the maximum stress of a ship 
it is customary to assume a wave height one-twentieth the length, 
the length of wave being taken the same as the length of the ship. 
This seems a reasonable average, but steeper waves have been 
often observed. Short waves are more apt to be steep than long 
waves. As to actual lengths it may be confidently stated that 
waves over 500 feet long are unusual, though a 4O-foot sea would 
probably be between 600 and 800 feet long, and lengths of 1000 
feet and more have been measured. 

For the development of maximum waves a great space of open 
water is essential. Major Gaillard concluded after investigation 
that "during unusually severe storms upon Lake Superior, which 
occur only at intervals of several years, waves may be encountered 
in deep water of a height of from 20 to 25 feet and a length of 
275 to 325 feet." It appears, then, that the 5oo-foot vessels navi- 
gating Lake Superior will probably never encounter waves their 
own length. This condition indeed is rapidly being reached by 
the enormously long Atlantic liners of the present day. 

12. Relations between Wind and Waves. The length of waves 
(or their speed of advance) is governed by the velocity of the wind 
creating the wave. The relation is not known. Waves have 
often been observed in advance of a storm and also waves in a 
storm that were traveling faster than the wind was blowing. It 
does not follow that a wave can travel faster than the wind that 
forms it. Severe storms are revolving or cyclonic, and the storm 
center does not move as fast as the wind blows. Hence a wave,. 


though traveling more slowly than the wind that formed it, may 
run entirely ahead of the storm or into a region where the wind is 
blowing less violently. 

Published observations upon the ratio of wave and wind velocity 
are not very concordant. Lieutenant Paris, of the French Navy, 
maker of very extensive and careful wave observations, gives the 
wave velocity as .6 that of the wind in a very heavy sea, and 
relatively greater as the sea becomes less heavy. Major Gaillard 
found at Duluth for waves in shallow water, which probably did 
not travel so fast as in the open lake, that the wave velocity as 
averaged from observations taken during fourteen storms was but 
.5 that of the wind. It appears probable that in a strong gale 
making a heavy sea the wave velocity is from .5 to .6 that of the 
wind, but that waves formed under these conditions often travel 
to regions where the wind is not blowing so fast as the waves are 

If we take the wave formed as moving with .5 the speed of 
the wind we have from the trochoidal formula for deep water the 
following relations: 

Speed of wind, statute miles 



60 ' 



Speed of wave, statute miles 






Speed of wave fs 






Length of wave, crest to crest, feet 


1 68 



IDs I 

It would seem, then, if the above ratio between speed of wind 
and speed of wave is approximately correct, that waves more than 
1000 feet in length should be very rare. As a matter of fact, they 
are very rare. 

The height of storm waves will evidently depend upon the 
violence of the wind and the " fetch" or length of open water 
over which the wind blows. Mr. Thomas Stevenson, the noted 
British lighthouse engineer, established from many observations 
the following empirical formula: 


where h is the wave height in feet, c is a coefficient depending upon 
the force of the wind, and / is the "fetch" in nautical miles. For 
strong gales the value of c is 1.5. 


From this formula we have the following : 

h = 10 15 20 25 30 35 40 
/ = 44 zoo 178 278 400 544 711 

At first sight these results might appear inconsistent with the 
fact that waves more than 40 feet high are very rare, even where 
there are several thousand miles of open water. As a matter of 
fact, however, violent gales are revolving storms, and the violent 
part of such storms is seldom more than five or six hundred miles 
in diameter, so that Stevenson's formula is consistent with the 
general facts. 

3. The Law of Comparison 

1. Principle of Similitude. Modern ideas of the resistance of 
ships are based largely upon the Law of Comparison, or Froude's 
Law, as it is generally called, connecting the resistance of similar 
vessels. By judicious application of this law we are enabled to 
determine, with fair accuracy, the resistance of a full-sized ship 
from the experimentally determined resistance of a small model of 
the same. 

Froude's Law is a particular case of the general law of mechan- 
ical similitude, defining the necessary and sufficient conditions that 
two systems or aggregations of particles that are initially geometri- 
cally similar should continue to be at corresponding times not only 
geometrically but mechanically similar. The principle of simili- 
tude was first enunciated by Newton, but the demonstration now 
generally accepted we owe to French mathematicians of the last 
century. Mr. William Froude appears, however, to have developed 
independently the particular form used to compare ships and 
models and to have been the first to use the Law of Comparison 
to obtain useful practical results. 

2. Deduction of Law of Comparison. Suppose we have a 
particle of a system whose coordinates referred to rectangular 
axes are x, y and z. Let m denote the mass of the particle. If the 

particle is moving, it will have at time t an acceleration parallel 


to the axis of x, an acceleration -r* parallel to the axis of y and 




similarly parallel to the axis of z. Let the components parallel 

to x, y and z of the external moving force upon the particle be 
denoted by X, Y and Z. Denote by 8x, dy and dz the resolved 
motions parallel to the axes due to a small motion of the particle 
along its path. 

Then using the well-known principle of Virtual Velocities, the 
differential equation giving the motion of the particle is 

Suppose, now, we have in a second system, mechanically similar, 
a corresponding particle of mass m' whose coordinates at time /', 
corresponding to time / in the first system, are #', y', z' and whose 
impressed force components are X', Y', Z', Its equation of motion 
will be 

dt' z 

If the motions of these two particles are geometrically and 
mechanically similar, the equations of motion must be the same, 
differing only by a constant factor. Now, for similar geometrical 
motions we have a constant ratio between x and x', etc. 

Suppose x' = \x, y' = \y, z' = Xz. 

Then d 2 x' = \d 2 x and so on. 

Let m' = /j.m, p being the constant ratio of masses of the two 

Let the corresponding times be in the ratio T or t' = Tt and 

Substituting for x f , etc., their values we have 

*/ A (/ *V 

This may be rewritten 


Evidently, in order that this may become identical with the equa- 
tion for the first system, we must -have 

x - 

* X ~ T* 

and similarly 

r = M = z^ 

Y ~~ T 2 Z' 

It follows, then, that the external forces on corresponding par- 
ticles must bear a constant ratio to each other. Let F denote this 
ratio. Then the necessary and sufficient relation for geometrical 

and mechanical similitude of motion of the two particles is F = ^- 

The same relation connects every corresponding particle of the 
two systems, and hence the systems as a whole. Now T, the 
relation ratio between corresponding times, is not very convenient 
for use in practical application. It is readily eliminated. Let 
v and v f be corresponding velocities. Then 

_ dx f _ dx' X dx 
~ dt' ~ dt' ~ T dt ' 

v f X X 2 

Whence = = c say. Then T z = 

1) JL C 

Whence F = ^ = ^- 

\ A 

We may further simplify the case by assuming a relation between 
c and X. Suppose we make the ratio of corresponding speeds such 
that c z = X or that the speed ratio is equal to the square root of 
the dimension ratio. Then F = /*. Now we know that whatever 
the speed ratio and dimension ratio, the external forces due to 
gravity must be in the ratio /* or the ratio of masses. We see from 
the above that for motions mechanically and geometrically similar, 
if the speed ratio is made equal to the square root of the dimension 
ratio, all external forces must be in the ratio of mass or weight. 
The application to the case of a ship and its model is obvious. 
If a certain portion of the resistance of a ship is due to a certain 
disturbance of the water and if, at a corresponding speed of the 


model, bearing to the speed of the ship a ratio equal to the square 
root of the dimension ratio between model and ship, there is a 
similar disturbance set up by the model, the resistances due to the 
similar disturbances will be proportional to the weights of ship and 

For the resistance of the ship or model, as the case may be, is in 
each case the external force, other than gravity, acting upon the 
system of particles involved in the disturbance, and the mass of 
disturbed water, if the disturbances are similar, is proportional to 
the displacement of the ship. 

It is apparent from the above that the applicability of Froude's 
Law to resistances of model and ship depends upon whether the 
disturbances at corresponding speeds are similar. This is a matter 
capable of reasonably close experimental determination as regards 
the wave disturbances of model and ship. It is found that these 
are similar at corresponding speeds, the wave disturbance set up 
by the ship being an enlargement to scale as closely as can be 
measured of that of the model at corresponding speed. 

Mr. William Froude estimated the actual resistance of the Grey- 
hound, a ship of over 1,000 tons displacement, by applying the 
Law of Comparison to carefully measured resistances of a small 
model in a manner to be explained later, and found the results thus 
obtained in very close agreement with the actual resistance as 
measured by towing experiments. But, perhaps, the strongest 
experimental confirmation of the Law of Comparison, and one fully 
warranting its practical application, is an indirect one. There are 
now a number of experimental model basins in existence engaged 
in estimating the resistances of ships by proper application of the 
Law of Comparison to results of model experiments. These are 
not able to verify their results directly, because, for the full-sized 
ship when tried, we ascertain not resistance but the indicated power. 
The efficiency of propulsion connects the indicated power with 
the resistance. But, using the actual indicated powers and the 
estimated resistances determined from model results by the Law 
of Comparison, there are obtained efficiencies of propulsion which 
are consistent and reliable as a basis for new designs of vessels. 

We are fully warranted, then, by numerous considerations, 


both theoretical and practical, in reposing especial trust and con- 
fidence in Froude's Law. The modern theory of ships' resistance 
is founded upon it, and since it has been understood and utilized 
the numerous crude and treacherous theories which preceded 
Froude have practically disappeared. 

It is possible to make a less general demonstration than the above 
of Froude's Law from the steady motion formula for stream lines. 
This, too, depends upon the similarity of stream lines around model 
and ship, a fact requiring experimental determination. 

3. Applications of Law of Comparison. Let us now determine 
the formulae, etc., needed in the application of the Law of Compari- 
son to ships' resistance. 

Put into symbols, let L, B, H denote the length, breadth and 
mean draft of a ship in feet, D its displacement in tons and V its 
speed in knots. Let /, b, h, d, v denote similar quantities for a 
model of the ship. Suppose R and r denote resistances following 
Froude's Law. If X denote the ratio between linear dimensions 
so that L = \l, B = \b and so on and if V and v are connected by 
the relation V = v V\, 

R r 

Since T= T = x 

a \// 

/L\ 3 
we may write R = ( y J r = XV. It is to be noted, too, that 

\ V I 

\=(-\ so V = v(^ 

The Law of Comparison is useful and applicable in connection 
with many problems besides that of the resistance of ships. Thus, 
it is directly applicable in comparing full-sized machines and their 
models of the same material. Here, too, since gravity is one 
external force always present, the speeds of corresponding parts 
must be in the ratio of the square roots of the linear dimensions. 
Thus consider a small and a large steam engine, similar and working 
at corresponding speeds. Let us find from the Law of Comparison 
the relations connecting pressures, revolutions, etc. Let R, T, 


/, S and P denote, respectively, revolutions per minute, torque, 
indicated power, piston speed, and steam pressure for the large 
engine, and r, t, i, s, p the same quantities for the similar small 
engine or model. Let X denote the ratio of linear dimensions. 
Then since the speeds must correspond, we have S = s V\. 
Now S = stroke of large engine X 2 R, 

s = stroke of small engine X 2 r. 

Also stroke of large engine = X stroke of small engine. Whence 

S* 7? 9 r 

dividing = X But also = VA. Whence R = = 
s r s VX 

The total steam pressures on the pistons being the external forces 
must be in proportion to X 3 and the piston areas are proportional 
to X 2 . Hence P = \p. The indicated horse-power is proportional 
to the piston area, varying as X 2 , the steam pressure varying as X 
and the piston speed varying as VA. Hence on combining these 
three factors we have I = *X 3 ' 5 . Now 7 is proportional to TR. 
Hence the torque is directly proportional to the indicated power 
varying as X 3 ' 5 and inversely proportional to the revolutions varying 

Hence T= 

The above relations apply directly to centrifugal fans. For 
steam pressure we substitute the pressure at which the air is de- 
livered. Also the quantity of air delivered will vary directly as 
the area of outlet pipe or as X 2 and directly as the speed or velocity 
or as X*, whence at corresponding speeds the quantities of air 
delivered will vary as X 2 ' 5 . 

The above relations for revolutions, torque, power and pressure 
apply too to the operation of propellers. It should be noted since 
P = \p that the pressure per square inch of the water in which a 
propeller works should be X times that of the water in which its 
model works. Model propellers are usually tested under a total 
head of 35 feet or so of water (equivalent to atmospheric pressure -+- 
one foot or so submersion below surface, say, 35 feet in all). For 
the pressure to vary linearly would require a full-sized propeller 


ten times as large as the model to work under a total head of 350 
feet, or, say, 316 feet submersion, if the 34 feet head due to air 
pressure were equivalent in all respects to 34 feet of water. While 
this is only approximately the case, it is evident that the pressure 
conditions for model and propeller are not those required by the 
Law of Comparison. But it does not necessarily follow that the 
Law of Comparison would not apply to the conditions of practical 
operation. If the action of propellers is such that the power, 
torque and efficiency are unaffected by depth of submersion, the 
Law of Comparison would apply fully. 

We shall see later that, under some conditions of operation, 
propeller action is but little affected by depth of submersion, 
while under others it is materially affected. Hence under some 
conditions the Law of Comparison applied to model propeller 
experiments may be expected to be a reliable guide, while under 
other conditions of operation it would certainly be fallacious. 

Valuable and even indispensable as the Law of Comparison is 
in dealing with resistance and propulsion of ships, it must be 
applied with discretion and an understanding of its limitations. 
Some of these limitations will be developed later. 

4. Simple Resistances Following Law of Comparison. In 
reducing any kind of resistance to rule the endeavor is usu- 
ally made to express it by a formula involving some power of the 
speed as V 2 or F" 3 . Unfortunately actual resistances of ships do 
not lend themselves to such simple formulae, but it seems worth 
while to determine how resistances which satisfy the Law of Com- 
parison and vary as definite powers of speed vary with displace- 
ment or dimensions. 

Suppose R = </>(Z)) V n expresses the law of variation of a ship 
resistance which satisfies the Law of Comparison, R being resistance 
in pounds, <f>(D} some function of displacement, V the speed in 
knots and n an index according to which resistance varies. 

For the similar models' resistance we have 

r = <}>(<) v n . 

, V /Z>Y R D 

For corresponding speeds = I ) and = 

v \d/ r d 


Then R 

Lfi d 6 

Whence </>(</) ~^r = $W T = a constant regardless of displacement 
D d 

= C say. 

Then <j>(D)=CD l ~^ 

or R = CV n D l ~\ 

For integral values of n we have the following results 

n = i resistance varies as (displacement) 1 or (linear dimensions) 2 *. 
n = 2 resistance varies as (displacement)* or (linear dimensions) 2 . 
n = 3 resistance varies as (displacement)* or (linear dimensions) 1 *. 
n = 4 resistance varies as (displacement)* or (linear dimensions) 1 . 
n = 5 resistance varies as (displacement)* or (linear dimensions)*. 
n = 6 resistance is independent of displacement or dimensions. 

The above results are not of much practical value since actual 
resistances even when following the Law of Comparison do not vary 
as simple powers of the speed, but they are of some use in connec- 
tion with approximate formulae. 

5. Dimensional Formulae. In connection with the Law of 
Comparison it is of interest to note the so-called dimensional for- 
mulae which are the functions of certain primary variables or 
units to which are proportional a number of things which we shall 
have occasion to use. Thus taking length or a linear dimension 
as a primary variable we have area varying for similar surfaces 
as (linear dimensions) 2 and similarly volume varies as (linear 
dimensions) 3 . Then if we denote length or linear dimension by / 
we have / 2 and / 3 as the dimensional formulae for area and volume 

Similarly, if / denote time, since velocity varies directly as the 
length traversed in a given time and inversely as the time required 
to traverse a given length and is dependent upon no other variables, 


we have - as the dimensional formula for velocity. Further, since 


acceleration varies inversely as the time required to gain velocity we 

have - as the dimensional formula for acceleration. 

The practical application of dimensional formulse is mostly 
in connection with conversion factors for the determination of 
the numerical magnitude or numbers representing definite things 
when the fundamental units are changed. Thus, suppose we have 
a length of 24 feet. If the yard were the unit of length this length 
would be expressed numerically by 8 instead of 24. Similarly, 
suppose we have a surface of 108 square feet. If the yard were the 

primary unit the number of units of surface would be -r-r-. = 12. 

(3) 2 

Since the dimensional factor for area is I? the conversion factor 
is the square of the ratio of the linear units. Similarly the con- 
version factor for volume is the cube of the ratio of linear units and 

135 cubic feet would be -r 3 = 5 cubic yards. These transforma- 

tions are puzzling in some cases and it will be well to give the 
general rule applicable. 

We will have in any given case the old number, or the number 
expressing something quantitatively in the old units, the ratios 
between the units or the numbers expressing the new units in 
the old units and vice versa, and the dimensional formula for the 
thing under consideration area, volume, velocity or what not. 

Then express the old unit of each kind in terms of the new and 
substitute in the dimensional formula for each primary variable 

the corresponding numerical ratio - ' The result is the con- 

new unit 

version factor, and we have 

New number = Old number X Conversion factor. 
Thus when converting square feet to square yards the ratio 

- en ^ - r = - The dimensional formula is / 2 . Then 
new length unit 3 


Conversion factor = (-] = - 

Old number = 108. 

New number = 108 X - = 12. 

Similarly, suppose we have a velocity of 69.3 feet per second and 
wish to convert it into statute miles per hour. 

For velocity the dimensional formula is - 


Old length unit i Old time unit i 

New length unit 5280 New time unit 3600 

Conversion factor = 


i 3600 is 
-. --- = - -- = 

5280 3600 5280 22 
New number = 69.3 X = 47.25 statute miles per hour. 


By following the above method strictly and systematically there 
is no difficulty in obtaining correct conversion factors no matter 
how complicated the dimensional formulae. 

It is usual to use as primary variables in dimensional formulae 
for things with which we are concerned length denoted by /, time 
denoted by t, and mass denoted by m. 

Since, however, velocity, denoted by v, is proportional to - or t 


is proportional to - , we may use m, I and v as primary variables. 

Further, if, as in the Law of Comparison, we assume certain 
relations to exist between / and m and / and v, we can express di- 
mensional formulae in terms of / alone. For the Law of Compari- 
son we assume m to vary as l z and v to vary as /*. The table 
below gives the dimensional formulae of importance for our 



Area or surface 

Angular velocity and revolutions per i 

Angular acceleration 

Linear velocity 

Linear acceleration 


Moment of inertia ml 2 



Moment of momentum or angular) m p 

momentum j 

Force or resistance _ 


Work, energy and torque 




Pressure or stress per unit area. 

Dimensional Formulae. 

In m. /, /. 

In m, I, v. 









In I alone when 
Law of Com- 
parison rela- 
tions between 
m, I and v 



It will be observed that the relations in the third column agree 
with those deduced in various specific cases when considering the 
Law of Comparison. 

4. Wetted Surface 

i. Importance of Surface Resistance. For all but a minute 
proportion of actual steam vessels the skin friction resistance, or 
the resistance due to friction of the water upon the immersed hull 


surface, is greater than the resistance due to all other sources of 
resistance combined. For some of the fastest Atlantic liners, for 
instance, the skin resistance at top speed, under ordinary smooth- 
water conditions, is about 64 per cent of the total resistance. For 
only the comparatively few vessels that are pushed to a speed very 
high in proportion to their length does the residuary resistance 
due to all causes surpass the skin resistance. 

Such extremely fast vessels are nearly all for naval purposes. 
They are seldom warranted by commercial conditions. 

In view of the great importance of the Skin Resistance it is 
advisable to make a careful investigation into the question of the 
wetted surface of ships. We need to know how to calculate it 
accurately, and how to estimate it with close approximation. We 
need, too, if the question of wetted surface is to be given its proper 
influence in design work, to understand the relations between 
wetted surface and size, proportions and shape of ships. 

2. Appendage Surface. The wetted surface of hull append- 
ages can be calculated as a rule without difficulty. Appendages of 
importance have nearly always plane or nearly plane surfaces, and 
their areas are readily determined by straightforward processes. 
Appendage surface, then, can be calculated by simple methods, 
the exact procedure varying with circumstances. In dealing with 
such appendages as bilge keels and docking keels, which cover or 
mask some of the surface of the hull proper, it is best to deduct 
from the gross area of the appendage the area masked by it, the 
net area resulting being the addition to the wetted surface of the 
hull proper due to the presence of the appendage. 

3. Surface of Hull Proper. When we undertake the accurate 
calculation of the wetted surface of the hull proper of a ship, we 
encounter at once a serious difficulty. It is not possible to develop 
or unroll into a single plane the curved surface of a ship's bottom. 
We can draw a section at any point and measure its girth, and if 
the ribbon of surface included between two sections a foot apart 
were equal in area to the girth in feet of the section in the middle 
of the ribbon, it would be very simple to determine accurately the 
wetted surface of the hull proper by applying Simpson's Rules or 
other integrating rules of mensuration to a series of girths at equi- 


distant stations, covering the whole length of the ship. Unfortu- 
nately, however, on account of its obliquity, the area of this ribbon 
of surface is in general appreciably greater than its mid girth, and 
for the best results we must devise a more accurate method. The 
simplest plan is to correct the mean girth in question, multiplying 
it by a suitable factor, so that the area of the ribbon will be equal 
to the corrected mid girth. Then we can apply the ordinary rules 
to the corrected mid girth and obtain accurate results. Let us see 
now how to determine the correction factor first for one point of a 
section and then for a whole section. 

4. Obliquity Factors. In Fig. 31 suppose AB, drawn straight 
for convenience, to represent a short portion of a section of a ship's 
surface by a normal diagonal plane. CD is parallel to the fore and 
aft line. Let AB cut the section FE in E and adjacent parallel 
sections each six inches from FE at L and K. Fig. 32 shows dia- 
grammatically the three sections and the diagonal plane on the 
body plan. The oblique line KL is an element of surface, and we 
want to connect its length with ML, the distance between stations. 
Now, KL = ML sec KLM. Hence sec KLM is the factor we 

r^TT,, KM mk in Fig. 12 T 

need. Now tan KLM = - = - ,, In practice, then, 


if we take a point on a section midway between two other or end 
sections, draw a line on the body plan at the point perpendicular to 
the section and measure the intercept (mk in Fig. 32) between the 
two end sections, we have 


Tangent of angle of obliquity = 

distance between end sections 

and correction factor for obliquity at the point = secant of angle 
of obliquity. 

We do not want to calculate tangents and secants, and we wish 
to work directly from the body plan. So we divide the sections 
on the body plan at six points into five equal parts. The most 
satisfactory method is to lay off small chords with a pair of dividers 
and thus determine the points of division by trial and error. Then 
we prepare a paper scale so divided that when set perpendicular 
to a section at a division point we read at once the correction factor 
for obliquity from the intercept between the two sections adjacent 


to the one for which we are determining obliquity factors. The 
paper scale can be laid off graphically, but can also be readily 
calculated. Let us suppose that the actual distance apart of suc- 
cessive sections in the sheer or half-breadth plan is i inch. Then 
the distance between two sections on either side of a middle section 
will be 2 inches. Suppose at a certain point the intercept of the 
perpendicular in the body plan between the two stations adjacent 
to the one we are considering is 0.25 inch. Then the tangent of 


the angle of obliquity is - = .125. Hence at this point the 


angle of obliquity is 7 y'i since tan" 1 .125 = 7 7' \. The 
correction factor at the point is sec 77 / i or 1.00778. Then 
for our scale \ inch corresponds to a correction factor of 1.00778. 
But to lay off our scale we want to determine the varying lengths 
corresponding to equal intervals of correction factor. 

The necessary calculations are shown in Table II which applies 
directly to i-inch section spacing. 

Of course, an actual set of lines would nearly always have sec- 
tions spaced more than i inch on the plans. For instance, a ship 
416 feet long between extreme stations, with 21 stations or 20 
spaces, would, if the plans were on the scale of \ inch to the foot, 

have the sections on the plans spaced - X - = 5.2 inches. For 

20 4 

such a ship the data for laying off the proper obliquity scale would 
be obtained by multiplying the figures in Column 4 of Table II 
by 5.2. 

5. Sample Calculations. Fig. 33 shows an actual body plan 
with each section divided into five equal spaces for the purpose 
of measuring obliquity and an obliquity scale in place measuring 
a correction factor of 1.015 for a point on section No. 15. Table I 
shows the calculations in standard form. It is seen that for each 
section the average correction factor for obliquity is calculated 
from the measurements at six points. The actual measured mean 
girths having been corrected, the wetted surface is readily calcu- 
lated. The trapezoidal rule is used for the work, being really as 
accurate as Simpson's for curves of the type to be handled, and 
much shorter. 


6. Average Correction Factors. It is seen that the correction 
factors for obliquity are always very close to unity. Advantage 
may be taken of this fact when dealing with ships of ordinary form 
to utilize average correction factors which, when multiplied into 
the product of the mean girth by the length, will give the wetted 
surface with great accuracy, i.e., within a small fraction of one per 

Fig. 34 gives contour curves of correction factors for obliquity 

plotted upon values of - > or ratio between length and beam, and 


> or ratio between length and draught. 

For vessels of ordinary form it will be found that by determin- 
ing the mean girth and applying the correction factor from Fig. 34 
the wetted surface is determined with substantially the same 
accuracy as if complete calculations had been made. Fig. 34 must 
be used with caution for vessels not of ordinary form, if very 
accurate results are wanted. 

7. Girths of Sections. Having seen how to determine with 
accuracy the wetted surface of a ship of which complete plans are 
available, I will now take up the determination of the approximate 
wetted surface of a vessel whose dimensions and displacement 
are known, but for which complete plans are not yet available. 
This is a calculation which must often be made. Consider 
first the question of the girth of a ship section below water. 
This varies with dimensions, proportions, and shape or fullness of 
section. The variation with dimensions is a very simple matter. 
For similar sections the girth varies as any linear dimension, 
such as beam, or draught or Varea. It is convenient to use 
Varea as governing quantity and express the girth G of a section of 
area in square feet = A by G = g V 'A . For all similar sections of 
varying dimensions the quantity g in the formula preceding is 
constant. It is in fact the girth of a section of one square foot area 
and similar in all respects to the section whose area is A. Being a 
measure as it were of the girth, let it be called the girth parameter. 
We want now to ascertain how the girth parameter of a section 
varies with proportions and shape. The girth parameters of a 


few simple sections are obvious. Thus, if we have a square sec- 
tion of one square foot area, the beam is equal to the draught and 
the girth is 3 feet, or the girth parameter is 3. If the section is 
rectangular of \ foot draught and 2 feet beam, the girth parameter 
is again 3. We can in fact express by a formula the girth parameter 
of a rectangular section of any proportions. Let B denote its 
beam, xB its draught. Then xB 2 is its area A , and B + 2 xB the 

*v r> AT *u *u G B + 2 XB I + 2 X 

girth G. Now the girth parameter g = -= = -- = = 

Fig. 35 shows a curve of girth parameter for rectangular sections 
plotted on x. The minimum value is 2.8284 for x = %, for which, 
if the section is one square foot in area, the beam is 1.4142 and the 


draught is .7071. For a semicircle of radius r the area = - - and 


the girth irr. Whence g = -- =\/Tr = 2.5066. This value 


2.5066 for a semicircle appears to be the minimum girth parameter 
possible. The sectional coefficient for a circle is .7854, and, as 
will be seen, this coefficient is close to that for a minimum girth for 
any proportion of beam and draught. 

8. Actual Girth Parameters. The best way to investigate the 
variation of girth parameter with proportions and fullness of 
section is to draw a number of sections of varying proportions 
and fullness and determine and plot their girth parameters. This 
has been done for a large number of sections covering a wide range 
of fullness and proportions. These sections were all calculated 
from the same basic formula, the variations of fullness, etc., being 
obtained by variation of coefficients. The details of the work are 
somewhat voluminous and need not be given. The results are 
fully summarized in Fig. 36, which gives contour curves of girth 


parameter plotted upon values of and sectional coefficient. 


Fig. 36 is not, of course, applicable to freak or abnormal sections, 
but throughout its range is believed to be practically exact for 
sections of usual type. 


For instance, Fig. 37 shows a series of sections of which No. i 
is a parabola and No. 6 is made up of two straight lines and the 
quadrant of a circle. The other four sections divide into five equal 
parts the intercepts between i and 6 of diagonal lines through O. 
Four other figures similar to Fig. 37, except that they had different 
proportions, were drawn, and the areas and girth parameters of 
the 30 sections thus obtained were carefully determined. Table III 
shows these actual girth parameters and girth parameters for the 
same proportions and fullness as taken from Fig. 36. The actual 
girth parameters were calculated to the nearest figure in the third 
place only. 

It is seen that Fig. 36 applies to the curves of Fig. 37 and the 
other derived figures with great accuracy. 

As instancing its application to actual ships' sections attention is 
invited to Table IV. This gives for 20 actual midship sections of 
vessels whose dimensions and proportions are stated, the actual 
girth parameters as measured and the girth parameters from 
Fig. 36 for sections of the same proportions and coefficients. The 
agreement is very close indeed. 

It is evident from Tables III and IV that Fig. 36 represents 
with great accuracy the variation of girth parameters of usual 
sections of ships as dependent upon ratio of beam to draught and 
coefficient of fullness. It follows that, substantially, these are the 
only variables. That is to say, if we settle the beam, draught and 
area of a section of usual type, we substantially settle the girth, 
which varies but little with possible changes of shape. Of course, 
this does not apply to sections that are very hollow, having coeffi- 
cients well below .5. Fig. 36 does not cover such sections, nor 
sections of extreme proportions of draught to beam, such as for- 
ward and after deadwoods. For such sections the girth parameters 
vary with great rapidity for small changes of beam. Fig. 36, 
however, covers nearly all the sections of actual ships of usual 
form and is worthy of careful study. We see from it that there 
is an actual minimum girth parameter a little greater than 2.5 


occurring for = 2 and coefficient of fullness a little below .8. 

Probably we may safely call the coefficient for minimum girth 


parameter .7854, the coefficient for a circle. Roughly speaking, 


as we vary the minimum girth parameter is always found for 

sectional coefficient in the neighborhood of .8 until we get to low 


values of , below 1.5, where the minimum girth parameters 

correspond to larger coefficients. Similarly, as we vary sectional 
coefficient only the minimum girth parameter corresponds very 


closely to = 2 until we reach coefficients greater than .9, when it 


corresponds to smaller values of The most striking feature of 


Fig. 36, however, is the comparatively small variation of girth 


parameter over a range of values of and sectional coefficient 


which covers the bulk of the sections of actual ships. This fact 
is of great importance in connection with the determination of a 
reliable approximate formula for wetted surface and the considera- 
tion of the influence of dimensions, proportions and shape upon 
wetted surface. 

9. Approximate Formula for Wetted Surface. Suppose we take 
n + i sections of a given ship, equally spaced at n + i stations 
o, i, 2, 3 ... n. For each section, with subscript denoting the 
station, denote the girth by G, the girth parameter by g and the 
area by A . Let L denote the length and G the mean girth. Then 

Q = go\/A , GI= gi VAi and so on. 
Using the trapezoidal rule we have 


Let S denote the wetted surface. Then neglecting obliquity, which 
will take care of itself later, when we determine coefficients from 
actual ships, we have 



If we keep the same sections and space them twice as far apart, 
we double length and displacement. We also, neglecting obliquity, 
double the wetted surface. If we keep length the same and double 
the area of each section, we double displacement. The girth param- 
eters of the individual sections are unchanged, so that the result 
is to multiply S by va. Now, what convenient expression in- 
volving only length and displacement will give us the same varia- 
tion? Evidently, if we write S = C \/DZ, where D is displacement 
in tons, L is mean immersed length in feet and C is a coefficient 
depending upon proportions, shape, etc., but not upon dimensions, 
we have an expression for S which will vary for similar vessels just 
as the almost rigorous expression deduced above. For, if we double 
length and displacement, we double 5; if we keep L constant and 
double D, we multiply 5 by W 

As regards primary variation, then, this expression is as accurate 
as the rigorous one. It should be carefully noted that L in this 
formula is the mean immersed length, or the average water line 
length. In many types of vessels the water line lengths are suffi- 
ciently close to the mean immersed lengths to be used without 
error, but in others, the stem and stern profiles are such that for 
accurate work the mean immersed lengths must be determined. 
For rough work and first approximations before we are in a position 
to determine from plans the mean immersed length, load water 
line length is used. Secondary variation in the rigorous expression 
given above can come only with variations of the girth parameters, 
go, gi, etc. The principal factors affecting the girth parameters 
are, as we have seen, variations of ratio of beam and draught and 
variations of sectional coefficient. Our formula S = C \/DL so 
far takes no direct account of these. They will show themselves 
in variations of the coefficient C from ship to ship. 

10. Variation of Wetted Surface Coefficient. Consider, first, 
the effect upon wetted surface coefficient of the ratio between 
beam and draught. This variation is most conveniently referred 


to the value of for the midship section. Fig. 38 shows the varia- 


tion of wetted surface with the variation of for the lines of the 



United States Practice Vessel Bancroft. Keeping length and dis- 
placement constant, a number of body plans were drawn from her 


lines with varying from i to 6. The wetted surface for each 

ratio was calculated and the resulting curve is shown plotted on 


in Fig. 38. It is seen that the minimum wetted surface is found 

D D 

at = 2.8; but as is changed the variation is slow until we 
H H 


reach small values of , when the wetted surface begins to increase 


rather rapidly. Such small values of , by the way, are below 


values found in practice. The general features of Fig. 38 could be 
inferred from Fig. 36. We see from the latter figure that for a 


single section the minimum value of g is found for = 2. Now, 



if for the midship section we had = 2, the girth parameter ot 


this one section would be a minimum, but for every other section 
the girth parameter would be above the minimum, since for every 


other section would be less than 2. Also for the smaller values 


of the girth parameters increase more rapidly than for the larger 

values. Henct, for actual ship lines of given length and displace- 


ment, but varying , the minimum wetted surface must correspond 


to a value of greater than 2, and the wetted surface would in- 

t? H 

crease, of course, on each side of the minimum. This minimum 


is found at = 2.8 in Fig. 38. 

It is not so easy to connect the variations of girth parameter of 
an actual ship with variations of sectional coefficient. Further- 
more, Fig. 36 shows such small variation of girth parameter for 
sectional coefficients ranging from .7 to .9 that we may expect to 


find in practice the variation due to sectional coefficient masked 
by other arbitrary causes impossible to reduce to rule, such, for 
instance, as unusual amount of deadwood or extreme reduction 
of deadwood. 

However, broadly speaking, the fuller the midship section, the 
fuller all the sections are likely to be, and, if the midship section is 
very fine, all sections are likely to be fine. These principles con- 
sidered with Fig. 36 would lead us to expect in practice, when using 
the formula S = C \/DL, to find rather high values of C associated 
with very fine midship sections, and possibly a minimum value of C 
for a fairly high midship section coefficient. 

In this connection attention is invited to Figs. 39 and 40, which 
show variation of wetted surface coefficient with midship section 
coefficient, Fig. 39 for fine ended models and Fig. 40 for full ended 
models. The four curves in each figure refer to different values 

/ L \ 3 

of the coefficient Z> -h -las indicated. The higher values of 

wetted surface coefficient are found with the higher values of the 

/ L \ 3 

coefficient D -f- [ - - I This is to be expected, since the greater 

the displacement on a given length the greater the obliquity. 
Figs. 39 and 40 refer to a single ratio of beam to draught, namely 
2.923, but they show distinct minimum values of wetted surface 
coefficient in the neighborhood of midship section coefficients of 
.90. As regards absolute values of the coefficients it is to be noted 
that at midship section coefficient .84 they are practically coincident. 
For higher values of the midship section coefficient the fine ended 
models have the smaller wetted surface. For smaller values of 
midship section coefficient the fine ended models have the greater 
wetted surface. The extreme variations of coefficients in Figs. 39 
and 40 are but about 3 per cent above and below the average, a 
fact which shows that the coefficient C in the approximate formula 
is nearly constant in practice. 

ii. Average Wetted Surface Coefficients. Figs. 39 and 40 
refer to models of only two types of lines. 

A large number of actual wetted surfaces for many types of lines 


have been calculated at the model basin from which Fig. 41, show- 


ing contour curves of the wetted surface coefficient C plotted on 


and midship section coefficient, has been deduced. 

The wetted surface coefficients of Fig. 41 were obtained from 
average results of vessels of ordinary form. For such vessels, if 
the mean immersed length is accurately known, they are correct 
within a small percentage. They apply to the hull proper only, 
exclusive of appendages, and should be used with caution for vessels 
of abnormal form, such as very shallow draught vessels, vessels 
with very broad, flat sterns, vessels with deadwood cut away to an 
unusual extent, etc. 

In practice Fig. 41 can be utilized to ascertain with a good deal 
of accuracy the wetted surface of a vessel of abnormal type, provided 
we have the correct value of C for one vessel of the type which does 
not differ too much in proportions and coefficients from the vessel 
whose wetted surface is needed. 

For, suppose that Fig. 41 is 4 per cent in error for the abnormal 
vessel whose wetted surface coefficient is known. It will continue 
to be very approximately 4 per cent in error for the type of lines 
under consideration as proportions and coefficients are changed, 
and its results corrected by 4 per cent may be relied upon for the 
abnormal type. In other words, Fig. 41 may be utilized in two 

a. To ascertain the approximate wetted surface of any vessel 
of ordinary type whose dimensions, displacement and midship 
section area only are known. 

b. To ascertain the approximate wetted surface of a vessel of 
extraordinary type of known dimensions, etc., provided we know the 
actual wetted surface of another vessel of the same extraordinary 

From a consideration of what has gone before, and especially 
of Figs. 36 to 41, we appear to be warranted in drawing a few broad 
conclusions as to the wetted surface of vessels of usual types. 

1. For a given displacement the wetted surface varies mainly 
with length, being nearly as the square root of the length. 

2. For a given displacement and length the wetted surface varies 


but little within limits of beam and draught possible in practice. 
As regards wetted surface the most favorable ratio of beam to 
draught is a little below 3. 

3. For given displacement and dimensions the wetted surface is 
affected very little by minor variations of shape, etc. Extremely 
full sections are somewhat, and extremely fine sections are quite 
prejudicial to small surface. 

4. After length, the most powerful controllable factor affecting 
wetted surface is probably that of deadwood. By cutting away 
deadwood boldly, we can often save more wetted surface on a ship 
of given displacement and length than by any practicable variation 
in ratio of beam to draught, or in the fullness of sections. 

5. Focal Diagrams 

1. Field for Focal Diagrams. In attempting to analyze experi- 
mental data it frequently happens that we know the general law 
which we think should govern, and we wish to examine whether 
the law does apply and, if it does, to determine suitable coefficients 
from the experiments for use in the formula expressing the law. 
Experimental data being at best an approximation, it is desirable 
to use a method which will not only give us an adequate approxi- 
mation to the coefficients or constants desired, but give us some 
idea as to how closely our results are going to represent the ob- 
served data. 

Mathematically, the problem is in general one of Least Squares. 
In practice, for many problems there is one coefficient or constant 
to be determined, the actual determination, of course, being made 
by taking average results. In a great many cases not so simple 
there are two coefficients or constants involved. For such cases, 
instead of applying the complicated and laborious methods of 
Least Squares, very satisfactory results can always be obtained 
from data not too much in error by the use of what I may call a 
Focal Diagram. 

2. Illustration of Focal Diagrams. This method may be 
readily comprehended from a concrete illustration. Fig. 42 shows 

a parabola whose equation is y = 3 x , the general equation 



being of the form y = ax bx~. At the point P, say, where 
x = 4, y = 8. Substituting these values of x and y in the general 
equation, we have 8=40 i6&. This is a linear relation between 
a and b, and laying off axes of a and b as in Fig. 43, we can draw 
a line representing this relation. If we take the simultaneous 
values of a and b for any point on this line and substitute them in 
the general formula y = ax bx 2 , the resulting parabola in x and 
y would pass through y = 8, x = 4. 

Fig. 43 shows ten lines in a and b corresponding to ten points 

x 2 
on the parabola y = 3 x These points are as follows: 


x,= i 23 45 67 89 10 

3^ = 2.75 5 6.75 8 8.75 9 8.75 8 6.75 5 

These ten lines all pass through the point a = 3, b = .25, forming 
a focus at this point. Evidently, if we know the x and y values of 
the ten points and the fact that they are on a curve whose formula 
is of the form y = ax bx 2 , we could determine a and b by drawing 
the ten lines as in Fig. 43 and taking the focal values a = 3, b = .25. 
If we knew the exact ordinates of but two spots, we could draw the 
two corresponding lines in Fig. 43 and determine the values of 
a and b. 

In practice, if we determined the spots on the curve by experi- 
ment or observation, we would have more spots than theoretically 
needed to determine the focus; but the line for each spot instead 
of passing through the focus would pass somewhat near it, its 
distance from the focus depending upon the nearness of our 
observations to exact truth. 

In Fig. 42, circles on the curve indicate ten exact spots, and 
adjacent crosses indicate spots of varying errors in location. The 
errors, both vertical and horizontal, vary by .05 from + .25 to 
.25, and the actual errors at any spot were assigned by lot. 

We have, then, for the approximate spots 

x = i 1.75 2.85 4.10 4.80 5.90 7.20 8.05 9.25 9.95 

y = 2.$5 4.85 6.65 8.15 8.75 9.20 8.85 7.75 7.00 5.05 

A focal diagram similar to Fig. 43 can be drawn with a line 

for each approximate spot, and this is done in Fig. 44. It is 


evidently possible in Fig. 44 to spot the focus with an accuracy 
ample for most practical purposes. 

3. Considerations Affecting Focal Diagrams. If the assumed 
law or general equation is materially in error, a good focus will not 
be formed, no matter how close the observations may be. Even 
with an exact law it may be difficult to locate the focus if the 
observations are poor, but when we do get a good focus we know 
at once that the corresponding values of the coefficients in our for- 
mula will cause the formula to represent the experimental results 
with great accuracy, indicating that the assumed formula is close 
to the truth and that the observations are good. 

In Fig. 44 the lines are straight. This need not necessarily be 
the case. The relation between a and b may not be linear, but can 
always be represented by a curve. Linear focal diagrams are, 
however, much the simplest and best and should always be sought 
for. Frequently, when the relation between the coefficients is not 
linear, it may be made so by adopting new coefficients of definite 
relation to the original ones. 

In a linear focal diagram we usually determine two points on 
each line. The exact methods best to use vary somewhat with the 
nature of the case. It is always desirable to determine the two 
points, one on either side of the focus. Below are given the 
detailed calculations for the case we have been considering from 
the results of which Fig. 44 was plotted. 

Formula : 

y = ax by?, a = - 


bx, b = o, a = z , b = .5, a = z + .5 x. 




2 -5S 










2 = a for b = o 












a for b . 5 











6. The Disturbance of the Water by a Ship 
The disturbance of deep water by a ship passing through it is 
a very complicated matter and the disturbance of shallow water 
even more complicated. Broadly speaking, all resistance is due to- 


disturbance of the water and before considering in detail the ele- 
ments of resistance it will be well to form some idea of the nature of 
the disturbances to which resistance is due. 

i. Comparison between Ideal Stream Motion and Actual Motion. 
Suppose we could apply on the surface of the water a rigid 
frictionless sheet as of ice surrounding to a great distance a moving 
ship and advancing with it. If the ship had a smooth and friction- 
less bottom and the water were a perfect liquid there would be 
perfect stream line motion, and we know from stream line considera- 
tions the salient characteristics of what may be called the stream 
line disturbance in the vicinity of the ship's hull. In the vicinity 
of and forward of the bow the water would be given a forward and 
outward motion, with pressure in excess of that of the undisturbed 
water. Passing aft, the water would continue to flow outward, 
but at a short distance abaft the bow would lose its forward motion 
and begin to move aft as well as outward. Its pressure, a maximum 
near the bow, would steadily fall off, soon becoming less than that 
of undisturbed water. 

Abreast the midship section, the sternward velocity would reach 
a maximum and the pressure a minimum. Passing sternward, as 
the water closed in it would lose its sternward velocity, and pressure 
would increase again until in the vicinity of the stern we would have 
excess pressure and the water would have motion forward as at the 
bow. Since there would be a deficiency of pressure over the greater 
portion of the hull, we must, in order to realize the ideal motion, 
assume that the rigid sheet surrounding the ship is strong enough 
to hold it firmly at the level at which it naturally floats when at 
rest. We must also assume that the pressure of the undisturbed 
water is such that the defect of pressure caused by the motion of 
the ship will not cause the water to fall away from the rigid sheet. 

Now the motion of the actual ship through actual water differs 
from the ideal conditions assumed above. 

1. The water is not frictionless, but is affected by the frictional 
drag of the surface of the ship. 

2. The ship is not constrained to remain at a fixed level, but may 
rise and fall bodily and change trim in response to the reactions of 
the water. 


3. The water surface is not constrained to remain at one level, 
but is free to rise and fall in response to the action of the ship. 

2. Changes of Level of Vessel and Water. Notwithstanding 
the differences between the actual circumstances of the motion and 
the ideal conditions assumed above, there is no doubt that the 
stream line action around an actual ship presents in a qualitative 
way nearly all the features of the ideal case considered. But in 
the actual ship the excess pressures at bow and stern result in 
surface disturbances, causing waves which spread away and absorb 
energy, and the defect of pressure amidships results in a lowering of 
the water level and a lowering of the ship bodily, accompanied by 
a change of trim. 

Figs. 45 to 49 show for two speeds of one model and three speeds 
of another changes of level and trim of model and of level of water 
against the side. The dimensions and displacements of the models 
are given in the legend just above Fig. 45. 

These figures are typical. They show elevations of the water 
at bow and stern, and show further two phenomena already de- 
scribed as to be expected from stream line action but not conspicu- 
ous or easy to determine for an actual ship. It is seen that there 
is a bodily settlement of the vessel and that in the vicinity of the 
mid length there is a bodily lowering of the water surface adjacent 
to the ship independent of the disturbance due to the wave created 
at the bow. 

3. Lines of Flow over Surface of Vessel. There have been a 
number of experiments made at the United States Model Basin 
upon the direction of relative flow of the water in the vicinity of 
models. The model surface being coated with sesquichloride of 
iron mixed with glue, pyrogallic acid is ejected at a point of the 
bottom through a small hole, which as it passes aft mingled with the 
water causes a gradually widening smear of ink upon the prepared 
model surface. The center line of this smear can be located with 
reasonable accuracy for some distance, and when it becomes uncer- 
tain a fresh hole is bored and the line traced on. When experiment- 
ing with flow not in the immediate vicinity of the model surface, 
meshes of fine string or wire coated with sesquichloride of iron are 
used and pyrogallic acid ejected at known points. 


The relative flow indicated in the immediate vicinity of the model 
is found to extend as regards type quite a distance from the skin, 
so as regards motion near the hull we need consider only the dis- 
turbance close to the bottom, or the lines of flow as they may be 

Figs. 50 to 59 show lines of flow past the bottom for five pairs of 
twenty-foot models of five widely varying types of midship section. 
The proportions, displacements and speeds of the models are given. 
The large and small models of each type of midship section are 
similar except as regards ratios of beam and draught to length. 
These figures are typical and confirmed by investigations of the 
lines of flow over a number of other models. Perhaps their most 
notable feature is the remarkably strong tendency of the water to 
dive under the fore body as it were. In fact, it seems as if the water 
near the surface forward dives down and crowds away from the 
hull the water through which the fore part has passed, while aft 
the water rising up crowds away from the hull the water which 
was in contact with it near the surface amidships. 

4. Kelvin's Wave Patterns and Actual Ship Wave Patterns. - 
It remains to consider the most striking of the disturbances caused 
by a moving ship. This is the surface or wave disturbance. 

The wave disturbance caused by a ship differs obviously from 
trochoidal waves, which we have considered. 

These latter were considered as an infinite series of parallel 
crests, each crest line extending to infinity. 

We owe to the genius of Lord Kelvin the solution of an ideal 
problem which applies reasonably well to ship waves. His work 
in this connection, which may be found in the Transactions of the 
Royal Society of Edinburgh (Vols. XXV (1904-5) and XXVI 
(1906)), bristles with difficult mathematics, but his results are 
comparatively simple. 

Suppose we have advancing in a straight line over the surface of 
a perfect liquid a point of disturbance. What will be the resulting 
waves? Lord Kelvin's conclusion is that there will be a number of 
crests, each crest line being represented by 

_ 20 ^. _ 


where the origin is supposed to travel in the direction of the axis 
of x with and at the point initiating the disturbance. 

The equation above is somewhat simpler in polar coordinates. 
Transformed it becomes 

r 4 - aV 2 (i + 18 sin 2 6-27 sin 4 6) + 16 a 4 sin 2 6 = 0. 

Fig. 60 shows a single crest line from the above equation. It 
starts always from o, where it is tangent to the axis of x. It spreads 
outward and backward to cusps CC, which are on a line making 
with the axis of x the angle of 19 28' whose tangent is Vf or sine 
is ^. The tangent at the cusp is inclined 54 44' to the axis of x, 
and the branch CAC of the crest line is perpendicular to the axis 
of x where it crosses it. The relative heights of various points on 
the crest as given by Lord Kelvin are indicated in Fig. 60. The 
fact that the heights at and CC are infinite shows simply that 
the formula cannot represent the physical conditions with exactness. 
It may, however, be an amply close approximation, for by the theory 
these infinite crest heights extend for but infinitely short distances. 

The physical interpretation of the formula is that at OC and C 
the heights are greatest and the crests the sharpest, so that at these 
points, if anywhere, breaking water will be found. This conclusion 
is fully borne out in practice. 

The whole wave disturbance due to the initiating point is made 
up by the super position of a series of crests such as are outlined in 
Fig. 60, with corresponding intervening hollows. Fig. 61 shows a 
series of such crest lines. The diverging crest lines cross the trans- 
verse crest lines, resulting in an involved surface disturbance. 
The distance between successive transverse crests along the axis 
of x is the same as the length of an ordinary trochoidal wave travel- 
ing in deep water at the speed of the point of initial disturbance. 
The heights of successive crests are inversely as the square roots 
of distances from the origin. 

That Lord Kelvin's solution agrees reasonably well with practical 
results is readily shown by careful scrutiny of the wave disturbances 
caused by ships and models, which makes it clear that the bow wave 
system and the stern wave system closely resemble Kelvin wave 


The differences are only such as might be expected from the fact 
that a Kelvin group is an ideal system initiated by forces at a single 
moving point, while an actual wave group is due to forces spread 
over the ship's hull. 

The heights of the later diverging waves close to the ship appear 
to be much less in practice than by the Kelvin formula, these crests 
frequently appearing as mere wrinkles of the surface, and the ship 
wave patterns vary with proportions of the vessel. Thus narrow 
deep ships have wave patterns whose transverse features are much 
more strongly accentuated than those of broad shallow ships. 

The wave patterns of ships appear to change somewhat with 
change of speed and the transverse features appear to be less promi- 
nent and important at high speed. According to observations 
made by Commander Hovgaard, formerly of the Danish Navy, and 
given by him in a paper before the Institution of Naval Architects 
at its spring meeting in 1909, the cusp line is usually at an angle 
less than 19 28', most observations of full-sized ships showing it 
between 16 and 19, though in one case, that of a Danish torpedo 
boat, Commander Hovgaard observed a cusp line angle as low 
as 11. 

Observations made on models by Commander Hovgaard in the 
United States Model Basin showed even smaller values of cusp line 
angles, particularly at relatively high speeds. 

But at such speeds the breadth of the basin is not sufficient to 
allow the cusp line determined with accuracy. 

For purposes of analysis the most important feature of the 
Kelvin wave group is the close agreement between its curved trans- 
verse crests and a series of transverse trochoidal crests extending 
from the cusp line on one side to the cusp line on the other. 

5. Havelock's Wave Formulae. Lord Kelvin's wave formulae 
given above are for deep water. Dr. T. H. Havelock has developed 
formulae for the wave patterns produced by a traveling disturbance 
in water of any depth. These will be found in a paper on waves, 
etc., in the Proceedings of the Royal Society, Vol. 81, 1908. 

In a paper on Wave-making Resistance of Ships, Vol. 82, 1909, 
Dr. Havelock has applied his formulae to produce practical results. 
For waves generated by a traveling disturbance in deep water 


Havelock's results agree with Kelvin's except that Havelock's 
formulae do not require infinite wave heights. 

But in shallow water Havelock finds that there is a critical speed, 
which is, in feet per second, VgA, where h is the depth in feet. This 
is, by the way, the speed of the solitary wave or wave of translation 
by the trochoidal formulas. 

As the speed increases up to the critical speed the cusp line angle, 
which was 19 28' in deep water, becomes greater and greater until 
at the critical speed it is 90. At this speed the wave disturbance 
reduces to a single transverse wave. 

Above the critical speed transverse waves cannot exist. Diverg- 
ing waves continue however, but instead of being concave the first 
one is straight at an angle which decreases from 90 with the axis 
as speed increases beyond the critical speed. 

The succeeding diverging waves are convex instead of concave. 
We shall see later that observed phenomena accompanying the 
motion of models in shallow water are in accordance with Have- 
lock's theoretical conclusions. 



7. Kinds of Resistance 

THERE are several kinds of resistance and usually all are present 
in the case of every ship. They will be enumerated here and then 
taken up separately in detail. 

1. Skin Resistance. In the first place, water is not frictionless. 
Its motion past the surface of the ship involves a certain amount 
of frictional drag, the resistance of the surface involving an equal 
and opposite pull upon the water. 

This kind of resistance is conveniently denoted by the term 
Skin Resistance. It is nearly always the most important factor 
of the total resistance. 

2. Eddy Resistance. While Skin Resistance is accompanied 
by eddies or whirls in the water near the ship's surface, the expres- 
sion Eddy Resistance is used for a different kind of resistance. 
The motion through the water of a blunt or square stern post or 
of a short and thick strut arm, etc., is accompanied by much resist- 
ance and the tailing aft of a mass of eddying confused water. Such 
resistance is designated Eddy Resistance. With proper design it 
is in most cases but a minor factor of the total resistance. 

3. Wave Resistance. A far more important factor, which 
though usually second to the Skin Resistance is in some cases the 
largest single factor in the total resistance, is the resistance due to 
the waves created by the motion of the ship. It is called for brevity 
the Wave Resistance. 

We have seen that the motion of a ship through the water is 
accompanied by the production of surface waves. These absorb 
energy in their production and propagation, and this energy is 
communicated to them from the ship, being derived from the Wave 



4. Air Resistance. Finally, we have the Air Resistance, which 
is, as its name implies, the resistance which the air offers to the 
motion of the ship through it. The Air Resistance is seldom large. 
It is, however, by no means always negligible. 

5. Comparative Importance of Skin and Wave Resistance. 
Considering the two main factors of resistance, namely, Skin 
Resistance and Wave Resistance, experience shows that for large 
vessels of very low speed the Skin Resistance may approach 90 per 
cent of the total. For ordinary vessels of moderate speed, it is 
usually between 70 and 80 per cent of the total. As speed increases, 
the Wave Resistance becomes a more and more important factor, 
until, in some cases of vessels pushed to speeds very high for their 
lengths, the Skin Resistance may be only some 40 per cent of the 
total, the Wave Resistance being in the neighborhood of 60 per 
cent. For such vessels as high-speed steam launches the Wave 
Resistance may be even more than 60 per cent of the total, but for 
vessels of any size it is seldom advisable to adopt a design where 
the Wave Resistance is as great as 50 per cent of the total. 

Features which tend to decrease Wave Resistance tend to in- 
crease Skin Resistance, and here, as in so many other matters, the 
naval architect must adopt a compromise dictated by the special 
considerations affecting the particular case. 

8. Skin Resistance 

i. William Froude's Experiments. The determination of the 
Skin Resistance of ships is based entirely upon the experimental 
determination of the frictional resistance of thin comparatively 
small planes moving endwise through the water. The classical 
experiments in this connection were made by Mr. William Froude 
many years ago and are recorded in the Proceedings for 1874 of the 
British Association for the Advancement of Science. 

Mr. Froude used boards i\ X 19 inches, of various lengths up to 
50 feet and coated with various substances, which were towed at 
various speeds not exceeding eight knots in a tank of fresh water 
300 feet long, their resistance being carefully measured. Mr. 
Proude summarized his experimental results in the following table: 







Length of Surface or Distance from Cutwater. 

2 feet. 

8 feet. 

20 feet. 

50 feet. 
















2. l6 





I. IO 

2 95 


i .92 














. 240 

2 37 



i'8 3 





45 6 



Fine sand 

Medium sand 
Coarse sand 

In the above, for each length stated in the heading 

Column A gives the power of the speed according to which the 
resistance varies. 

Column B gives the mean resistance in pounds per square foot 
of the whole surface for a speed of 600 feet per minute. 

Column C gives the resistance in pounds, at the same speed, of 
a square foot at the distance abaft the cutwater stated in the 

It appears, then, that the Frictional Resistance of a plane surface 
can be represented by the formula R f = JSV n , where 5 is the total 
surface of the plane, / is its coefficient of friction, V is its speed, 
and n an index giving the power of V according to which Rj is 
increasing. The table below repeats the values of n in Froude's 
table above and gives the values of / from columns B and C when 
we express speed in knots, 5 being expressed in square feet. 







Length of Surface. 

2 feet. 

8 feet. 

20 feet. 

50 feet. 

















Varnish .... 






















i. '83 








Fine sand 

Medium sand. 
Coarse sand 

2. Variation in Coefficient and Index of Friction. The coefficient 
of friction / is affected by a number of circumstances. The tables 
preceding show a variation with nature and length of surface. 
It also varies slightly with temperature, falling off as temperature 
increases, and it varies, of course, with the nature of the fluid. 
For the small variations of density from fresh to salt water / is 
taken to vary directly as the density. 

As to the index n, it is seen that for rough surfaces it remains at 
the value 2.00, while for smooth hard surfaces it falls off with 
increase of length, reaching the value 1.83 for planes 50 feet long. 
The diminution in / and that in n as length is increased are both 
due to the same cause, namely, the fact that the rear portion of a 
plane moves through water which has a forward motion caused by 
the friction of the front portion of the plane. Froude's conclusion 
that for a plane with smooth hard surface the index n has a value 
of 1.83 or thereabouts has been fully confirmed by other experi- 
ments. Prior to Froude it was always considered that the frictional 
resistance of a plane surface would vary as the square of the speed. 


This seems natural, and most experiments on the loss of head of 
water flowing through pipes show that the resistance to flow varies 
as the square of the speed. The conditions are, however, very 
different. In the case of the pipe we consider the average velocity 
of flow over the cross section of the pipe, which is necessarily the 
same from end to end, and its ratio to the rubbing velocity of the 
water close to the walls of the pipe is practically constant. In 
the case of the plane, the rubbing velocity steadily falls off along 
the plane. 

While the frictional index 1.83 for long smooth surfaces does not 
differ greatly from 2, the corresponding curve is far below the par- 
abola corresponding to the index 2. Thus the ratio F 1 ' 83 -4- F 2 , 
which is unity for V = i, is .761 for V = 5, is .676 for V = 10 and 
.609 for V = 20. This ratio falls off more and more slowly as 
speed is increased. Thus, in passing from V = i to V = 20 it 
falls off from to .609, while to reduce it to .500 the speed must 
increase to V = 59. 

3. Frictional Resistance of Ships Deduced from Plane Re- 
sults. In order to apply the results for friction of planes to the 
frictional resistance of ships, it is necessary first to extend the 
experimental results for short planes to long surfaces, the lengths 
of actual ships. This has been done by Froude and Tideman, by 
extending the curves of index, coefficient, etc., for the short planes 
experimented with. While this extension is speculative to some 
extent, it does not appear that it is likely to be seriously in error. 

Then it is assumed that the frictional resistance of the wetted 
surface of a ship is the same as the frictional resistance of a plane 
of the same length and total surface moving endwise through the 
water with the speed of the ship. This assumption is necessarily 
an approximation. The water level changes around a ship under 
way, changing the area of wetted surface; and, owing to stream 
line action, the velocity of flow over the surface is at some places 
less, at others greater, than it would be over the plane surface. 
The assumption made, however, is practically necessary, and is a 
reasonably close approximation to actual facts. 

Finally, it is necessary to assume that the frictional quality of 
the ship's surface is the same as that of our experimental planes. 


From experiments made in the Italian model basin and else- 
where it may be concluded that the frictional resistance of a 
smooth hard surface is not materially affected by the variety of 
paint with which it is covered. But Froude's experiments show 
that friction is powerfully affected by roughness of surface. For 
a 5o-foot plane covered with calico or medium sand and towed at 
600 feet per minute, or about 6 knots, Froude found a frictional 
resistance nearly double that of a varnished plane of the same 
size. The calico surface had an index but little greater than the 
varnished surface, so its friction would remain in nearly constant 
ratio to that of the varnished surface. The medium sand, how- 
ever, had a greater index. This results in a much greater rela- 
tive increase at high speeds. Thus, using Froude's coefficients, 
the ratio between medium sand and varnish, which is 1.43 at one 
knot, becomes 2.12 at 10 knots, 2.38 at 20 knots, and 2.56 at 30 

The relatively enormous increase of. frictional resistance with 
fouling is well known, but we have very little quantitative infor- 
mation as to the difference as regards frictional quality even be- 
tween the smoothest possible steel ship and one whose bottom, 
while acceptably fair, is not ideally smooth. 

It would be very desirable to narrow the gaps which we must 
now bridge by assumptions in connection with frictional resist- 
ance from the results of experiments on large and long planes of 
various surfaces made in open water at high speeds. Such ex- 
periments would, however, be very difficult. It would be very 
hard to tow such planes straight. 

Pending such experiments, we must rely upon coefficients 
deduced from the small scale experiments. 

4. R. E. Froude's Frictional Constants. Mr. R. E. Froude, in 
a paper in 1888, before the Institution of Naval Architects, has 
supplemented the British Association paper of his father, Mr. 
William Froude, by data of coefficients and constants used by him, 
from which Table V of Froude's Frictional Constants has been 

It will be noted that as regards paraffin surfaces the table 
differs slightly from Mr. William Froude's results, obtained in 1872. 


Mr. R. E. Froude states that as regards the paraffin in use in 
1888 it appeared identical in frictional quality with a smooth 
painted or varnished surface. 

5. Tideman's Frictional Constants. Closely following the elder 
Froude's classical experiments of 1872, Herr B. Tideman, Chief 
Constructor of the Dutch Navy, made a number of similar 
experiments, from which he deduced a complete set of frictional 
constants. These are given in Table VI. The most important 
are those for " Iron Bottom Clean and Well Painted." These 
are comparable with Froude'c constants, and it will be noted that 
they are slightly greater. 

For varnished planes 20 feet long, Froude's constants agree 
very closely with results of careful experiments at the United 
States Model Basin; but for full-sized ships it is considered pref- 
erable to use Tideman's coefficients, simply because they are slightly 
larger, and hence make some allowance for the imperfections of 
workmanship found in practice. At the United States Model 
Basin, it is the practice, when dealing with vessels more than 100 
feet long, to use the Tideman values of /, but the index 1.83 
instead of 1.829, as given by Tideman. This increases Tideman's 
results by negligible amounts. 

6. Law of Comparison not Applicable to Frictional Resistance. 
Having concluded, then, that we should represent the fric- 
tional resistance of a ship by R f = fSV 1 ' 83 , where R f is frictional 
resistance in pounds, / is a coefficient varying slightly with length, 
S is wetted surface in square feet and V is speed in knots, let us 
see whether we can apply the Law of Comparison to resistance 
following the formula. 

Let Rif,f\, Si, Vi refer to one ship, R 2 /, f z , S 2 , V 2 to a similar ship. 

Then R\/= /iSiFV' 83 . RZ/= fzSzVz 1 ' 83 . Let the ratio of linear 
dimensions of the two ships be X and let F 2 and V\ be in the 
ratio Vx, as required by the Law of Comparison. 


Now = X 2 




Then at corresponding speeds 

and we have made ~ = V\ = X*. 


1.83 f 
k 2 = J* X 2-915. 

But to satisfy the Law of Comparison we should have at corre- 


spending speeds -^ = X 3 . We see, then, that frictional resistance 

does not follow the Law of Comparison, and hence we cannot 
deduce the frictional resistance of a full-sized ship from that of 
a model. Thus suppose we had a vessel 500 feet long of 12,500 
tons displacement and 39,000 square feet wetted surface. A 
similar 2o-foot model would have 62.4 square feet of wetted 
surface. If the speed of the ship were 20 knots, the correspond- 

/ 20 

ing model speed would be 4 knots = 20 y 

V 500 

Using Froude's coefficient and 1.83 index, the frictional resist- 
ance of the 2o-foot model would be .01055 X 62.4 X 4 1-83 = 
8.3218 pounds. If the Law of Comparison held, this would make 
Rf for the full-sized ship at 20 knots 8.3218 (25) 3 = 130,028 pounds. 
But using Froude's coefficient of friction we have for the full- 
sized ship R f = 39,000 X .00880 X 2O 1 ' 83 = 82,495 pounds, and using 
Tideman's coefficient Rf = 84,745 pounds. 

It is seen, then, that the Skin Friction, as we calculate it, falls 
far short in practice of what it would be if the Law of Compari- 
son were applicable to it. 

7. Air Disengaged around Moving Ships. There is one phe- 
nomenon generally accompanying the motion of a full-sized ship 
which seldom manifests itself in model experiments. As a fast 
ship moves through the water, it is seen that the water in the 
immediate vicinity of the skin plating, particularly aft of the 
center of length, has a great many air bubbles. The air is either 
disengaged from water in which it is entrained by the reduction of 
pressure in frictional eddies, or it is carried down and along the 
ship as a result of breaking water toward the bow. However 


produced, its presence must reduce the density of a layer of water 
covering a large portion, if not all, of the surface of the bottom, 
and it would seem, at first, that there should be a corresponding 
reduction of friction. It is in fact a favorite dream of inventors 
to deliver air around the outside of a ship so that the immersed 
surface will be surrounded by a film of air instead of water. Could 
this result be accomplished, it would undoubtedly result in a great 
reduction of skin friction. But air released under water persists 
in forming into globules, not films. Experiments have been made 
at the United States Model Basin by pumping air around a model 
through a number of holes near the bow and out through narrow 
vertical slots in the forward portion of a 2o-foot friction plane. 
The results of these experiments were that for the model the 
resistance was always materially increased when the air was 
pumped out. In this case the air came out through holes and 
promptly formed globules. In the case of the friction plane the 
air came out in a thin film which spread aft. At speeds of 12 to 
1 6 knots, when the films of air on each side visibly extended over 
perhaps a third of the plane, the resistance was almost exactly the 
same as when no air was pumped. At speeds below 12 knots the 
resistance was greater when the air was pumped. 

It is possible that for vessels of the skimming-dish or other 
abnormal type the efforts of inventors to reduce resistance by 
means of air cushions may be successful, but there is little doubt 
that no matter how much air may be forced into the water around a 
ship of ordinary type, practically none of it remains in contact with 
the ship's surface. That is covered always by a film of solid water. 
The air forms globular masses or bubbles and never touches the 
surface of the hull. While in an actual ship the air bubbles 
naturally appearing must somewhat reduce the density of some of 
the liquid around the bottom, it appears likely that, to reduce 
skin friction materially, this reduction of density would have to 
extend to a much greater distance from the hull than is usually the 
case and that in practice the evolution of air found probably in- 
creases the resistance by an uncertain amount. This uncertainty 
could be removed by friction al experiments upon planes of such size 
and nature of surface as to be closely comparable to actual ships. 


8. Effect of Foulness upon Skin Resistance. In design work 
we usually deal primarily with clean bottoms. When vessels 
become foul by the accumulation of marine growths such as grass 
and shellfish the Skin Resistance is much increased. Fig. 62 
illustrates the effect of change of surface upon Skin Resistance. 

Froude's experimental results for five surfaces are extended by 
his formula to high speeds. The two smooth hard surfaces 
varnish and tinfoil are nearly the same. But a surface covered 
with calico shows about double as much resistance, and surfaces 
covered with fine or medium sand show more than double the 
resistance of the varnished surface at speeds above 20 knots. 
When we reflect that in the most extreme cases of fouling a ves- 
sel's bottom may have a complete incrustation of shellfish it is 
easy to realize that fouling may result in Skin Resistance four or 
five times that of the clean ship. 

Of course in practice such fouling is permitted only under ex- 
ceptional circumstances, vessels in service being docked at inter- 
vals. But even in cool waters where fouling usually goes on 
rather slowly a vessel three or four months out of dock is liable to 
have an increase of 20 per cent or more in Skin Resistance, and in 
tropical waters the increase of resistance is greater. 

Foulness is usually gauged by the loss of speed, which tends to 
mask the great increase of Skin Resistance. Thus a loss of two 
knots of speed for the same power means in the case of a vessel 
originally of moderate speed an increase of about 100 per cent in 
Skin Resistance. 

When in design work it is necessary to allow for the effect of 
fouling it is usually done indirectly by providing a margin of 
speed with a clean bottom equal to the loss to be expected from 
fouling. This loss must be estimated from previous experience 
with vessels in the service under consideration. 

9. Eddy Resistance 

As already stated, Eddy Resistance is a minor factor in the 
case of most ships and cannot be determined separately by ex- 
periment. It is possible, however, to get a reasonably good idea 


of the laws of Eddy Resistance by experiments with planes, sec- 
tions of strut arms, and similar appendages. 

1. Flow Past a Thin Plane Producing Eddy Resistance. Fig. 
63 shows a section through a plane AB and a stream of water 
flowing past it, and indicates, diagrammatically, what happens. 
The plane is inclined at an angle a to the direction of undisturbed 
flow; K is the dividing point of the stream. On one side of K 
the water flows around the corner at A. On the other side it 
flows by B. The position of K depends upon the angle a. In 
front of the plane there is practically perfect stream motion, as 
indicated. The velocity of the water is checked, with corre- 
sponding increase of pressure, but there is no discontinuity. In 
the rear of the plane, however, the conditions are different. The 
water breaks away at A and B, and there is found behind the 
plane a mass of confused eddying water, whose pressure must be 
reduced below the normal pressure due to depth below the sur- 
face, but in a more or less erratic manner. 

2. Rayleigh's Formulae for Eddy Resistance. The total Eddy 
Resistance of the plane would then be due to a front pressure and 
a rear suction. These are evidently but little dependent upon 
each other. The front pressure has been investigated theoreti- 
cally by assuming a smooth solid inserted behind the plane, so that 
the water has perfect stream motion throughout. The resulting 
formulae as deduced by Lord Rayleigh are as follows : 

2 TT sin a. w 

n . 

4 + TT sin a 2 g 

AK _ 2+4 cos a 2 cos 3 a + (if a) sin a 
AB 4 + TT sin a 

In these formulae P n ' is normal pressure or total pressure per- 
pendicular to the front face of the plane, a is the angle the plane 
makes with the direction of motion, w is the weight per cubic 
foot of the water, g is the acceleration due to gravity, A is area of 
plane in square feet and v is its velocity in feet per second. 

It may be noted that at K, where the water is brought com- 


pletely to rest, the excess pressure is v z . If this pressure were 


over the whole plane, the total normal front pressure would be 

W A 2 

Av 2 . 

The fraction - is, then, the ratio between the front pres- 

4 + TT sin a 

sure and the pressure due to velocity multiplied by the area of the 
plane. This fraction is, as might be expected, a maximum for 

a = go . Its value, then, is or .88. This is materially less 

4 + 7T 

than unity, and as a decreases the fraction soon begins to fall off 

rapidly. Fig. 64 shows curves of the ratio - and the 

4 + TT sin a 

ratio - plotted on a. 

The front pressure by Rayleigh's formula follows the Law of 
Comparison. For suppose we have two similar planes at the 
same angle. If P\ denote the front pressure on No. i and P 2 the 
front pressure on No. 2, 

2 TT sin a w 9 n 2 TT sin a w 


4 4- TT sin a 2 g 4 + ?r sin a 2 g 

Whence - '-* Now if X denote ratio of linear dimensions, 

AI= \ 2 A% and for corresponding speeds Vi z = \v z 2 . Then at corre- 

spending speeds -=^ = X 3 , or Froude's Law is satisfied. 



For salt water -- = i practically. Furthermore it is desirable 


to reduce ail speeds to knots, denoted by V. When this is done 
Rayleigh's formula for front face pressure may be written 

/y_ 5-705 sin Ar 

1.273 + sm a 

3. Joessel's Experiments and Formulae for Eddy Resistance. 
When we come to consider the total normal resistance of an in- 
clined plane moving through water we are compelled to rely upon 
semi-empirical formulae derived by experiments. 
It is impossible to reduce the resistance due to confused eddy- 


ing behind the plane to mathematical law. The ground has never 
been adequately covered experimentally, and it is as a matter of 
fact a question whose accurate experimental investigation pre- 
sents many difficulties. 

M. Joessel made experiments with small planes 12 inches by 
16 inches in the river Loire at Indret, near Nantes, about 1873. 
The maximum current velocity was only about 2? knots. Joes- 
sel's results may be expressed as follows: 

If I denote the breadth of a plane in the direction of motion 
making the angle a with the direction of flow and x the distance 
of the center of pressure from the leading edge, 

x = (.195 + .305 sin a) I. 

If P n denote total normal force due to pressure in front and 
suction in rear, we have for area A in square feet and velocity V 
in knots 

_ 7.584 sin a . , n 

* n , . AY , 

.639 + sin a. 

4. John's Analysis of Beaufoy's Eddy Resistance Experiments. 

- Mr. A. W. John in an interesting paper on " Normal Pressures 
on Thin Moving Plates," before the Institution of Naval Archi- 
tects in 1904, has analyzed Colonel Beaufoy's experiments of 
1795 with square plates of about three square feet area (double 
plates abreast one another about 8 feet apart and 3 feet below the 
surface) and shown that the results present the following peculiar 
features. Up to about 30 degrees inclination the normal pressure 
increases linearly, and from 30 degrees to 90 degrees it remains 
almost constant. The same result has been found by various 
recent experiments with planes in air. It appears to be charac- 
teristic of squares, circles and rectangles approaching the square, 
and is not so pronounced in the case of long narrow rectangles 
moving perpendicular to the long side. 

Beaufoy's results as plotted by John may be approximately 
expressed by a semi-empirical formula of the same form as Ray- 
leigh's formula, 

r> _ A sin a , T/2 

n ~ D I ' " V ' 

B + sin a 


This may be made to coincide at two points with the experimen- 
tal results. We have 

For coincidence at a = 90 and a = 10 A = 5.20, B = .557 

For coincidence at a = 90 and a = 15 A = 4.63, B = .389 

For coincidence at a = 90 and a = 20 A = 4.08, B = .223 

It is reasonable to take the values for a = 15. We then have 

formula derived from Beaufoy, 

P _ 4-63 sin a . , 

n . " ' 

.389 + sm a 

5. Stanton's Eddy Resistance Experiment. Dr. T. E. Stan- 
ton has recently made experiments with very small plates of 
2 square inches area in an artificial current of water of 4 knots 
velocity. His results are published and discussed in a paper of 
April 2, 1909, before the Institution of Naval Architects. He 
found the same phenomenon developed by John's analysis of 
Beaufoy's experiments, namely that the normal pressure on a 
square plate rises almost linearly to an angle of 35 or so and 
then does not change much from 35 to 90. For a plate whose 
length in the direction of motion was twice its width there was 
a pronounced hump at about 45, the normal pressure at this 
inclination being 13 or 14% greater than at 90. For a plate of 
length in the direction of motion but one-half its width the hump 
feature was not so pronounced and was strongest at an inclina- 
tion below 30. 

6. Formulae for Eddy Resistance of Normal Plates Compared. 
When a = 90, or the plane moves normally to itself, we have 
By Rayleigh's formula: Pressure on front face = P n ' = 2.51 A F 2 
By JoessePs formula: Total normal force = P n = 4.63 A V 2 
By formula from Beaufoy's results, P n = 3.33 AV 2 
From Stanton's results, P n = 3.42 AV 2 

It is probable that Rayleigh's formula expresses quite closely 
the resistance of a square stem for instance. If we adopt Joessel's 
formula, which gives the largest resistance, and deduct the front 
face pressure, we would have for rear suction P r = 2.12 A V 2 . 
This formula will probably give an outside value for resistance 
such as that of a square stern post. 


7. Formulae for Eddy Resistance of Inclined Plates Compared. 

For small values of a it is convenient to use a formula of the 
form P n = C sin a A V 2 . If we choose C to correspond to P n 
from the complete formula for an angle of 15 degrees we can 
simplify Rayleigh's formula, etc., for use up to angles of 30 or so. 
Stanton's results are already expressed in this simple form, and 
William Froude has a formula of this type expressing normal 
force for small angles of inclination. 

Rayleigh's formula becomes P n ' ' = 3.73 sin a AV 2 

JoessePs formula becomes P n = 8.45 sin a A V 2 

Formula from Beaufoy becomes P n = 7.15 sin a AV 2 

Froude's formula becomes P n = 4.85 sin a AV 2 

Stanton's formula for a square ) r 

[P n = 5.13 sin aAV 2 
plate becomes ) 

Stanton's formula for a plate ) r 

, , [P n = 7-70 sin aAV 2 

twice as broad as long becomes ) 

The above formulae are not very consistent with each other. 
The question of planes advancing at various angles through water 
is in need of a complete and accurate experimental investigation. 
It may be noted that Stanton's plane twice as broad as long 
approaches somewhat the proportions of an ordinary rudder of 
barn-door type, and his coefficient for such a plate agrees well 
with Joessel's results, which have been used a good deal for rudder 
work in France. In England, the so-called Beaufoy's formula has 
been much used for rudders. This gives P n = 3.2 sin a AV 2 , a 
value much below that from Joessel's formula. But in using this 
formula, the center of pressure is assumed to be at the center of 
figure instead of forward of it as by Joessel's formula for center 
of pressure. The net result is that the English formula gives a 
twisting moment on the rudder stock at usual helm angles only 
about 30 per cent less than that derived from Joessel's complete 
formulae. This is for ordinary rudders. For partially balanced 
rudders the difference is somewhat less. 

Experiments with rudders have indicated normal pressures on 
them materially less than and sometimes but a fraction of what 
would be given by Joessel's formula when V was taken as the 


speed of the ship. But the true speed of a rudder through the 
water in its vicinity is nearly always less and often much less than 
the speed of the ship, and there are other conditions wherein a 
rudder differs very much from a detached plate. 

8. Eddy Resistance Formulae Applicable to Ships. All things 
considered, it seems well, pending more complete experimental 
investigation, to use for a plane Rayleigh's formula for front face 
resistance and Joessel's for total resistance. 

Then we would have for a square stem, the end of a bow tor- 
pedo tube, and similar fittings having head resistance only, 

P n '=2. 5 AV\ 

For square stern posts and similar objects P r = 2.1 AV 2 , and for 
scoops, square or nearly square to the surface of the ship, and 
similar fittings, P n = 4.6 A V 2 . In these formulae A is area in 
square feet, V is speed of the ship in knots and P n , etc., are in 

It is probable that these formulas would nearly always over- 
estimate the resistance concerned, but as the resistances to which 
they apply constitute a very small portion of the total in most 
cases, it is not necessary to estimate them with great accuracy 
and it is advisable to overestimate rather than underestimate 

The resistance of struts is largely eddy resistance, but methods 
for dealing with them will be considered in connection with ap- 

9. Formula for Eddy Resistance behind Plate has Limitations. 
In connection with the formula suggested for rear suction, namely 
P r = 2.1 AV 2 , it should be pointed out that this cannot apply as 
speed is increased indefinitely. 

Consider a plane of one square foot area immersed 10 feet say. 
The pressure on its rear face, allowing 34 feet of water as the 
equivalent of the atmospheric pressure and taking water as sea 
water weighing 64 pounds per square foot, would be 44 X 64 = 
2816 pounds. Evidently there is maximum rear suction when 
there is a vacuum behind and no pressure on the rear face. Hence 
2816 pounds is the maximum possible rear suction. By the for- 


mula,if P r = 2816 = 2.1 F 2 , F 2 = 6 = 1341, V = 36.62. Then 

the formula obviously cannot apply beyond V = 36.62. Even if 
the constant 2.1 is too great we will still in time reach a speed 
where any formula of this type will give a rear suction equal to 
the original forward pressure. Any formula which assumes that 
suction increases indefinitely as the square of the speed must then 
be regarded as expressing not a scientific fact but a convenient 
semi-empirical approximation to the actual facts over the range of 
speeds found in practice. 

10. Wave Resistance 

In discussing the disturbances of the water by a ship we have 
given some consideration to the waves produced. To maintain 
these waves, energy must be expended which can come only from 
the ship. That portion of the ship's resistance which is absorbed 
in raising and maintaining trains of waves is conveniently called 
Wave Resistance. 

i. Bow and Stern System. The tendency is toward the for- 
mation of two distinct series of waves one initiated at the bow 
and conveniently called the Bow Wave System and the other in- 
itiated at the stern and called the Stern Wave System. The Stern 
Wave System, however, makes its appearance in water already 
more or less disturbed by the Bow Wave System and hence the 
ultimate wave disturbance is compounded of the two systems. 

When considering Kelvin's wave system as illustrated diagram- 
ma tically in Figs. 60 and 61, we saw that it was made up of trans- 
verse crests and diverging crests, the transverse crests being but 
little curved and extending to the cusp line on each side. For 
a given speed the length between successive transverse crests is 
the same as the trochoidal wave length for the same speed. 

It is evidently a reasonable approximation under the circum- 
stances to substitute for the actual wave systems ideal systems 
composed of traverse trochoidal waves extending out to the cusp 
lines of Kelvin's waves and each wave of uniform height such 
that energy of the ideal systems is the same as that of the actual 


Consider first the bow system. To maintain this system there 
must be communicated to it while the ship advances the length 
of one wave energy proportional to the energy of one wave 

If we denote by / the length from crest to crest of the tro- 
choidal wave, by b its mean breath and by H its height, w being 
the weight of water per cubic foot, we know from the trochoidal 
wave formulae that the energy per wave length is proportional to 
wblH 2 . Now the external energy communicated to the system by 
the wave resistance R w while the ship traverses a wave length / 
is proportional to RJ. Hence RJ, is proportional to wblH 2 
or R w oc wbH 2 . A similar formula applies to the stern wave 

2. Resultant Wave System. The actual wave resistance is 
due to the wave system formed by compounding the bow and 
stern wave systems. To determine the resultant system we com- 
pound the bow and stern wave systems by the formulae for com- 
pounding trochoidal waves. 

In order to determine the resultant of the two separate wave 
systems of the same length advancing in the same direction, we 
need to know the distance between crests, and it is advisable to 
consider the first crest of each system. The first crest of the 
bow wave system will be somewhat abaft the bow and the first 
crest of the stern wave system somewhat abaft the stern. Their 
positions and the distance between them will vary with speed. 
Call the distance between them the wave-making length of the 
ship and denote it by mL, where m is a coefficient varying slightly 
with speed and, as we shall see, somewhat greater than unity. 
Now, if V is the speed of the ship in knots, the bow wave length I 
in feet is .5573 F 2 . The distance between the first stern system crest 
and the bow system crest next ahead of it is evidently the remainder 
after subtracting from mL the lengths of the complete waves, if 
any, between the first bow crest and the first stern crest. Let there 
be n such waves and let the distance between the first stern sys- 
tem crest and the bow system crest next forward of it be ql, where 
/ is the wave length. Then mL = (n + q) I = (n + q) .5573 F 2 , 
where w is a whole number and q is a fraction. In the compound 


wave formula we need to know cos - or cos - Now, a in the 

J\. I 

u j j.i At 7 T-L. 2 ira 2 irql 

above is evidently the same as ql. ihen cos = cos r 3 - = 

I L 

cos 2 irq. Now w being a whole number, cos 2 irq = cos 2 TT (<7 + n) 

mL 2 ira 2 irmL 360 m 

Hence cos = cos 

5573 V 2 I -5573 V 2 -5573 Z_ 


^ 2 U 2 T-U 2 ^ m * <0 

Denote by c 2 . Then cos - = cos 646 . 
L I c 2 

The whole bow system is not superposed upon the stern system, 
but only the inner portion, since the natural bow system extends 
transversely to a greater distance than the natural stern system. 

Let HI denote the height of the natural bow system when it has 
spread to a given breath b, H 2 the height of the natural stern 
system when it has spread to the same breath. Let kHi denote 
the height of the natural bow system where the stern system has 
spread to the breath b. Suppose its breath then is b'. Since it 
has lost no energy bH* = b'k^H^. 

Then the energy per wave length of the compound system re- 
sulting from the superposition of a portion of the bow system of 
breath b upon the whole stern system of breath b is measured by 

Ib \&Hi*+ # 2 2 + 2 #!# 2 cos 646 I. 

The energy of the portion of the bow system beyond the stern 
system and not compounded is measured by 

/ (b'- b) &H? = l (b'&Hf- bVHfi = Ib (Hf- PHfi, 

since b'k 2 Hi 2 = bHi 2 . Adding the above expressions for partial 
energies the total energy per wave length is measured by 

Ib ( H ! 2 + # 2 2 + 2 kHtHt cos 646). 
\ c 2 I 

Whence the wave-making resistance is proportional to 
b (H?+ H 2 Z + 2 kHtHt cos ^ 646). 


Now, b being an arbitrary convenient constant width, we can say 
that the wave-making resistance R w is proportional to 

#i 2 +# 2 2 + 2 kHA cos 646. 


3. General Formula Connecting Wave Resistance and Speed. 

- The above expression for wave resistance is of little quanti- 
tative value without knowledge of coefficients appropriate to all 
cases, and for practical use in estimating wave resistance there are 
methods more desirable than the use of a formula, but the expres- 
sion is of value in enabling us to realize the general nature of the 
variation of wave resistance with speed. 

As a step in this direction we need to know the connection 
between HI and HI and the speed. 

We know that in perfect stream motion the excess of pressure 
near the bow is proportional to the square of the speed. If, then, 
the wave height were proportional to the excess pressure, which 
it must be approximately, since the surface pressure does not 
change, we would have HI proportional to F 2 . Similarly H 2 
would be proportional to F 2 , and we would have as the general 
expression for R w the wave resistance, 

R W = W^t+^BH- 2 kAB cos 646). 

The coefficients A and B are not constant. There are two 
main sources of variation. If the bow wave height were always 
proportional to the excess bow pressure as speed increases, A 
would not vary on this account. It seems probable that at mod- 
erate speeds when wave resistance first becomes of importance the 
bow wave height does vary as the excess pressure, but as speed 
increases a greater proportion of the stream line pressure is absorbed 
in accelerating the water aft in stream line flow and a less pro- 
portion in raising the water level. The same reasoning applies to 
the stern wave, so, from this point of view, we would expect A 
and B to be approximately constant at low and moderate speeds 
and to fall off steadily at high speeds. 

There is another important source of variation in A and B. 
Suppose we have a vessel 400 feet long. Then the length of the 


fore body is 200 feet. At 13 ^ knots the length of the bow wave 
from crest to crest is very nearly 100 feet; at 27 knots it is 400 
feet. Then at 13^ knots the bow wave is formed by the forward 
quarter of the ship, as it were, while at 27 knots the whole for- 
ward half of the ship must come into play. The result is, of 
course, a modification of A and B with speed. There appears to 
be a critical speed at which the wave length and the wave motion 
and pressures are in step, as it were, with the ship, and the wave is 
exaggerated. This may be called the speed of wave synchronism. 
Broadly speaking, we may say that for fine models of cylindrical 
coefficient below .55 the speed of wave synchronism in knots is 
above \/Z, while for full-ended models of cylindrical coefficient 
above .6 the speed of wave synchronism is below Vz. We may 
expect to find a rapid rise of A and B as we approach the speed of 
wave synchronism and a less rapid falling off as we pass beyond it. 
Consider now the coefficient k in the formula 

R w = F 4 f^ 2 + B 2 + 2 kAB cos ^ 
\ c 

At low speeds k is evidently zero, since observation shows that at 
low speeds the bow disturbance has spread out abreast the stern 
to a distance where it is not affected one way or the other by the 
stern disturbance. As the speed increases, however, more and 
more of the bow wave energy is found in the vicinity of the stern 
and k may be expected to become greater and greater. It is also 
a matter of observation that for narrow deep models the trans- 
verse features of the bow wave are accentuated, and hence for 
such models k will, other things being equal, be greater than for 
broad shallow models, since it is the transverse portion of the bow 
system which is available for combination with the stern system. 


Consider, now, finally, the term cos 646. This expression is 


equal to +i when 646= 360 or any multiple of 360. It is 


equal to i when 646= 180 or 180+ any multiple of 360. 

The quantity m is approximately constant for a given ship, though 
it increases somewhat with the speed. It also appears to increase 


somewhat with fullness from ship to ship. A fair average value of 
m would seem to be about 1.15 for speeds where humps and hol- 
lows are of importance. For lower speeds m approaches i. Fig. 


65 shows for m = 1.15 a curve of cos 646 plotted upon c or 



It is seen that at low speeds maxima and minima succeed one 


another very rapidly. Each maximum corresponds to a " hump " 
in the curve of residuary resistance and each minimum to a 
" hollow." 

Humps and hollows on actual resistance curves do not manifest 
themselves, however, in accordance with Fig. 65. The varying 


term is kA B cos 646, and since in most cases at low speeds k 


is so small as to be practically negligible, we find in practice that 
the first important hump usually appears for full models at about 

= = i , while for fine models this hump is imperceptible or shows 


itself only as an unfair portion of the curve and the first important 


hump is at about = 1.4 to 1.5. 

For quite full models, especially those with parallel middle 

body, the hump for = .8 is often important, and for such models 

the hump for - = .67 to .7 is frequently detected though not of 




The values of = above refer to the centers of the humps or the 


points where the percentage increase of resistance above an aver- 
age curve is a maximum. Of course, the departures from the 
average begin and end some distance before and beyond the hump 

Fig. 66 shows graphically the relations between speed of ship, 


length in feet and values of = By using a varying scale for 



length, the abscissae being proportional to \/length, the contours 
of =r are straight lines. By shading the regions corresponding to 

humps and leaving clear those corresponding to hollows the rela- 
tive locations of humps and hollows are indicated. It will be 
observed that the two lower humps of Fig. 66 are indicated at 

slightly lower values of = than in Fig. 65. This is because Fig. 65 

is for a constant value of m, namely 1.15, while in practice we find 
for the lowest hump m = i.oo very nearly, and for the next 

V V 

m = i. 08 or so. For the region from -= = .9 to = = 1.2, embrac- 


ing a hump and a hollow, m = 1.15 very nearly while beyond this 
speed m is somewhat greater on the average. 

It might seem at first sight very important to adopt such length 
for a desired speed as to be sure of landing in a hollow rather than 
on a hump, but, though this point should always be considered, in 
comparatively few cases is it a matter of serious practical impor- 
tance. In most cases it is desirable to adopt proportions and form 
such that the humps and hollows up to the speed attained are not 
prominent, so there is no material saving to be had by landing in 
a hollow rather than on a hump. 

4. Curves of Residuary Resistance and of Coefficients. Hav- 
ing discussed generally the characteristics of wave-making re- 
sistance as indicated by the formula 

R w = F 4 ^ 2 +5 2 + 2 kAB cos 646), 

it is well to consider some concrete examples. 

Fig. 67 shows curves of residuary resistance determined from 
model experiments for ten 4oo-foot ships without appendages. 
The residuary resistance is practically all wave-making. The 
proportions, etc., are tabulated on the figure. 

It is seen that there are five displacements in all, there being 
two vessels of each displacement differing in midship area or lon- 
gitudinal coefficient. All vessels were derived originally from the 
same parent lines, so the variations of resistance are essentially 
due to variations of dimensions and of longitudinal coefficient. 


The curves of Fig. 67 are not very encouraging to the develop- 
ment of an approximate formula for wave resistance. 

For instance the variation with longitudinal coefficient is a 
very difficult feature. The models of .64 longitudinal coefficient 
all show pronounced humps at about 21 knots, while their mates 
of .56 longitudinal coefficient show no hump there. But at 25 
knots or so the wave resistances for the two coefficients come 
together again, and for higher speeds the models of .64 coefficient 
have the smaller resistances. At 30 knots or so there is a second 
hump which shows for both the full and the fine coefficients. 

Resistance curves are frequently analyzed by assuming them of 
the form R = AV n and determining suitable values of n, the 
power of the speed according to which the resistance is varying, 
and of a, the corresponding coefficient. The curves of Fig. 67 are 
analyzed in this way without much trouble by plotting them upon 
logarithmic section paper. For a curve so plotted the exponent n 
at a point is proportional to the inclination of the curve. 

Fig. 68 shows curves of the exponent n for the 10 curves of 
wave resistance of Fig. 67. It is seen that the variations of n 
are enormous. As to a in the formula R = aV n the values cor- 
responding to the curves of n in Fig. 68 vary too rapidly and radi- 
cally to be adequately represented graphically. 

Suppose now we attempt a slightly different analysis. We have 
deduced a qualitative formula for wave resistance as follows: 

m , , \ 
? 6 4 6j. 

Then curves of - will also be curves of 

2 kAB cos 646 


and might be expected not to vary very much. Fig. 69 shows 


curves of ^ for the 10 curves of Fig. 67, the residuary resistance 

R r being taken as identical with R w . It is seen that up to 18 
knots or so these curves are reasonably constant. Here they 
begin to rise. For the full coefficients there is a maximum at 21 


knots or so, a minimum at 23 to 24 knots and a second maximum 
at 29 to 30 knots. For the fine coefficients there is only one pro- 
nounced maximum at 29 to 30 knots. 


It is evident from Fig. 69 that the curves of ~ are somewhat 

systematic in their variations and that it might be possible to for- 
mulate values of A, B, k and m such that in a given case we could 
determine R w with reasonable approximation from the basic for- 

R w = F 4 (,4 2 + J B 2 + 2 kAB cos 646). 

\ C I 

It is equally evident that the formulae for A, B and c involved 
would be difficult and complicated. It will be shown later that by 
graphic methods the residuary resistance in a given case can be 
readily approximated and hence the task of devising approximate 
formulae need not be undertaken. 

It is interesting to note for ships i to 4 the relative reduction in 
wave resistance beyond 30 knots. 

The reason will be made clear upon reference to Fig. 66. It is 
seen that for a 4oo-foot ship the last hump occurs at about 30 
knots. In this condition the wave length corresponding to the 
speed is somewhat greater than the length of the ship, so that the 
second crest of the bow wave is superposed upon the first crest of 
the stern wave. Hence the hump. At a speed of about 40 knots 
there would be a final hollow corresponding to the conditions 
when the first hollow of the bow wave is superposed upon the 
first crest of the stern wave. This is the main cause of the ap- 
parent relative falling off of wave resistance in Figs. 68 and 69 
between 30 and 40 knots. 

Fig. 66 would indicate that some distance beyond 40 knots the 
wave resistance of these 4OO-foot ships would again begin to in- 
crease relatively, but there is some reason to believe that at excessive 
speeds say 120 knots for the 400-foot ships the wave resist- 
ance would be decreased by the bodily rise of the ship, which 
would begin to approach the condition of a skipping stone and 
tend to glide along the surface. Of course, the speed of 120 knots 


is unattainable by any 4oo-foot ship at present, but it corresponds 
to 36 knots for a 36-foot boat, which is not very far beyond the 
speed-launch results now attainable. Consideration of such ex- 
treme cases is, however, beyond the scope of this work. 

ii. Air Resistance 

The above water portions of a ship may be regarded as im- 
mersed in the air, and air, like water, offers resistance to the motion 
of a body surrounded by it. Air is, roughly, only from one-ninth 
to one-eighth of one per cent of the weight of water, the actual 
weight depending on the pressure and temperature, and air re- 
sistances compared with those of water are, roughly, as the rela- 
tive densities. But air resistance is by no means always negli- 
gible. Sailing vessels are driven by the resistance of sails to the 
motion of air past them, and any one who has attempted to stand 
on the deck of a vessel exposed to a gale of wind will admit that a 
strong head wind opposes a good deal of resistance to a vessel with 
even a moderate amount of top-hamper. 

i. Zahm's Experiments upon Air Friction. Air resistance can 
be separated into two classes frictional and eddy resistance. 
Careful investigations of the friction of air upon plane surfaces 
have been made by Prof. A. F. Zahm, of Washington, who in a 
paper of February 27, 1904, before the Philosophical Society of 
Washington (Bulletin, Vol. XIV, pp. 247-276) has given experi- 
mental results for air friction upon thin planes somewhat similar 
to those tried in water by Froude. 

Prof. Zahm's air planes were 25? inches wide, one inch thick, and 
of varying lengths up to 16 feet. While rather smaller than 
Froude's planes, they were tried up to a high air velocity of 25 
statute miles per hour, or 2if knots. 

Prof. Zahm summarizes his most important conclusions upon 
the subject of air resistance as follows: 

1. The total resistance of all bodies of fixed size, shape and 
aspect is expressed by an equation of the form R = av n , R being 
the resistance, v the wind speed, a and n numerical constants. 

2. For smooth planes of constant length and variable speed,, 
the tangential resistance may be written R = fa; 1 ' 85 . 


3. For smooth planes of variable length / and constant width 
and speed the friction is R = c/' 93 . 

4. All even surfaces have approximately the same coefficient of 
skin friction. 

5. Uneven surfaces have a greater coefficient of skin friction, 
and the resistance increases approximately as the square of the 

These conclusions as to air friction are in striking agreement 
with those deduced by Froude for surface friction in water. 

The coefficients given by Zahm are readily reduced for speeds 
in knots instead of feet per second or statute miles per hour. 

Upon doing this, if R denote frictional air resistance in pounds, 
A denote whole area of surface in square feet, / denote length of 
surface in feet and V denote speed through the air in knots, we 

have R = .0000122 I' 93 A F 1 ' 85 . 

It should be remembered that this formula is based upon ex- 
periments with planes no longer than 16 feet tested up to speeds 
of 25 statute miles per hour. So, while it may be used with con- 
fidence for short planes up to any velocity reached by ships, it 
must be regarded as only a fair approximation for long surfaces. 
Fortunately for the purpose of the naval architect a fair approxi- 
mation to frictional air resistance is all that he ever need know in 
practice. It is very seldom indeed that he will need to take any 
account of it at all. 

For convenience in calculation Table VIII gives values of F 1 ' 85 

and of r^- We have /' 93 = y^' and hence can readily obtain / -93 if 
I I 

we know A table of l' g3 would not admit of easy interpolation, 

while ' which varies comparatively slowly, lends itself to inter- 


Comparing the results of his experiments on air friction with 
those of Froude on water friction, Zahm states: 

" With a varnished board 2 feet long, moving 10 feet a second, 
the ratio of our coefficients of friction for air and water is 1.08 


times the ratio of the densities of those media under the con- 
ditions of the experiments." 

Froude, however, found that the coefficient of friction fell off 
more rapidly with length than as /~' 07 , so that for longer planes the 
above ratio is greater than i .08 times the ratio of densities. Thus, 
for 2o-foot planes the ratio of coefficients would be some ii times 
the density ratio, that is, the friction in air would be i? times that 
deduced from water friction by dividing it by the density ratio. 

Zahm states that in his experiments " no effort was made to 
determine the relation between the density and skin friction of 
the air, partly for want of time, partly because, with the apparatus 
in hand, too great changes of density would be needed to reveal 
such relation accurately. Doubtless the friction increases with 
the density." 

It appears probable that we may assume Zahm's formula for 
frictional resistance of air to apply to air at 60 F. and a barome- 
ter pressure of 30 inches. 

2. Eddy Resistance in Air. Results of Experiments with 
Planes. While the frictional resistance of air is of importance in 
connection with flying machines, for ships the most important air 
resistance is the eddy resistance. 

The eddy resistance of air seems to follow the same general laws 
as the eddy resistance of water. Within the limits of the speed 
attained by the wind, say up to 100 miles per hour, it varies for a 
given plane as the square of the speed. Observations made under 
the direction of Sir Benjamin Baker during the construction of 
the Forth Bridge indicated that small planes exposed to the 
wind offered greater resistance per square foot than larger planes 
exposed to the same wind. M. Eiffel found for planes not over 
i meter square falling through still air that the larger planes showed 
slightly greater resistance per square foot. 

For rectangular planes the resistance varies somewhat with the 
ratio of the sides, a long narrow plane offering greater resistance 
than a square of the same area. 

For our purposes it is not necessary to consider closely these 
minutiae, and it will suffice to use an average coefficient and ex- 
press the resistance in pounds of a plane of area A in square feet 


moving normally through the air with velocity V knots by a single 
formula 7? = CAV\ 

The values of the coefficient C which have been obtained by 
various experimenters vary a good deal. The more recent experi- 
menters seem to obtain the lower values, but coefficients obtained 
by experimenters within the last 30 to 40 years range from .0035 
to .005 about. 

In England, Stanton, with very small planes exposed to a cur- 
rent of air through a large pipe or box, has obtained a coefficient 
of .0036. Dines with rather small planes on a whirling arm has 
obtained .00384. Mr. William Froude with good-sized planes mov- 
ing through still air at rather low velocities obtained .0048. In 
America, Langley, by whirling-arm methods, obtained somewhat 
variable coefficients averaging about .0047. In France, quite 
recently, M. Eiffel, with planes up to 10 square feet or more in 
area, falling through still air, conducted very careful and elab- 
orate experiments and obtained a coefficient of .004. (See " Re- 
cherches Experimental sur la Resistance de PAir Executees a la 
Tour Eiffel par G. Eiffel." This was published in 1907.) 

All things considered, in the light of our present experimental 
knowledge on the subject it appears reasonable to adopt the 
coefficient .0043 as suitable for practical use. Then our formula 
for the resistance in pounds of a plane moving normally to itself is 
R = .0043 A V 2 , where A is area in square feet and V is speed in 
knots. For speed in statute miles the coefficient above should be 
divided by 1.326; for speed in feet per second by 2.853. 

When it comes to the normal pressure on an inclined plane 
moving through the air the results obtained by experimenters are 
somewhat peculiar. For square planes and rectangular planes 
whose sides are not too dissimilar the normal pressure increases 
rapidly from zero at zero inclination up to an inclination of 30 
degrees or so. At this inclination the normal pressure is nearly 
the same as at 90 inclination, and from 30 to 90 inclination the 
normal pressure, while varying somewhat irregularly, does not 
change much. 

The simplest formula is that of M. Eiffel. For inclined planes 
he proposes to take the normal pressure as constant from 30 


to 90, and from o to 30 to take it as varying linearly. The 
Eiffel formula is a sufficiently close approximation for practical use. 
The formula, then, for practical use expressing the normal pres- 
sure in pounds P n on an inclined plane moving through the air at 
an angle of degrees will be 


From o to 30, P n = .0043 A V 2 ; 


above 30, P n = .0043 A V 2 , where A is area in square feet 
and V is speed in knots. 

The normal pressure is, of course, different from the resistance 
in the direction of motion, which is P n sin 6, or the component of 
P n parallel to the direction of motion. 

3. Determination of Air Resistance of Ships. There is no prac- 
tical method recognized at present for determining the air resist- 
ance of a ship. Mr. William Froude made some experimental 
investigations of the matter about 1874, in connection with the 
Greyhound, a vessel 172.5 feet X 32.2 feet X 13 feet draught, of 
about 1000 tons displacement. The vessel was tried without masts 
or rigging. He concluded that in this condition at 10 knots, the 
air resistance of the Greyhound was nearly 150 pounds, or about 
i per cent of the water resistance. 

For steamers without large upper works, the air resistance, 
when the air is still, is, without doubt, too small as a rule to re- 
quire much consideration. With a strong head wind the air 
resistance is, of course, very much increased, but under such con- 
ditions the increase of water resistance due to the head sea is 
probably in most cases far greater than the air resistance. In 
cases where air resistance is important, it can be investigated by 
exposing a model with the upper works complete to a current of 
air of known speed. The law of the square applies, and it will be 
possible to determine the air resistance of the model at the actual 
speed, not the corresponding speed of the ship. Then the air 
resistance of the full-sized ship, being practically all eddy re- 
sistance, may be estimated by multiplying the resistance of the 
model at the speed of the ship by the square of the ratio between 
the linear dimensions of the ship and the model. 


For a rough approximation, we may take the area of the por- 
tion of the ship above water projected on a thwartship plane and 
assume that the air resistance is that due to a plane of this area 
denoted by A advancing normally through the air, using the 
formula already given for the resistance of a plane. This would 
give us 

Air Resistance in pounds = .0043 A V 2 , where V is speed through 
the air in knots and A is area of upper works projected on a thwart- 
ship plane. 

12. Model Experiment Methods 

In view of the very large use now made of model-basin experi- 
ments there will be given a brief description of the methods used 
in deducing from the model experimental results the resistance or 
effective horse-power of the full-sized ship. 

At a model tank or basin there are facilities for making to scale 
models of ships representing accurately the under-water hulls and 
a sufficient amount of the above-water hulls. Most model basins 
work with models from 10 to 12 feet long. Some use models as 
long as 20 feet. A complete model can be towed through the still 
water of the basin, the speed and corresponding resistance being 
measured for a number of speeds covering the range desired. 

i. Treatment of Model Results. By plotting each resistance 
as an ordinate above its speed as an abscissa we obtain a number 
of spots through which a fair average curve is drawn, giving the 
total resistance of the model. Fig. 70 shows for an actual model 
a number of experimental spots and the resistance curve drawn 
through them. When reducing the results the first step in practi- 
cally all cases is to determine the estimated frictional resistance 
of the model. 

The wetted surface of the model has been calculated and we 
have recorded from experiments with planes the length of the 
model, the resistance of a square foot of surface for each tenth of a 
knot extending up to any speed to which a model is likely to be 
tested. Fig. 70 shows a curve of r f or frictional resistance of 
model, its ordinates having been determined for various speeds by 
multiplying the model surface by the resistance of one square foot. 


2. Deduction of Ship Resistance, Using Model Results. For 

the most common case the model represents some full-sized ship 

- actual or designed and we wish by the aid of the model 

results to determine a curve of estimated effective horse-power 

for the full-sized ship. 

Table IX herewith gives the calculations for the Yorktown, for 
whose model the resistance curve is given in Fig. 70. The object 
of much of the form is obvious. The " Mean Immersed Length," 
L, of the ship is usually the length on the load water line. For 
models of peculiar profiles there is a correction applied by judg- 
ment, the object being to obtain the average immersed length. 
The mean immersed length of the model is usually made 20 feet 
at the United States Model Basin, though moderate departures 
from this length are made when desirable for any reason. Also, 
as it is difficult to get satisfactory observations above a speed of 
17 knots of model, it is necessary to make models shorter than 20 
feet if the maximum corresponding speed would be over 17 knots 
for a 20-foot model. 

The model is so weighted that if it is exact it will float in the 
fresh water of the basin at exactly the corresponding water line of 
the ship in salt water. Hence the ratio at corresponding speeds 

/ZA 3 ^6 (L\ 3 
of resistances which follow Froude's La.w is not : j but a - ( )i the 

factor *7 being introduced on account of the passage from fresh 

water to salt water. 

Coming now to the tabular form, there are entered in the first 
column values of v or the speed of the model in knots, and in the 
second column corresponding values of r or the total resistance of 
the model in pounds as taken from the curve in Fig. 70. In the 
third column is entered r f or the frictional resistance of the model 
calculated as already described. In the fourth column we 
enter the residuary resistance, r r , which is equal to r r { . It is 
this resistance to which Froude's Law applies, and we wish to de- 
duce from it in the shortest and simplest manner the correspond- 
ing residuary effective horse-power. While r r is mostly Wave 
Resistance, it includes the Eddy Resistance and Air Resistance 


of the model. Both are taken as following the Law of Com- 

Now for the full-sized ship the residuary resistance in pounds at 

^6 /ZA 3 
corresponding speed is r r X (y) = -^r say. The speed of ship, 

V, corresponding to a speed of model, v, is v l/y, and the effective 

horse-power absorbed by R r is R r X .0030707 V. Then, if the 
residuary effective horse-power for the full-sized ship is denoted 
by EHP r we have 

36 /ZA 3 I~L 

EHP r = R r X .0030707 V = r r > ( y J X .0030707 v t/ 

35 \t / v I 


36 /ZA 3 /Z 

We denote by a the quantity (y) .0030707 Uy and calculate it 

35 \ / / 

once for all, as indicated in the heading. Then in the fifth column 
of the table we enter av and in the sixth column EHP r , which is 
simply r r multiplied in each case by av. In the ninth column we 
enter V, the corresponding speed for the ship, obtained by multi- 
plying each value of v by V/y We have now for a number of 

values of V the values of EHP r or residuary effective horse-power. 

We need to determine the frictional portion of the effective 
horse-power. This is denoted by E f or EHP f . To determine 
frictional resistance we take from Table VI of Tideman's Con- 
stants the coefficient of friction appropriate to the length of the 
vessel and the nature of bottom. The area of wetted surface 
has been calculated. 

We have seen that frictional resistance in pounds = R f = 
wetted surface X frictional coefficient X V 1 ' 83 
and E/= .0030707 R f X V 

= .0030707 X wetted surface X frictional coefficient X F 2 ' 83 . 
Taking from Table VII the values of F 2 ' 83 we readily determine 
and enter in column n the values of EHP f . These values are 
plotted as in Fig. 71 and a fair curve run through. Then from 


this curve for the values of V cor in column 9 we take off the 
values of EHP f and enter them in column 7. Column 8, which 
is the sum of columns 6 and 7, gives the values of the total EHP, 
which, spotted in Fig. 71 over the values of F cor , enables us to 
draw the final curve of E.H.P. for the condition of the ship defined 
in the heading of the table. 

From this curve it is possible to fill in column 12, which gives 
the values of E.H.P. corresponding to the even values of V in 
column 10. Column 12 is, however, seldom needed. 

3. Residuary Resistance Plotted for Analysis. When we are 
dealing with an actual ship or design it is generally desirable to 
deduce from the model results the final E.H.P. curve as soon as 
possible. When, however, it is a question of analysis of residu- 
ary resistance it is desirable to express it in a slightly different 
form. A very convenient and instructive method is to use the 

values of = as abscissae and of Resistance -H Displacement as 



For convenience the value of Resistance H- Displacement is 
expressed as Resistance in Pounds per Ton of Displacement. 
Fig. 72 shows the curve of Residuary Resistance in Pounds per 
Ton plotted on V +vL for the model to which Figs. 70 and 71 
refer. Fig. 72 is applicable to any size, and it is this elimination of 
the size feature which renders this method of plotting of value 
for purposes of analysis. 

13. Factors Affecting Resistance 

The problem of resistance in its most general form involves too 
many variables to be capable of experimental solution. For a 
vessel of given displacement and speed the resistance varies with 
variations of (i) The dimensions, (2) The shapes of water lines 
and sections. For a vessel of given displacement we may have 
an infinite number of variations of dimensions and shape, so even 
if we could deduce the resistance of a vessel with mathematical 
accuracy from model experiments, it would be a formidable under- 
taking to investigate all admissible or likely variations of dimen- 
sions and shape for but a single vessel of a fixed displacement. 


I. Derivation of Models from Parent Lines. If, however, we 
adopt a single definite shape or set of parent lines, deducing all 
models from these lines by variations of dimensions and coeffi- 
cients of fineness, the problem is enormously simplified. By test- 
ing a practicable number of models we can determine, not for 
one displacement only, but for any displacement within a certain 
range and for any dimensions and fineness likely in practice, the 
approximate resistance at any practicable speed. 

In connection with fineness the expression " longitudinal coeffi- 
cient" will be used to denote the ratio between the volume of dis- 
placement of a vessel and the volume of a cylinder of section the 
same as the submerged midship section and of length the same as 
the length of the vessel preferably the mean immersed length. 
This coefficient is sometimes called the "cylindrical coefficient" and 
very commonly the "prismatic coefficient." While cylindrical coef- 
ficient is descriptive and correct, it is thought that the designation 
''longitudinal coefficient" is preferable as emphasizing the fact that 
this coefficient measures and expresses the fineness of the vessel in 
a longitudinal direction. The expression "prismatic coefficient" is 
slightly in error, since strictly speaking the section of any prism 
is bounded by a straight-sided polygon and not by a curve. 

Given a set of parent lines, the deduction from them of lines of the 
same coefficients but of different proportions or relative values of 
length, beam and draught, is a simple matter. If length alone is 
changed, we need only change the spacing of stations in propor- 
tion to the change of length. If draught alone is changed, we 
need change only in a corresponding way the spacing of water 
lines. If beam alone is changed, we need change only the ordi- 
nates of water lines. 

Since the changes caused by change of length, beam and draught 
are independent we may simultaneously change all three, if we 
wish, without difficulty. 

Suppose, however, we wish to keep dimensions unchanged and 
make changes in shape and fullness. We cannot change the 
midship section without departing from the parent lines, but we 
can change in a comparatively simple manner the longitudinal 
coefficient or curve of sectional areas. Thus in Fig. 73, suppose 


the curve numbered i is the curve of sectional areas for the parent 
model and the curve numbered 2 the desired curve of sectional 
areas. Through , the point on curve 2 corresponding to the 
station AB, draw EF horizontally to meet curve i at F. Through 
F draw CD, then the proper section at AB of the derived form is 
the section at CD of the parent form. Having the two curves of sec- 
tional area and the half -breadth plan of the parent form, any desired 
section of the derived form can be determined without difficulty. 

From a single parent form then, we can derive forms covering 
all needed variations of displacement, of proportions and of fine- 
ness as expressed by "longitudinal coefficient." By contour curves 
from the results of a number of models derived from one parent 
form we can deduce diagrams enabling us to ascertain the resist- 
ance at any speed of any vessel upon the lines of the parent form. 
This applies, of course, to residuary resistance only, since the fric- 
tional resistance can always be estimated without model results. 
or experiments in the manner already indicated. 

2. Classification of Factors Affecting Resistance. It would 
require experiments with models derived from an infinite num- 
ber of parent forms to trace the effect of all possible variations of 
shape, but if we can determine the major factors affecting resist- 
ance and their approximate effect we need seldom concern ourselves 
with the minor factors. 

While it is necessary to be cautious in laying down from past 
experience a hard and fast line of demarcation between the major 
and minor factors of resistance, since novel developments in the 
future may convert one into the other, yet so far as can be judged 
from trials at the United States Model Basin of over a thousand 
models we appear warranted in drawing some conclusions as ta 
the principal factors affecting the resistance of ships not of abnor- 
mal form and the relative importance of these factors. We need 
consider only frictional and wave-making or residuary resistance. 

Given the displacement, speed and frictional quality of the 
surface, the only other factor of importance as regards frictional 
resistance is the length. The greater the length for a given dis- 
placement the greater the frictional resistance. This because 
frictional resistance is proportional to surface or \^DL. 


As regards residuary resistance for a given displacement, the 
principal factors arranged in their usual order of importance are 
as follows : 

1. The length. 

2. The area of midship section or, conversely, the longitudinal 

3. The ratio between beam and draught. 

4. The shape of midship section or midship section coefficient. 

5. The details of shape toward the extremities. 

It is seen that factors i, 2 and 3 can be investigated from a 
single parent form. The complete investigation of factors 4 and 5 
would require investigations involving a very large number of 
parent forms. Fortunately, however, these factors are those of 
least importance. 

3. Details of Shape Forward and Aft. In placing factor 5 as 
of small importance, it should be understood that this is the case 
only as regards the variations found in good practice. If abnormal 
shapes for the extremities are adopted, abnormal resistance is 
liable to follow. The dictum of William Froude many years ago 
appears to be still our best guide. He stated that, broadly speak- 
ing, it was desirable to make the bow sections of U shape and the 
stern sections of V shape. This amounts to saying that at the 
bow it is^ advisable to put the displacement well below water and 
make the water line narrow, and at the stern it is advisable to 
bring the displacement up towards the surface and make the 
water line broad. Carried to an extreme, this would give us 
hollow water lines at the bow and the broad flat stern of the 
torpedo boat type. As a matter of fact, model basin experi- 
ments appear to indicate that for smooth water, up to quite a 
high speed, this type of model is about the fastest. For extreme 
speeds, even in smooth water, hollow bowlines are seldom 
adopted, but there is not sufficient experience in this connection 
to say positively that they are or are not desirable from the point 
of view of speed alone. 

In this connection it may be pointed out that experiments show 
a ram bow of bulbous type to be favorable to speed, even apart 
from the fact that the ram bow usually involves a slight increase 


in effective length. This is simply because the ram bow, which is 
the extreme case of the U bow, is much fuller below water than at 
the water line. 

The excess pressures set up around the ram being well below 
the surface are more absorbed in pumping the water aft, where it 
is needed, and less absorbed in raising the surface and producing 
waves than if the same displacement were brought close to the 

There appears to be a reasonable explanation of the advantages 
as regards resistance of the broad flat stern. In wake of the 
center of length, the water is flowing aft to fill up the space being 
left by the stern, the greatest velocity of the water being under 
the bottom. As the vessel passes, the water flows aft and up, 
losing velocity all the while and increasing in pressure. 

With a U stern there is little to check the upward component of 
the velocity which is absorbed in raising a wave aft. With the 
broad flat stern against which the water impinges, as it were, more 
or less of the upward velocity is absorbed by pressure against the 
stern, which will have a forward component, the result being a 
closer approach to perfect stream motion and less wave dis- 

While the broad flat stern is slightly superior as regards resid- 
uary resistance in smooth water, it is apt to have unnecessary 
wetted surface and is objectionable from a structural and sea- 
going point of view. With model basin facilities it is generally 
possible to determine upon a stern of V type which is almost as 
good as the broad flat type as regards resistance, and distinctly 
preferable to it from a structural and sea-going point of view. 

In connection with the details of shape forward and aft the 
effect of change of trim upon resistance may be considered, since 
the principal effect of change of trim is to modify the shape 
towards the extremities. 

Any change of trim, no matter how small, necessarily pro- 
duces some effect upon resistance, and there are many sea-going 
people who ascribe great virtue to some particular trim and great 
influence upon resistance to change of trim, generally considering 
trim by the stern as advantageous for speed. 


Trim by the stern has some advantages in that it generally 
improves the steering of the ship or its steadiness on a course, and 
in rough weather it is generally advantageous to secure greater 
immersion of the screws and more freeboard forward; but as 
regards resistance in smooth water changes of trim occurring in 
practice generally produce changes of resistance of little or no 

In 1871 Mr. William Froude investigated the effect of trim upon 
the resistance of the Greyhound, a vessel 172 feet long and towed 
at displacements from 938 to 1161 tons and at trims varying 
from 1.5 feet by the head to 4.5 feet by the stern. The maxi- 
mum speed at which the vessel was towed was about 12 knots. 
These experiments showed that for the Greyhound trim by the 
head was beneficial at low speeds, below 8 knots, and trim by the 
stern was beneficial at the upper speeds, above 9 knots. The 
differences, however, were comparatively small for quite large 
changes of trim. Mr. Froude's conclusion from these full-sized 
towing experiments was, " As dependent on differences of trim, the 
resistance does not change largely; indeed, at speeds between 8 
and 10 knots it scarcely changes appreciably, even under the maxi- 
mum differences of trim." The results from the Greyhound 
were corroborated by model experiments which agreed quite 
well with the full-sized results, and since these classical experi- 
ments of Mr. Froude, model experiments investigating this ques- 
tion have been repeatedly made. 

Many experiments made at the United States Model Basin 
appear to indicate that, broadly speaking, for the majority of 
actual vessels at full speed a slight trim by the stern is beneficial, 
but that in the vast majority of cases the benefit is too small to 
be of practical importance. With a well-balanced design, the 
fineness forward and aft being properly distributed, the effect 
upon resistance of change of trim is practically nil. 

4. Shape of Midship Section. Let us now consider the in- 
fluence upon resistance of midship section fullness or the midship 
section coefficient. Figs. 50 to 54 show body plans of five models, 
all having the same length, the same displacement 3000 pounds 
the same curve of sectional areas, the same area of midship 


section and practically the same load water line. Figs. 55 to 59 
show similarly body plans of five 1000 pound models. 

Each group of five models has midship section coefficients vary- 
ing from .7 to i.i, the models with fine midship section coefficients 
having greater values of B and H since the actual midship sec- 
tion areas are the same for all models of a group. The ratio 
B -f- H for all ten models is 2.92. The models are of moder- 
ately fine type, the longitudinal coefficient being .56 for all ten. 
Fig. 74 shows curves of residuary resistance in pounds per ton 
for the five 3000 pound models and Fig. 75 shows similarly the 
resistances of the five 1000 pound models. 

It is seen that while the models with full midship section 
coefficients drive a little easier up to F-S-vZ = i.i to 1.2 and 
the models with fine coefficients have a shade the best of it at 
higher speeds, the differences for such variations of fullness as 
are found in practice are remarkably small. The results given 
above are taken from a paper by the author before the Society of 
Naval Architects and Marine Engineers in November, 1908, on 
" The Influence of Midship Section Shape upon the Resistance of 
Ships." This paper contained many other results similar to those 
given, and its conclusion was that " for vessels of usual types and 
of speeds in knots no greater than twice the square root of the 
length in feet, the naval architect may vary widely midship section 
fullness without material beneficial or prejudicial effect upon speed." 
Of course, it follows that the minor variations in shape of midship 
section that can be made in practice without changing fullness have 
practically no effect upon resistance. 

It should be most carefully borne in mind that the above 
applies to the shape and coefficient of a midship section of a given 
area, not to the area of the section. 

5. Ratio between Beam and Draught. Consider now the 
effect of the ratio between beam and draught. Figure 76 shows 
curves of E.H.P. as determined by model experiment for 6 vessels, 
all derived from the lines of the U. S. S. Yorktown but varying in 
proportions of beam and draught from a very broad shallow model 
to a very narrow deep one. 

It is seen that the broader and shallower the model the greater 


the resistance. This result is typical and confirmed by many 
other experiments at the United States Model Basin. It may at 
first sight seem opposed to many cases of experience where beamy 
models proved easy to drive. But in these cases it will be found 
that the increase of beam carried with it increase of area of mid- 
ship section. Had beam been increased and draught decreased 
in proportion, the area of midship section remaining unchanged, 
the results would have been different. 

However, the variations of resistance with variations of the ratio 
of beam to draught are not very great as a rule. 

6. Longitudinal Coefficient or Midship Section Area. Take 
up now the effect upon resistance of the variation of midsnip sec- 
tion area or longitudinal coefficient. This is a factor of prime 
importance in some cases and quite secondary in others. Thus, 
Fig. 67 shows curves of residuary resistance for five pairs of 400- 
foot ships, each pair having the same displacement and derived 
from the same parent lines but differing in midship section area or 
longitudinal coefficient. It is seen that at 21 knots No. 10 with 
.64 longitudinal coefficient has 2.3 times the residuary resistance 
of its mate No. 9 with .56 longitudinal coefficient. But at 24^ 
knots they have the same residuary resistance. 

Again, No. 4 of .64 coefficient at 21 knots has nearly twice the 
residuary resistance of No. 3 of .56 coefficient. At 255 knots they 
have the same residuary resistance and at higher speeds No. 4 
has the best of it, having but .9 of the residuary resistance of No. 3 
at 35 knots. These results, which are thoroughly typical, are sus- 
ceptible of a very simple qualitative explanation. A small longi- 
tudinal coefficient means large area of midship section and fine 
ends. A large longitudinal coefficient means small area of mid- 
ship section and full ends. At moderate speed the ends do the 
bulk of the wave making and the fine ends make much less wave 
disturbance than the full ends. Hence the enormous advantage of 
the fine ends at 21 knots in Fig. 67. But at high speeds the whole 
body of the ship takes part in the wave making and the smaller 
the midship section the less the wave making. It follows that for 
a ship of given dimensions, displacement, type of form and speed 
there is an optimum longitudinal coefficient or area of midship 


section. Data will be given later by which this can be deter- 
mined with close approximation. 

7. Effect of Length. There remains finally to .consider the 
factor which, broadly speaking, has more influence upon residuary 
resistance than any other. This is the length. We have seen 
that for a given displacement the greater the length the greater 
the frictional resistance it varying as V ' L. Residuary resist- 
ance, on the contrary, always falls off as length increases, though 
not according to any simple law. Fig. 77 shows curves or residu- 
ary resistance of five vessels, all of 5120 tons, derived from the 
same parent lines and having the lengths given. Of course the 
longer Vessels have beam and draught decreased in the same ratio 
sufficiently to keep the displacement constant. Fig. 77 illustrates 
very clearly the enormous influence of length upon residuary 
resistance. Since frictional resistance increases and residuary 
resistance decreases with length, it is reasonable to suppose that 
for a given displacement and speed there will be a length for which 
the total resistance will be a minimum. There is such a length, 
but in the vicinity of the minimum the increase of resistance with 
decrease of length is slow, and since length in a ship is usually 
undesirable from every point of view except that of speed, ships 
should be made of less length than the length for minimum resist- 
ance. For men-of-war particularly it is good policy to shorten the 
ship, put in slightly heavier machinery and accept the increased 
coal consumption upon the rare occasions when steaming at full 
speed, rather than to lengthen the ship, carry greater weight of 
hull and armor necessitated thereby, and consume more coal at 
ordinary cruising speeds. 

14. Practical Coefficients and Constants for Ship Resistance 

i. Primary Variables Used. The first thing to do when we 
wish to establish methods for the determination of ship resist- 
ance is to fix the primary variables to be used. In a given case 
we may have dimensions, displacement, etc., all fixed, and need to 
determine the resistance at a given speed, or we may wish to de- 
termine dimensions to bring resistance below a certain amount, or 
the problem may present other aspects. The primary variables 


adopted should enable the data available to be applied simply 
and directly to the problems arising. 

It is convenient to express resistance as a fraction of displace- 
ment, and a suitable measure is the resistance in pounds per ton 
of displacement. Then a resistance of one pound per ton of dis- 
placement means a resistance which is uu 1 ^ of the displacement. 
At corresponding speeds for similar models, resistances which 
follow Froude's Law are proportional to displacement, and hence 
the pounds per ton are constant. 

Speed is conveniently expressed not directly but in terms of > 

the speed length ratio or speed length coefficient. For similar 

models at corresponding speeds - is constant. 

When it comes to size we need a variable which does not change 
for similar models whatever the displacement. Since the dis- 
placement varies as the cube of linear dimensions, such a quan- 
tity would be Displacement -j- (any quantity proportional to the 
cube of linear dimensions). As length is much more important 
in connection with resistance than beam or draught, a suitable 

quantity would be This would usually be a very small frac- 

tion, however, and it is desirable to use a function which in prac- 
tical cases assumes numerical values convenient for consideration 

and comparison. Such a function is > called the displace- 


ment length ratio or displacement length coefficient. It is the 
displacement in tons of a vessel similar to the one under con- 
sideration and i GO feet long. 

2. Skin Resistance Determination. It is necessary to con- 
sider separately the two elements of resistance, Skin Resistance 
and Residuary Resistance. 

The former is the greater in most practical cases and its inde- 
pendent calculation is very simple. We have seen that the for- 
mula for Skin Resistance is R f = fSV 1 ' 83 , where / is coefficient of 
friction from Tideman or Froude, S is wetted surface and V is 



speed in knots. For a complete design 5 may be accurately cal- 
culated. For a preliminary design it may be closely estimated 
from the formula S = c \/DL, where c is the wetted surface 
coefficient and may be taken from Fig. 41. 

If we were concerned with Skin Resistance only, it would prob- 
ably be the best plan always to determine E.H.P./ by formula as 
was done when calculating the E.H.P./ of a full-sized ship from the 
results of model experiments. But it is necessary to use a more 
complicated system of variables in order to handle Residuary 
Resistance, so it is desirable to express R f in the same variables. 

Wejiave seen that R f = /SF 1 ' 83 and S 1 = c \ / T)L. 
fc \/DLF 183 . 

TTT <. D o-u 1000000 D L 3 

Write y = . , Then y = - - or D = 

Hence R f = 

. , 
/ L 

i oooooo 

Also write x = -^=- Then V = x \/Z F 1 ' 83 = 



D D 

Whence finally 


3.1. 83 0-915 

In the above / varies slightly with length, L' 085 varies slowly 
with length, and c is an almost constant coefficient. 

Evidently then for a given length and value of c we can plot 

contours of on and . r ., as primary variables. Fig. 78 

D VL fJL] 

shows such contours for a length of 500 feet, the value of / 

being taken from Table VI of Tideman's constants. But y^r does 
not vary very rapidly with length and it varies with length only. 


So Fig. 78 can be applied to all lengths and values of c by the 
use of simple correction factors. The correction factors for length 
are given on the scale beside the figure to the right. In Fig. 78 
the standard value assumed for c is 15.4. If we are dealing with 
a vessel for which we know that c is 16.0 for instance, it is obvious 

that we should multiply the values of * from Fig. 78 by- 

D 15.4 

3. Residuary Resistance from Standard Series. Take up now 
the question of Residuary Resistance. Here we are driven to the 
use of model results. 

Fig. 79 shows the lines used for a series of models which may be 
called the Standard Series. 

Fig. 79 shows a model having a longitudinal coefficient of .5554, 
a midship section coefficient of .926 and a displacement length 
ratio of 106.95. The stem was plumb and the forefoot carried 
right forward in a bulbous form. From these parent lines a num- 
ber of models were constructed with various values of beam 
draught ratio, etc. 

There were two values of beam draught ratio used, namely 
2.25 and 3.75. 

There were five values of displacement length ratio used, namely 
26.60, 53.20, 79.81, 133.02 and 199.52. 

There were eight values of longitudinal coefficient used, namely 
.48, .52, .56, .60, .64, .68, .74 and .80. 

Fig. 80 shows relative curves of sectional area used for the 
ight values of the longitudinal coefficient. 

Each of the 80 models was run, its curve of residuary resistance 
in pounds per ton determined and from the results of the two 
groups of different beam ratios after cross fairing, Figs. 81 to 120 
were plotted. 

B V 

Each figure refers to a fixed value of and of =. It shows 

" v L 

contours of residuary resistance in pounds per ton over the range of 
values of longitudinal coefficient and j - most likely to be found 

( ) 


in practice. In applying the results of Figs. 81 to 120 for approxi- 


mate estimates of E.H.P. for beam draught ratios other than 
2.25 and 3.75, interpolation of resistance is linear. This is war- 
ranted by results of experiments with models from the same 
parent model and of intermediate beam draught ratio. While 
not quite exact, it seems sufficiently close to the truth for practi- 
cal purposes. 

4. Estimates of E.H.P. from Standard Series. We are now 
prepared to calculate curves of E.H.P. for a vessel of any size 
beam ratio and length within the range covered by Figs. 81 to 
1 20 and from the parent lines of the Standard Series. Table X 
shows the complete calculations for a vessel of the size, beam 


ratio and length of the U. S. S. Yorktown. For each value of = 


the corresponding figures for -the two beam ratios are consulted 


and columns 2 and 3 filled with the values of ^ for longitudinal 

coefficient =.592 and = 138.1. Then in succession columns 

( ) 


5, 4 and 8 are filled as indicated in the headings. Column 6 is 
filled from Fig. 78. 


The correction factor (&) for -=* is obtained as clearly indicated 

in the heading and column 7 is column 6X6. 

The total residuary resistance in pounds per ton is entered in 
column 9, and column 10 contains the E.H.P. factor by which this 
must be multiplied to determine at once the E.H.P. 

This E.H.P. factor is .00307 DV, but it is convenient to call it 

.00307 D VL X - = Then (a) or .00307 D Vl, is calculated and 

entered in the heading and the values of -=. are found in the 


first column. Column n contains the E.H.P. and column 12 the 
corresponding values of V. Column 10 could be obtained by 


multiplying column 12 by .00307 D, but the methods indicated in 
the table will usually be found more convenient in practice. 

5. Comparison of Standard Series Estimates with Yorktown 
Model Results. As illustrating the application of the Standard 
Series results to estimates of E.H.P. attention is invited to Fig. 121. 
This shows the E.H.P. curve of the Yorktown as determined by 
experiment with a model of the vessel and the curve of E.H.P. from 
the Standard Series as calculated in Table X. It is seen that the 
Standard Series E.H.P. is less than the actual model E.H.P. up to 
the speed of 18 knots, which is higher than the trial speed of the 
Yorktown. This simply shows that the Standard Series lines are 
better than those of the Yorktown. As a matter of fact, hardly 
any models of actual ships tried in the Model Basin have shown 
themselves appreciably superior as regards resistance to the Stand- 
ard Series and very few have been equal to it. Figs. 76 and 122 
show further comparison between actual models and Standard 
Series results. Fig. 76 shows six E.H.P. curves calculated from 
six actual models for the Yorktown and five variants having the 
same length and displacement and derived from the Yorktown 
lines but having varying proportions of beam and draught as indi- 
cated in the table with Fig. 76. 

Fig. 122 shows E.H.P. curves for the same six vessels estimated 
from the Standard Series results. It is seen that the agreement is 
reasonably close. The Standard Series generally shows less power 
than the vessels on Yorktown lines, and the curves from it are 
more closely bunched, but the general features of the two figures 
are markedly similar. 

6. Effect of Longitudinal Coefficient. Figures 81 to 120, show- 
ing the residuary resistance for vessels on the lines of the Standard 
Series, are worthy of the most careful and attentive study. Atten- 
tion may be called to one or two of the most obvious features. 
It is seen that for nearly every speed there is for a given displace- 
ment length ratio a distinct minimum of resistance correspond- 
ing to a definite longitudinal coefficient. For low and moderate 

speeds up to = i.i the best longitudinal coefficient is between 

.5 and .55. Above this point, however, the optimum longitudi- 


nal coefficient rapidly increases, reaching about .65 when - = 1.5 


and being a little greater still when = = 2.00. 


The influence of variation of longitudinal coefficient is greatest 
below extreme speeds, and it is very great indeed at some speeds. 

Thus, in Fig. 91, for = 2.25, = i.i, = 100, the resid- 

tl VL 


uary resistance in pounds per ton for a longitudinal coefficient of .55 
is about 6j. But for a longitudinal coefficient of .65 the residuary 
resistance in pounds per ton is more than doubled being over 14. 

7. Effect of Displacement Length Ratio. The change in type 
of the figures with increasing speed length ratio is notable. Thus, 
for speed length ratio of .75 the contours are nearly vertical in 
wake of the rather full coefficients which such slow ships would 
usually have. This means that if we keep length and speed con- 
stant and increase displacement, the residuary resistance per ton 
remains practically constant or the residuary resistance varies as 
the displacement. Consider now Fig. 100, where the speed length 
ratio is 2.0. For displacement length ratio = 30 the optimum lon- 
gitudinal coefficient is about 63 and the residuary resistance in 
pounds per ton about 51. For the same longitudinal coefficient 
and a displacement length ratio of 50 the residuary resistance in 
pounds per ton is about 77. This 77 applies not only to the 20 
increase above 30 but to the original 30 as well as that. Though 
the relative displacements are as 50 to 30, the relative residuary 
resistances are as 50 X 77 to 30 X 51 or as 3850 to 1530. So an 
increase of displacement of 66 per cent means an increase in 
residuary resistance of about 165 per cent. 

8. Optimum Midship Section Area. The displacement, length 
and longitudinal coefficient being fixed, the area of midship sec- 
tion can be calculated without difficulty. For convenient refer- 
ence, however, Fig. 123, derived from a series of 2.92 beam 
draught ratio on the lines of the Standard Series, gives contours of 

/ L \ z 

(midship section area) -5- ( - '-) for minimum residuary resistance 



plotted on speed length ratio and displacement length ratio. From 
this diagram there may be readily determined in a given case 
the optimum midship section area as regards residuary resistance. 
Of course, in practice there are many considerations affecting 
midship section area besides that of minimum residuary resist- 
ance, and the midship section cannot be fixed from considerations 
of resistance only. 

9. Effect of Length. Figs. 81 to 120 do not show directly the 
effect of variation of length but may be readily utilized to do this. 

Thus, suppose it is required to design a vessel of 30,000 tons 
displacement to be driven at 29 knots. For preliminary work 


assume = 3.75. 

Assuming various lengths we use Fig. 78 to determine the 
corresponding values of the frictional E.H.P. and the Standard 


Series figures for = 3.75 to determine the residuary E.H.P. 

It is assumed in this preliminary work that it is possible to adopt 
the optimum cylindrical coefficients. 

Fig. 124 shows for the case under consideration separate curves 
of frictional and residuary E.H.P. and a curve of their sum, or the 
total E.H.P. all plotted on L. The slow growth of frictional 
E.H.P. and the rapid falling off of residuary E.H.P. with length 
are evident. It is seen that the minimum total E.H.P. corre- 
sponds to a length of 950 feet. It has already been pointed out 
that in practice the length should be made less than that for mini- 
mum resistance. 

Thus, if the vessel were made 850 feet long the increase of E.H.P. 
would be infinitesimal, and if made 750 feet the increase would be 
only from 36,500 to 40,200. As the length is made shorter, however, 
the E.H.P. begins to rise very rapidly. This figure illustrates 
clearly the enormous effect of length upon residuary resistance. 
Thus the residuary E.H.P. is a little over 5000 for a length of 
950 feet and is 50,000 for a length a little below 600 feet. 

It may be noted here that for a case such as that shown in Fig. 
124 it would usually be advisable to adopt a longitudinal coeffi- 
cient above that for minimum resistance. This for several reasons, 


among which may be mentioned the better behavior in a sea way 
associated with the fuller ends, and the better maintenance of 
speed in rough water associated with the smaller midship section. 


For a vessel where - is large, however, it is usually advisable to 


make the longitudinal coefficient less than that for minimum re- 
sistance. Such vessels are nearly all torpedo boats or destroyers, 
which cruise usually at speeds below their maximum, and it is 
advisable to save power at cruising speeds by using a longitudi- 
nal coefficient a little below that best for maximum speed. 

10. Parallel Middle Body Results. The Standard Series re- 
sults of Figs. 8 1 to 1 20 do not apply to one important type of 
vessel, namely, the slow vessel of speed length coefficient from .5 
to .8 with a parallel middle body. Two questions arise in this 
connection. First, whether as regards resistance it is advisable to 
use a parallel middle body, and second, what is the most desirable 
length for the parallel middle body in a given case ? 

Experiments were made with models having a midship section 
coefficient of .96, a ratio of beam to draught of 2.5, various values 
of displacement length coefficient and three values of longitudinal 
coefficient, namely, .68, .74 and .80. For each longitudinal coeffi- 
cient and displacement length coefficient one model was made 
without parallel middle body and four with parallel middle body. 
The lengths of parallel middle body expressed as fractions of whole 
length were as follows: 

For .68 longitudinal coefficient, .09, .18, .27, .36. 
For .74 longitudinal coefficient, .12, .24, .36, .48. 
For .80 longitudinal coefficient, .15, .30, .45, .60. 

Curves of residuary resistance were deduced somewhat as in Figs. 
81 to 120. 

It was found that at low speeds there is a distinct advantage in 
using parallel middle body. This means, of course, that at these 
speeds for a given longitudinal coefficient it is advisable to place 
as much displacement as possible amidships and to fine the ends. 

It was found too that when contours of residuary resistance 
were plotted for a given longitudinal coefficient and speed length 


coefficient, the abscissae being percentages of parallel middle body 
and the ordinates displacement length coefficients, the contours were 
practically vertical in the vicinity of the optimum length of paral- 
lel middle body or that for minimum residuary resistance. In other 
words, under these conditions the residuary resistance in pounds 
per ton does not vary much with displacement length coefficient 
and the latter can be practically eliminated as a variable. Hence, 
for the purpose in hand the results of the experiments with the 
models of parallel middle body may be summarized in Figs. 125, 
126 and 127 which apply to the three cylindrical coefficients used, 

namely, .68, .74 and .80. Thus, consider Fig. 126. The abscissae 

are values of One curve shows percentage length of parallel 

middle body for minimum residuary resistance. The correspond- 
ing residuary resistance is given. For convenience, two other 
curves are given, which show approximately the percentages of 
parallel middle body greater and less than the optimum, which 
correspond to residuary resistance ten per cent greater than the 
minimum. These give an idea of the variations of length of par- 
allel middle body permissible without great increase of residuary 

That the saving by the use of parallel middle body is real is 
evident from Fig. 128. This gives the three curves of residuary 
resistance in pounds per ton for the optimum length of parallel 
middle body from Figs. 125, 126 and 127 and average curves for 
the same longitudinal coefficients for the Standard Series with no 
parallel middle body. The lines of the Standard Series appear to 
be slightly superior to those used for the models with middle body, 
but even so the saving by the use of the optimum length of par- 
allel middle body is appreciable. 

While three coefficients are not enough to fair in exact cross 
curves on longitudinal coefficient, an approximation can be made 
from them of ample accuracy for practical purposes, and Fig. 129 
shows plotted on speed length coefficient and longitudinal coeffi- 
cient by full lines contours of optimum length of parallel middle 
body and by dotted lines corresponding residuary resistance in 
pounds per ton. It should be understood that the optimum 


length of parallel middle body shown in Fig. 129 can be materially 
departed from, as indicated in Figs. 125, 126 and 127, without 
much increase of residuary resistance. 

Particular attention is invited to Fig. 129 which shows how 
rapidly residuary resistance increases with speed for full models 
and also how rapidly at speeds above the very lowest it increases 
with increase of longitudinal coefficient. A judicious selection of 
a longitudinal coefficient suitable for the speed is just as impor- 
tant for slow vessels as for fast. While hard and fast rules cannot 
be laid down, experience appears to indicate that few good de- 
signers adopt coefficients and proportions for slow ships such that 
the residuary resistance is much over 30 per cent of the total; and 
though it is as low as 20 per cent of the total in but few cases, 
this figure, if it can be attained for low-speed ships, results in 
vessels which are very economical in service. 

15. Squat and Change of Trim 

In discussing the disturbance caused in the water by a ship, 
this question has been touched on, Figs. 45 to 49 showing changes 
of trim and level for two models at several speeds. 

i. Changes of Level of Bow and Stern. It is the practice at 
the United States Model Basin when towing models for resistance 
to measure the rise or fall of bow and stern and then plot curves 
showing the relation between speed and change of level of bow and 
of stern. These results apply linearly to model and ship at corre- 
sponding speeds; that is to say, if the ship dimensions are / times 
those of the model, the rise of bow of the ship at a given speed will 
be I times the rise of the model at corresponding speeds. 

This fact is taken advantage of in plotting the curves of Figs. 130 
to 139, which show for 10 models curves of change of level of bow 
and stern, the departures of bow and stern from original level being 

expressed as fractions of length L and plotted not on actual speeds 

but on values of -=. These curves are then applicable to any size 

of ship upon the lines of the model from which they were deduced. 
Actual values of rise and fall can be determined promptly for any 


speed and length of ship by multiplying by L the values of the 

curve ordinates for the - values of the ship. Change of trim in 


degrees can be determined with sufficient approximation by multi- 
plying the difference between the scale values of bow and stern 
levels by the constant 57.3, the value in degrees of a radian or 
unity in circular measure. There are given on the face of each 
figure the values of the displacement length coefficient, the longi- 
tudinal coefficient and the midship section coefficient of the corre- 
sponding model, thus enabling adequate ideas of its general type 
to be formed. 

The curves of Figs. 130 to 139 show what would happen to vessels 
that are towed. The propeller suction in the case of screw steamers 
would cause such vessels when self-propelled to sink more by the 
stern than indicated, but the difference would not be great. 

2. General Conclusion as to Level and Trim Changes with Speed. 

- The results of Figs. 130 to 139 are typical of results shown by 

hundreds of other models which warrant the general conclusions 

below upon the subject of the change of level and trim of vessels 

under way in deep smooth water. 

i . At low and moderate speeds below - = i .o both bow and 

stern settle. For short full vessels this bodily settlement is much 
greater than for long fine vessels. 


2. Below = i.o about, there is little or no change of trim. 


In the majority of cases the bow settles a little faster than the stern, 
particularly for rather full vessels. 


3. As speed is increased beyond = = i.o the bow settles more 

*^ Li 

slowly, reaches an extreme settlement at about - = 1.15, and 


soon begins to rise rapidly, reaching its original level when - = = 


1.3 to 1.4, and continuing to rise. The stern settles more and more 


rapidly beyond about = =1.2, and settles much more rapidly 


than the bow rises, so that the ship as a whole continues to settle 
while rapidly changing trim. 


4. At about -=. = 1.7 to 1.8 the stern is settling less rapidly than 

the bow is rising, so that bodily settlement reaches its maximum. 

The stern does not change its level much beyond = = 2.0, while the 


bow rises always with increase of speed, the result being that the 

vessel is rising again at speeds beyond - = 2.0 about. The 


center of ordinary vessels will never rise to its original level at any 
practicable speed; but, since the effect of the passage of the vessel 
is to depress the immediately surrounding water, it may seem at 
very high speeds as if the vessel had risen above its original level. 

Vessels of special forms and skimming vessels if driven to extreme 
speeds may rise bodny. 

3. Critical or Squatting Speed. The most striking feature of 

change of level curves is the abrupt change at about - = = 1.2, 

the critical speed at which the bow begins to rise and the stern to 
settle abruptly, causing rapid change of trim. 

This "squatting" is often thought to be a cause of excessive resist- 
ance. As a matter of fact, it is simply a result of large bow wave 


resistance. At- - = i.i to 1.2 the first hollow of the bow wave 


is somewhere near amidships and the second crest somewhere for- 
ward of the stern holding it up, as it were. With increase of speed 
the crest moves aft clear of the stern and the hollow moves aft 
toward the stern. The stern, of course, drops into this bow wave 
hollow, causing the "squatting" or rapid change of trim noticed. 
As speed is increased the hollow in turn moves beyond the stern 
and the vessel advances on the back of its own bow wave, as it were. 
The higher the speed, the longer the bow wave and the closer the 
vessel is to the crest. 

It is perfectly true that marked squatting generally means great 
resistance, because it is the result of an excessive bow wave with a 
deep first hollow. With no bow wave there would be no squatting, 


and with slender models having small bow waves squatting is much 
less marked than for short full models. In every case, however, 
it is a symptom rather than a cause of resistance. 

4. Perturbation below Critical Speed. Figs. 131, 132, 133 and 

139 show perturbation in the change of level curves below the 

critical speed - =r = 1.2. These models are very full ended and 

have such strong bow waves that as the hollow corresponding to 


= 1.0 passes the stern it drops into it and the bow rises. 

V J^j 

Reverse operations take place as the next bow wave crest passes, 
and then we reach the critical speed, when the stern drops into the 

bow wave hollow corresponding to -=. = 1.2 and over. 


Instead of the pronounced perturbations of quite full models 
we find for moderately full models the wave hollows and crests 
passing the stern at speeds below the critical speed cause the curves 
of change of level to have flat or unfair places. Fig. 135 is a case 
in point. 

For fine models the bow wave is generally so small and the change 
of level also so small that no effect of the bow wave can be traced 

in the curves until we reach the critical speed = = 1.2. 

In considering Figs. 130 to 139 we should bear in mind that the 
large variations of level and trim shown are for speeds reached by 
very few vessels. 

The curves of Figs. 130 to 139 show changes of level with reference 
to the natural undisturbed water level, and not with reference to the 
level of the water in the immediate vicinity of the ship. We have 
already seen in discussing the disturbance of the water by a ship 
that, as illustrated in Figs. 45 to 49, the passage of the ship causes 
disturbances of water level in its vicinity the net result being that 
on the average there is depression of the water immediately sur- 
rounding the vessel. 

The changes of level, trim, etc., shown by vessels under way in 
shallow water differ somewhat from those found in deep water, and 
will be taken up when considering other shallow-water phenomena. 


16. Shallow-Water Effects 

1. Changes in Nature of Motion from that in Deep Water. - 
It is to be expected that as the water shoals the resistance of a 
ship moving through it will become greater. When the water can 
move freely past the ship in three dimensions the pressures set up 
by the ship's motion would naturally be less than when shallowness 
compels the water to motions approaching the two-dimensional 
character. Referring to Fig. 21, the greater stream pressures for 
plane or two-dimensional motion are evident. In shallow water 
these extra pressures cause waves larger than those in deep water, 
and in shallow water the lengths of waves accompanying a ship 
at a given speed are greater than for the same speed in deep water. 
These are the principal factors differentiating shallow-water resist- 
ance from deep-water resistance. There is a third factor, namely, 
the change in stream velocities past the surface of the ship when in 
shallow water. This factor would increase resistance somewhat, 
but its effect would seem to be so small that it is not necessary to 
consider it since we cannot at present determine with much accuracy 
the effect of the dominant factor, namely, the change in wave 
production. We can, however, as a result of experiments with 
models and full-sized boats get an excellent qualitative idea of the 

2. Results of Experiments in Varying Depths. Figs. 140 to 
144 show a series of curves of resistance or indicated horse-power. 
The data from which these curves were constructed came from 
widely separated sources. The information regarding the German 
torpedo boat destroyer came originally from a paper by Naval 
Constructor Paulus in the Zeitschrift der Vereines Deutsche 
Ingenieure of December 10, 1904. Data for the Danish torpedo 
boats was given by Captain A. Rasmussen, one of the first experi- 
mental investigators in this field. The " Makrelen " data was 
given in Engineering of September 7, 1894, and the "Sobjornen" 
data in a paper read before the Institution of Naval Architects in 
1899. Data for the torpedo boat model was given by Major 
Giuseppe Rota, R. I. N., in a paper read in 1900 before the Insti- 
tution of Naval Architects, the experiments with the model having 


been made in the Experimental Model Basin at Spezia, Italy. 
Information from which the curves for the Yarrow destroyer were 
deduced was given in a paper before the Institution of Naval 
Architects in 1905 by Harold Yarrow, Esq. In Mr. Yarrow's 
paper curves of E.H.P. were given as deduced from model 
experiments in the North German Lloyd experimental basin at 

Each curve refers to a definite depth of water, which has been 
expressed as a fraction of the length of the vessel. Furthermore, 

speed has been denoted not absolutely but by values of 


3. Deductions from Experimental Results. Examining the 
curves, which range from those for a 145-pound model to those for 
a 6oo-ton destroyer, and bearing in mind the varying depths ex- 
pressed as fractions of the length, we seem warranted in concluding 
that in a depth which is a given fraction of the length the perturba- 

tions occur at substantially the same values of - regardless of 

the absolute size. The reason for this must be sought in the rela- 
tion between the length of a wave traveling at a given speed in a 
given depth of water and length of vessel. 

By the trochoidal theory the formula giving wave speed in shallow 
water is 

4 *f _ 

4*^ 27T 

6 *+I 

where / is length of wave in feet, d is depth of water in feet and v 
is speed of wave in feet per second. 

Now let L denote length of ship in feet and put I = cL. 

Also let V denote common speed of ship and wave in knots. 

Then V = v ~; Substituting, reducing and putting g = 32.1 6 


we have 


Fig. 145 shows contour curves of equal values of c plotted on axes of 

7 and -=' Fig. 145 also shows in dotted lines curves deduced 

somewhat arbitrarily from Figs. 140 to 144 and other data showing 
the loci of the points at which increase of resistance due to shoal 
water becomes noticeable, attains its maximum and dies away. 

The data is not thoroughly concordant, and the dotted curves of 
Fig. 145 should be regarded as a tentative attempt to locate regions, 
rather than points. The broad phenomena, however, are clear. 
A high-speed vessel in water of depth less than her length will at a 
given speed in a given depth begin to experience appreciably in- 
creased resistance as compared with its resistance in deep water. 
The increase of resistance above the normal becomes greater and 
greater as speed increases until it reaches a maximum. This maxi- 
mum appears to be at about a speed such that a trochoidal wave 
traveling at this speed in water of the same depth is about ii times 
as long as the vessel. As the vessel is pushed to a higher speed the 
resistance begins to approach the normal again, reaches and crosses 
the normal at about the speed indicated in Fig. 145, and for 
higher speeds the resistance in shallow water is less than in deep 

It was at one time supposed that the speed for maximum increase 
in resistance was that of the wave of translation. This, however, 
as illustrated in Fig. 145, holds only for water whose depth is less 
than .2 L. For greater depths the speed of the wave of translation 
rapidly becomes greater than the speed of maximum increase of 

There are obvious advantages in the model-basin method of 
investigating this subject. Consider, for instance, Fig. 144 showing 
actual falling off of resistance beyond the critical speed in the 
curves for the Yarrow destroyer which were obtained by model- 
basin experiment. This remarkable feature would never be detected 
on a full-scale trial of an actual destroyer, because if such a vessel 
were forced to surmount the hump it would leap the gap, as it 
were, and show a sudden jump in speed. Theoretically if the depth 
of water were absolutely uniform it would be possible after the 
jump in speed to gradually throttle down until the boat would be 


working in the hollow, but the chance of this ever being done, unless 
it were known that the hollow should be there, is infinitesimal. 

4. Shallow- Water Experiments at United States Model Basin. 
That the hollow really exists, as shown in the curves for the Yarrow 
destroyer, is confirmed by published results of other model-basin 
shallow-water experiments and by a number of carefully made ex- 
periments in the United States Model Basin. 

Fig. 147 shows curves of resistance and change of trim of the 
model of a fast scout in various depths of water. The model was 
20 feet long on L.W.L., with 2'. 268 beam and o / .842 mean draught. 
It displaced in fresh water 996 pounds. The corresponding speed 
of the model for 30 knots speed of the full-sized ship would be 
only 6. 6 1 knots, but the experiments were carried to a much higher 
speed as a matter of interest. 

The sudden and peculiar drops in the shallow-water curves are 
very marked. It is seen that they are accompanied by peculiar 
corresponding perturbations in the curves showing change of trim 
or change of level of bow and stern. We have from Fig. 147 : 

Depth of water 

1 8" 


? 6" 

Speed of maximum % increase of resistance, knots . . . 
Trochoidal wave lengths above speed and depth. . . 
Speed of hollow in resistance curve, knots 


2 5-5' 
4. 60 



C .Os 


c o? 

Speed of wave of translation or trochoidal wave of 
infinite length in the depth of water, knots 


J 7C 

; &-> 

The general features of Fig. 147 agree closely with results of 
trials of other models in shallow water at the United States Model 
Basin. Some peculiar wave phenomena appear in such trials. 
In running such models in deep water or in shallow water at speeds 
well below that of the hump the disturbance set up in the water 
is inappreciable a short distance ahead of it. But at about the 
speed of the hump the wave at the bow tends to manifest itself as 
a crest extending straight across the basin and well ahead of the 
bow as much as 8 or 10 feet. As the speed is increased this 
singular manifestation disappears, and again there is no appre- 
ciable disturbance ahead of the model. These phenomena have 
not been given careful investigation. A reasonable explanation of 
the sudden drop of the resistance curve would be that it corre- 


spends to the wave of translation, which advances with less de- 
mand upon the model for energy to maintain it than was the case 
at a slightly lower speed when the wave system was being built 
up even ahead of the model. 

At the higher speeds the waves are forced waves, necessarily 
departing widely from trochoidal waves. It should be remarked 
that the high "deep water" resistance of the model at speeds in 
the vicinity of 8 knots may be in part due to the limited depth 
(14 feet) of the basin, but is probably mostly due to the appear- 
ance of the last normal deep-water hump of resistance curves. 
The hump which appears below 6 knots in 46 inches depth is 
found at about 8 knots in 14 feet depth. 

5. Shallow-Water Resistance for Moderate and Slow Speed 
Vessels. The case of greatest practical interest is that of the 
vessel of moderate speed say capable of a deep-water speed in 
knots of .9 \/L or less. Such a vessel in shallow water cannot 
be pushed beyond the last hump of her resistance curve, and hence 
always loses speed in shallow water. For such vessels we would 
like to know the least depth of water in which resistance is not 
appreciably increased or speed appreciably retarded and the 
amount of increase of resistance in water that is shallower. 

Results of experiments bearing directly on the first question 
were published in 1900 in a paper before the Institution of Naval 
Architects by Major Giuseppe Rota. Major Rota experimented 
with models of five vessels, one being the torpedo boat model, 
whose results are given in Fig. 143. Each model was run in vari- 
ous depths of water and the results carefully analyzed for the pur- 
pose of determining the depth at which increased resistance began. 

For the purpose of analysis and deducing results applicable to 
other vessels it is important to determine in connection with such 
experimental results the fundamental variables, as it were. For 
instance, in this case shall we connect the depth of water with the 
length, the beam or the draught of the ship? We have seen that for 
high-powered vessels we were led to the use of the ratio between 
depth of water and length of vessel, which gives satisfactory re- 
sults as regards determination of critical points, etc. Considera- 
tion, however, appears to indicate that for the vessel of moderate 


speed it would probably be better to use the ratio between depth 
of water and mean draught of ship, allowing the length factor to 
come in through the speed-length coefficient. 

While Rota's models could, of course, each be expanded to rep- 
resent any number of ships, he gives one size of ship for each as 
shown in the table below. 

Model No . 






Displacement of ship in tons. . 
Length of ship in feet 






Beam of ship in feet 

7c . e 



40. 3 


Mean draught of ship in feet . 
Block coefficient 


. =u 


. so 

2O. 2 





Taking Rota's curves giving the depths for no increase of re- 
sistance for various speeds of the above ships and replotting them 


to express in each case a relation between and the depth of 


water expressed in draughts of the ship, we have the results shown 
in Fig. 146. It is seen that for each model the locus thus plotted 
is reasonably close to a straight line and that the dotted line is 
reasonably close to the average of the five up to the speeds not 

greater than -=. = .9. Curiously enough, the two finer models fall 

above the dotted line. This, however, is probably due to the 
fact that they are vessels of distinctly shallow-draught type, and 
because of that, in spite of their fineness, need a depth of more 
draughts than vessels of deeper-draught type. A scrutiny of Rota's 
results, however, indicates that for models 4 and 5 the decrease of 
depth from that of lines 4 and 5 in Fig. 146 to that of the dotted 
line will involve in practice an increase of resistance barely percep- 
tible. Then Rota's experiments may be fairly summarized by the 
straight line of Fig. 146. If H denotes the draught, it is seen from 
the diagram that this line gives us the relative minimum depth for 


no increase of resistance = 

This formula giving mini- 

mum depth for no increase of resistance applies, strictly speaking, 
only to Rota's five models, but it is seen that they cover the range 
of usual proportions for models of a fine block coefficient. 


The formula, however, has been found to apply satisfactorily to 
models of block coefficient higher than .5 tested in the United 
States Model Basin. One model of block coefficient slightly above 
.65 was tried in various depths and the formula found to apply 

To sum up, I think that the above formula from Rota's experi- 
ments may be confidently applied : 

1. To vessels not of abnormal form or proportions up to a 
block coefficient of .65. 


2. For speeds for which ~=is not greater than .9. 

The formula may be of use beyond the limits indicated above, 
but in such cases needs to be applied with caution and discretion. 

6. Trial Course Depths. As illustrative of the little impor- 
tance attached to this question until a comparatively recent date, 
Major Rota in his 1900 paper states: "Stokes Bay, where British 
ships used to undergo their speed trials, is only 59 feet deep; the 
official measured mile at the Gulf of Spezia, Italy, is about 62 feet 
deep; the measured miles at Cherbourg and Brest are 49 and 59 
feet respectively." Such depths are now regarded as entirely 
inadequate and no speed trials of large ships are regarded as 
accurate unless made in deep water. Curiously enough, however, 
as indicated in Fig. 145, the shallow course exaggerates the speed 
of the very fast vessel, and there are many torpedo craft in exist- 
ence whose full-speed trials were held on shallow courses with 
resulting speeds greater than would have been attained in deep 

7. Percentage Variations of Resistance in Shallow Water. - 
Coming now to the question of the actual increase of resistance 
of a given vessel in water of a given depth, it is necessary again 
to make a distinction between the vessel of very high power and 
speed and the vessel of moderate speed. For the former it is 
probably best, as before, to use as the governing variable the ratio 

between depth and length, y- For the latter it still seems best 

to use the ratio between depth and draught,^- For either type, 




expressing the speed by > we are able for each vessel or model 

for which there is adequate experimental information to draw con- 


tours on = and ratio between depth and length or depth and 


draught as the case may be, which show percentages of increase 
over deep-water results. For the very high-speed vessels percent- 
ages of decrease will also appear. This work at best can be only 
a tolerably good approximation, and hence we assume in it that 
the law of comparison applies fully to the total model resistance. 
Figs. 148 to 153 are percentage increase diagrams, the type of 
vessel being indicated in each case. 

The diagrams for the high-speed vessels show percentages of 
decrease. For the moderate-speed vessels the percentage increase 
of resistance goes up rapidly with increase of displacement length 
coefficient. While Figs. 151, 152 and 153 cannot be said to cover 
the ground as would be desirable, they will be better than nothing 
and of help in many cases. 

Inland navigation is mostly smooth-water, shallow-water navi- 
gation, and there is great need of a complete investigation into 
the features of form affecting shallow-water resistance. While we 
know quite well the general features of the form best adapted to 
speed in deep water in a given case we do not know the same thing 
for shallow water. It appears probable, however, that if we 
wish to make 12 knots in shallow water and are considering vari- 
ous models, that one which will drive easiest in deep water at a 
higher speed say 1 5 knots or so will drive easiest in shallow 
water at the i2-knot speed. If high speed is to be attempted in 
inland navigation there are practical advantages in length which 
would be excessive for deep-water work. Wave making, with the 
resulting wash at banks and piers, should be kept as low as possi- 
ble for boats in river service. 

8. Shallow- Water Influence upon Trim and Settlement. Fig. 
147 shows the curves of the settlement of bow and stern of a scout 
model in shallow water. It is seen that the shallower the water 
the lower the speed at which marked change of trim begins, and 
within the limits of practicable speed the greater the change of trim. 


For speeds above those at all possible the trim changes would not 
very greatly depart from those for deep water. We are more con- 
cerned in practice, however, with settlement and change of trim 
at low speeds, corresponding to those at which shallow channels 
would be traversed. Fig. 147 shows that at such speeds the effect 
of shoal water is simply to increase the settlement of both bow 
and stern. In its broad features, Fig. 147 is fairly typical of 
change of trim results in shoal water for a number of other models. 
We may say that the effect of shoal water upon a vessel under 
way is to increase the natural settlement of both bow and stern 
at low speed. The shallower the water the lower the critical speed 
at which squatting or excessive change of trim begins and the 
greater the change of trim. At high speeds the shallower the 
water the more the stern settles and the more the bow rises. At 
extreme speeds, however, the stern does not appear to settle or 
the bow to rise so far as in deep water. It is interesting to note 
in Fig. 147 the peculiar perturbations in the change of level curves 
and the evident close connection between them and the remark- 
able drops in the resistance curves. 

9. Increase of Draught in Shallow Channels. In practice 
there are very few vessels of sufficient power to attain high speed 
in shallow water, and those that have the power would very sel- 
dom use it in shallow water, so that the behavior of vessels as 
regards settlement under way at moderate speed in shallow chan- 
nels is of more practical importance than their possible behavior 
at excessive speeds. 

A very interesting investigation of this question was made in 
connection with the channel of New York Harbor, and was de- 
scribed in detail by Mr. Henry N. Babcock in Engineering News 
for August 4, 1904. This channel was constantly used by large 
steamers passing in and out with very little to spare between their 
keels and the bottom of the channel. There were repeated com- 
plaints from such vessels that they had touched bottom in places 
where the officers in charge of the channels were unable to dis- 
cover spots shoaler than the still-water draught of the steamers. 
The observations were confined to large transatlantic steamships 
passing out of New York, averaging over 550 feet in length. They 


were made at three points, one where the channel was 80 to 100 
feet deep, one where the low- water channel depth was from 31.1 to 
32.5 feet, and a third where the low- water depth was from 31 to 
34.5 feet. 

The general scheme of most of the observations was to deter- 
mine the height above water of marks on the bow and stern before 
the steamer left her pier. Then as the steamer passed the observ- 
ing station the level of these marks was determined with reference 
to the station, and as soon as possible after the passage of the 
vessel the water level was determined with reference to the observ- 
ing station. Considering all the circumstances, exact observations 
are obviously not possible, but after making ample allowance for 
possible errors of observation Mr. Babcock's report demonstrates 
conclusively that vessels of the type considered when under way 
in channels settle both at bow and stern, and the shoaler the water 
and higher the speed the more they settle. It was not practicable 
from the results to formulate fully conclusions connecting amount 
of settlement with size and type of vessel, speed and depth of 
water, but Mr. Babcock, upon analyzing the results, concluded that 
for vessels of the large transatlantic steamship type the increase 
of draught in feet, when still water clearance under their keels was 
less than about 10 per cent of the draught, would be i the speed 
of the ship in miles per hour. For a natural clearance of some 30 
per cent of the draught the increase in feet would be about iV the 
speed of the ship in miles per hour, and for intermediate clearances 
intermediate fractions should be used. 

Further observations of the character reported by Mr. Babcock 
on the settlement of vessels under way, not only in shallow channels 
but in canals, would be of much interest and practical value. 

17. Rough- Water Effects 

i. Causes of Speed Reduction. The effect of rough water upon 
speed is like the effect of foulness of bottom almost impossible to 
reduce to quantitative rules. The very real and material reduc- 
tion of speed of vessels in rough weather is of universal experience. 

This, however, is not always due to increased resistance alone. 
The motion of the ship may render it impossible to develop full 


power. The danger of racing may render it inadvisable to use 
full power. The disturbance of the water reduces the efficiency of 
the propellers. The conditions may render it impossible to use 
full speed without risk of dangerous seas coming on board. 

2. Features Minimizing Speed Reduction. The increase of 
resistance in rough water is under practical conditions largely a 
question of absolute size. Waves 150 feet long and 10 feet high 
would not seriously slow a 4o,ooo-ton vessel 800 feet long. 

A vessel of a few hundred tons 120 feet long would find them a 
very serious obstacle to speed. Pitching enters into the question 
of rough-water speed as a very important factor. 

When conditions are such as to produce severe pitching, speed goes 
down very rapidly. Pitching exaggerates nearly all causes of speed 
loss. Not only is actual resistance rapidly increased but racing is 
caused, the propeller loses efficiency and more water comes on board. 

If it were possible to devise a vessel which would not pitch it 
would lose much less speed in rough water than one that does 
pitch; but though many naval architects have strong opinions on 
the subject there is no agreement among them as to the features 
of model which minimize pitching. The preponderance of opinion 
is probably in favor of the U-bow type and rather full bow water 
lines. But pitching is unfortunately largely a question of condi- 
tions. Under certain conditions of sea, course, and speed one type 
may be superior and under slightly changed conditions distinctly 

Apart from absolute size there appears, however, to be one 
broad consideration which is of some value as a guide. Suppose 
we have two 2o-knot vessels, A and B, of about the same power 
and such that at 22 knots A offers distinctly less resistance than B. 
There is little doubt that on the average A would lose less speed 
in rough water than B. 

When for a vessel intended for a certain service it is necessary to 
allow in the design for the effect of rough water upon speed there 
is only one safe method to follow namely, to allow a reduction 
from smooth-water trial conditions to rough-water service condi- 
tions based upon actual experience with previous vessels in the 


1 8. Appendage Resistance 

1. Appendages Fitted. Substantially all that has been said 
about resistance hitherto refers to the resistance of the main body 
or hull proper. There are found on actual ships appendages of 
various kinds, such as rudders, bar keels, bilge keels, docking keels, 
shaft swells, shafts, shaft struts, propeller hubs and spectacle 
frames, or shaft brackets or bosses. Shaft tubes, or removable 
tubes around the outboard shafts, are seldom fitted nowadays. 

The appendages fitted vary. Thus, a single-screw merchant 
ship with flat keel will have practically no appendage except 
the rudder, the slight swell around the shaft having hardly any 
effect. For such a vessel the appendage resistance would seldom 
be as much as 4 or 5 per cent of the bare hull resistance. 

A twin screw vessel with large bilge and docking keels and 
perhaps two pairs of struts on each side may have an appendage 
resistance as much as 20 per cent of the bare hull resistance. 

Appendage resistance is largely eddy resistance and can be kept 
down to the minimum only by very careful attention to details and 
the application of adequate fair waters wherever needed. 

2. Resistance of Bilge and Docking Keels. Bilge keels and 
docking keels should follow lines of flow and be sharpened at 
each end. When this is done it is generally found in experiments 
upon models that the additional resistance due to them is not 
greater than that due to the additional surface alone. In fact 
the additional resistance is sometimes found to be less than that 
due to the additional wetted surface. Mr. Froude found a similar 
result in his full-sized Greyhound experiments. While if bilge 
keels and docking keels are properly located and fashioned the 
additional resistance may be taken as that due to their wetted sur- 
face only, the wetted surface they add is often very considerable. 

In models bilge keels may be located at appreciable angles 
with the natural lines of flow without greatly augmenting resist- 
ance beyond that due to their surface, but it does not follow that 
the same result would be found in the full-sized ships. It is 
necessary to be cautious in applying the Law of Comparison to 
eddy resistance. There is little doubt that the law applies to the 

I2 4 


Eddy Resistance behind a square stern post, for instance. Here 
the eddying for model and ship is found in each case over cor- 
responding areas. 

But in the case of a bilge keel located across the lines of flow 
we may readily conceive that there may be but little eddying 
around the model bilge keel and a great deal around the full- 
sized bilge keel. This because the pressure of the atmosphere re- 
maining constant the total pressure around the full-sized bilge 
keel is not increased in the proportion required to insure com- 
pliance with the Law of Comparison. 

3. Resistance of Struts. Probably struts and spectacle frames 
are the appendages to which the most careful attention must be 
paid from the point of view of resistance. Experiments with a 
number of strut arms of elliptical section appear to indicate that 
the resistance in pounds per foot length may be expressed with 
fair approximation for areas from 40 square inches to 175 square 
inches by the following semi-empirical formula: 

R=^-(A - 


Where R is resistance in pounds per foot length, V is speed 
through the water in knots and A is area of cross section of strut 
in square inches. The coefficient C depends upon the ratio be- 
tween B, the thickness of the strut section, and L, its width in 
direction of motion. The table below gives values of C for vari- 

ous values of 














i. 880 








. 720 


From the point of view of resistance only, the best ratio of 
breadth to thickness would be 10 or over, but as the wide, thin 
strut requires more area for a given strength, it follows that 
the best all-round ratio would be somewhat smaller, say from 
7 tog. 


Even this ratio is not very often reached in practice, the tend- 
ency apparently being to make strut arms much narrower and 
thicker than they should be. 

As regards shape of section, model experiments indicate that a 
pear-shaped section, or a section of rounding forward part and 
sharp after part, offers the least resistance. Such a section may 
show model resistance as much as 10 per cent below the elliptical 

There is doubt, however, whether this holds for full-sized struts 
for high-speed vessels. Study of Fig. 16 would seem to indicate 
that at sufficiently high speeds there must be eddying over all the 
rear half of any strut, in which case the thickness of the strut 
should be reduced to a minimum. From this point of view, if a 
strut of given width and area is to have the minimum thickness 
for a given type of head the rear portion should be made of paral- 
lel thickness and cut off square. Furthermore, from this point 
of view, if air were piped to the rear of a strut the resistance 
would be decreased. This question of strut resistance is worthy 
of further careful experimental investigation. Pending this, the 
approximate formula and coefficients above for elliptical struts 
may be used, and it may be assumed that the elliptical form is 
about as good as any. For moderate speeds the rear portion of 
the strut may be brought to a sharp edge, but for high speeds 
this refinement will probably be of little use. 

4. Resistances of Propeller Hubs. Behind the strut hub the 
propeller hub is fitted, and for propellers with detachable blades 
is usually larger than the strut hub. About all that can be done 
for the propeller hub is to fit a conical fair-water behind it. Model 
experiments show that a long fair-water, say of length about twice 
the diameter of the propeller hub, offers materially less resist- 
ance than a short fair-water of length say about one-half the 
diameter of the propeller hub. 

While there is some doubt whether the long fair-water would 
show up so well in comparison on the full-sized ship, the length of 
fair-water should not be skimped. 

With quick running propellers the objections to large hubs have 
become more evident and there is a tendency to use solid pro- 


pellers with small hubs. From the point of view of appendage 
resistance, these are distinctly preferable to large hubs. 

5. Resistance of Spectacle Frames or Propeller Bossing. In 
merchant practice, struts are not much used for side screws, 
being replaced by spectacle frames or propeller bossing. 

These appendages, if well formed, offer less resistance than thick 
struts with the bare shafts, etc., but in many cases wide, reason- 
ably thin struts would offer less resistance than shaft bosses. 
Shaft bosses are, however, usually regarded as giving better secu- 
rity to the shaft, and certainly give access to a greater portion of 
its length. They absorb much more weight than struts. The 
angle of the web of a shaft boss may vary a good deal from what 
may be called the neutral position, or position where it is edge- 
wise to the flow over the hull without very great effect upon the 
model resistance, but there is a little doubt that the full-sized 
ship will be prejudicially affected if the shaft boss webs depart too 
far from the neutral position. Eddying is liable to appear in the 
case of the full-sized ship which does not occur in the case of the 

The angle of such webs has a powerful influence upon the stream 
line motion in the vicinity of the stern. A vertical web or a 
horizontal web tends seriously to obstruct the natural water flow 
and drag more or less dead water behind the ship. It seems to 
be usually the tendency from structural considerations to work 
the shaft boss webs somewhere near the horizontal. From the 
point of view of resistance alone a 45 angle for the rear edge may 
not be too great. This is another case where conflicting consider- 
ations necessitate a compromise. The determination of after lines 
of flow over the hull will greatly facilitate the determination of 
the most suitable shaft boss arrangements. 

6. Allowance for Appendages in Powering Ships. In esti- 
mating from model experiments the effective horse-power of a 
ship with appendages the methods are the same as for the bare 
hull. From the total model resistance the frictional resistance for 
the total wetted surface including appendages is deducted and the 
remaining or residuary resistance treated by the Laws of Compari- 
son. From what has been said in discussing appendage resist- 


ance, it is evident that estimates of E.H.P. with appendages are 
apt to be less accurate than estimates of the net or bare hull 
E.H.P. unless care has been taken so to shape appendages that 
they do not develop in the full-sized ship eddies which have no 
corresponding eddies in the case of the model. 

In practice, it is customary and almost necessary to power a 
new design from model experiments with bare hull only. This is 
readily done by using for the ratio between the bare hull E.H.P. 
and the I.H.P. of the ship with appendages a conservative coeffi- 
cient of propulsion based upon coefficients of propulsion actually 
obtained from past experience with vessels reasonably similar as 
regards appendages to the case under consideration. 



19. Nomenclature Geometry and Delineation of Propellers 

i. Definitions and Nomenclature. A screw propeller has two 
or more blades attached at their inner portions or roots to a hub 
or boss, which in turn is secured upon a shaft driven by the pro- 
pelling machinery of the ship. Figs. 154 to 157 show plans of a 
three-bladed propeller for a naval vessel. This is a true screw - 
that is, the face or driving face is a portion of a helicoidal surface 
of uniform pitch. A helicoidal surface of uniform pitch is the 
surface generated by a line the generatrix at an angle with 
an axis which revolves about the axis at a uniform angular rate 
and also advances parallel to the axis at a uniform rate. A 
cylindrical surface concentric with the axis will cut such a heli- 
coidal surface in a helix. The pitch of the helicoidal surface is 
the distance which the generatrix moves parallel to the axis dur- 
ing one complete revolution. Figs. 154 to 157 show a three- 
bladed right-handed propeller that is, a propeller which, viewed 
from aft, revolves with the hands of a watch when driving the 
ship ahead. The various portions of a propeller are indicated in 
the figures, such as the face and back of the blades, the leading 
edge and the following edge, the tip and the root. Since in prac- 
tice the back of each blade is its forward surface, care must be 
taken to avoid confusion. 

This result will be obtained by avoiding such expressions as 
" forward face," " after face," etc., and adhering to the terms 
"face" and "back." The word "face" will always denote the 
driving face or the face which pushes the water astern when the 
propeller is in action, while the word " back " naturally denotes 
the surface opposite the face. 

While a true screw as already indicated is a screw propeller 



whose blade faces are all portions of helicoidal surfaces of the 
same pitch, there are many variants from the true screw. 

Each point of the face may have its own pitch, which may be 
denned as the distance parallel to the shaft axis which an ele- 
mentary area around the point would move during one revolu- 
tion around the shaft if it were connected to the shaft by a rigid 
radius and working in a solid fixed nut. Fig. 158 shows two views 
of a small elementary area LL connected to the shaft axis O by a 
radius r. This area makes an angle with the perpendicular to the 
axis called the pitch angle and denoted by 6 in Fig. 158. If p 
denote the pitch of LL, during one revolution in a solid nut its 
center would advance along the helix OCCD, to the point D at 
a distance p along the axis from 0. If then we unroll the cylinder 
of radius r, upon which has been traced the helix OCCD, this 
helix will become the straight line OP of Fig. 158, while PM = p, 
the pitch. 


OM = 2 wr and tan 6 = 

2 irr 

There are several typical variations of pitch which are used 
more or less for actual propellers. Thus if the pitch increases as 
we pass from the leading to the following edge, the blade is said to 
have axially increasing pitch. If the pitch increases as we go out- 
ward, the blade is said to have radially increasing pitch. If the 
pitch decreases as we go outward, the blade has radially decreasing 
pitch. A blade may have pitch varying both axially and radially. 

Pitch of the blade face only has been considered in the above, 
and in an ideal blade of no thickness that is all that need be con- 
sidered; but for actual blades we need to consider the pitch of the 
back of the blade as well. Evidently each point of the back of 
an actual blade has a distinctive pitch. For blades such as shown 
in Figs. 154 to 157, where the face has uniform pitch and the blade 
sections are of the usual ogival type, the pitch of the center of 
the blade back is the same as the pitch of the face. The pitch of 
the leading portion of the back is less; and of the following por- 
tion greater than the face pitch. These pitch variations over the 
blade back have important effects upon propeller action. 

The ratio between pitch and diameter is called pitch ratio, and 


the ratio between diameter and pitch is called diameter ratio. 
Each point of a blade has, of course, its own pitch ratio and 
diameter ratio, but these expressions are also used in reference to 
the propeller as a whole. When so used the diameter referred to 
is the diameter of the screw or of the tip circle, and the pitch is 
the uniform pitch of the face for a true screw and an assumed 
average face pitch for a screw of varying pitch. 

There are two other ratios which it is convenient to define here. 
Fig. 159 shows a radial section through the center of a blade of 
very common type by a plane through the axis. This plane in- 
tersects back and face of the blade in two straight lines, which, 
prolonged through the hub to the axis, cut it at C and A respec- 


The ratio - - is called the blade thickness ratio and is 


evidently constant for similar propellers, whatever their size. 

The blade section in Fig. 159 is shown raking aft, the total rake 
reckoned along the mid-thickness of blade sections being in the 

figure BO. Then - is called the rake ratio. It is reckoned 

positive for after rake and negative for forward rake. 

Propellers do not in practice move through the water as through 
a solid nut. They advance a distance less than their pitch for 
each revolution. Under given conditions of operation the distance 
advanced is the same for each revolution, hence the path of each 
element is a helix and can be developed into a straight line. Recur- 
ring to Fig. 158, = OC\C\D\ is the helical path of LL with slip and 
OS the development of this helix. As before, POM is the pitch 
angle 6. The angle POS is called the slip angle and will be denoted 
by <f>. Fig. 158 may also be regarded as a diagram of velocities, 
OM being the transverse or rotary velocity of the element and MS 
its velocity parallel to the axis. MS is often called the speed of 
advance, and MP, or the speed for no slip, is called the speed of 
the propeller, being the pitch multiplied by the revolutions. Then 
PS is the speed of slip or the slip velocity. Slip is usually char- 

acterized, however, by the ratio --L , or the ratio between the speed 


of slip and the speed of the propeller. This is properly called the 
slip ratio, or slip fraction. It is also commonly and conveniently 
called simply the slip and expressed as a percentage instead of a 
decimal fraction. Thus when we say, for example, that a propeller 
works with a slip of 15 per cent we mean that 

Speed of Propeller Speed of Advance _ 
Speed of Propeller 

Sometimes we need the ratio 

Speed of Advance 
Speed of Propeller ' 

and this may conveniently be designated the speed ratio. 

2. Delineation. In practice a propeller is usually delineated 
as in Figs. 154 to 157, by projections of the blades in at least two 
directions, an expansion of a blade and sections of a blade. 
Views and sections are also shown as necessary to determine the 
hub of propeller with solid hubs and the hub and blade flanges 
and bolting of propellers with detachable blades. 

It will be observed that the faces of the sections in Fig. 155 all 
radiate from a fixed point on the axis, called the pitch point. This 
is a more or less convenient arrangement. Referring to Fig. 160, 
suppose p is the pitch of a blade at the radius OA = r. Lay off 

OP = - Then tan OAP = ~ + r = -- But from Fig. 158 

2 TT 2 TT 2 irr 

= tan 6 where 6 is the pitch angle or the angle which the 
2 irr 

element makes with a transverse plane. Hence in Fig. 160 OAP 
and the corresponding angles at the other radii are the pitch angles 
at the radii in question. 

Figs. 154 to 157 refer to an ordinary true screw of oval blade 
contour with a rake so small that it is practically negligible. 
Much more complicated forms are used sometimes, the complica- 
tions involving varying pitch, curved radial sections, extreme rake 
forward or aft, lopsided or unsymmetrical blade contours, and 
various types of blade sections. Some forms of propellers are 
difficult problems in descriptive geometry. There does not seem 
to be any benefit in practice from complicated forms of propellers 


and no attempt will be made to take up the problems of their 

3. Area and its Determination. The question of propeller area 
is a very important one. There are various areas considered in 
connection with a propeller. When we speak of the blade area of 
a propeller we generally mean what is called the helicoidal area, or 
the actual area of the helicoidal faces of the blades. As it happens, 
however, a helicoidal surface cannot be developed into a plane so 
the helicoidal area of a propeller cannot be determined exactly. 
The area we determine is what is called the developed area, the 
blade face being developed into a plane by a more or Jess approxi- 
mate method. 

The disc area of a propeller is the area of the circular section of 
its disc or the area of the circle touching the blade tips. 

The projected area is the area of the projections of the blade 
faces upon a transverse plane perpendicular to the axis. 

The ratio between the developed and disc areas of a propeller 
is sometimes called the disc area ratio. 

The ratio Projected Area -j- Disc Area is also frequently used 
and is of more practical value than the ratio Developed Area -4- 
Disc Area. 

While the helicoidal face of a propeller blade cannot be developed 
exactly into one plane it can be so developed with such slight 
distortion that the resulting surface is an approximation amply 
close for practical purposes. 

Suppose we cut the helicoidal surface of a blade face by a cylin- 
der concentric with the axis. It will cut a helix from the helicoidal 
surface. If now we pass a plane tangent to the helicoidal surface 
at its center, it will cut the cylinder in an elliptical arc. If then 
we take that portion of this elliptical arc whose rearward projec- 
tion is the same as that of the actual helix of the blade face we will 
have an arc of very nearly the same length as the helix. Then if 
we take a series of such arcs, swing them into a common plane and 
join their extremities by a bounding curve, we shall have a devel- 
oped surface which is very close to the actual helicoidal surface 
in area. 

Fig. 1 60 shows the construction, is the center, P the pitch 


point, OA the radius of a cylinder. Let BBB be the projected 
blade. Then the cylinder of radius OA cuts BB at C. The plane 
at A tangent to the helicoidal surface makes with the axis the angle 
OP A the complement of the pitch angle. The minor semiaxis 
of the ellipse which it cuts from the cylinder is OA. The major 


semiaxis is = ~r~ 7 = AP. Draw the elliptical arc AD with 
sin OP A 

major axis of length AP and minor axis OA in length and 
position. Then draw the horizontal line CD meeting the ellipti- 
cal arc at D. D is a point on the developed blade, and by deter- 
mining a series of such points and drawing a line through them 
we obtain the developed contour EDBEE. Suppose now we draw 
AF horizontal through A and make AF equal in length to the 
elliptical arc AD. A line through a series of points such as F will 
give what may be called the expanded contour. It is denoted in 
the figure by HFBHH. The developed area is usually taken as 
BEEKEDB. The expanded area, BHHKHFB, is very close to 
the developed area. 

The developed area obtained by the above method is slightly 
smaller than the true area. The elliptical arcs are not very easy 
to draw in practice and a simple method is to use arcs of circles 
with radii which are the radii of curvature of the ellipses. Thus 
draw PM at right angles to AP and cutting AO produced at M. 
Then M is the center of curvature of the ellipse at A , and instead 
of drawing the ellipse we may draw a circular arc of radius MA. 
The developed area thus determined is slightly greater than the 
exact helicoidal area, the area using the exact ellipses being 
slightly less. But the area determined using the circular arcs is 
a closer approximation to the true area, particularly for broad 

In practice we generally assume the developed contour, making 
it any desired shape, deduce the projected contour by reversing 
the method of development described above, and from the pro- 
jected contour deduce by the methods of descriptive geometry the 
other projections desired. A very common and very good con- 
tour for the developed blade is an ellipse touching the axis, having 
the radius as major axis and the expanded breath of blade at 


mid-radius as minor axis. In the vicinity of the hub the ellipse is 
departed from as necessary to make a good connection. 

4. Coefficients of Area for Elliptical Blade. Fig. 161 shows an 
elliptical developed blade contour with major axis equal to the 
propeller radius. The radius of hub is tV that of the blade. 
There is shown dotted a rectangular area touching the hub and 
tip circle and of width such that its area is the same as that of the 
elliptical blade outside the hub. Then the width of this rectangle 
is called the mean width of the blade. 

It is convenient usually to use the diameter as the primary 
variable when dealing with propellers, so we naturally express the 
mean width as a fraction of the diameter. 

The ratio (mean width of blade) -r- (diameter of propeller) is 
called the mean width ratio and is denoted by h. 

This mean width ratio characterizes a blade very definitely and 
it is convenient to express many other features by its use. For the 
elliptical blade with hub diameter & of the propeller diameter let 
I denote the maximum width or minor axis of the ellipse. Then 

we have mean width ratio = h =.842 - , or / = 1.188 hd. 


If n denote the number of blades we have the total blade area 
or Developed Area = .4 n<Ph. 

The projected area for a given developed area depends upon the 
pitch ratio, which denote by a. For values of a found in practice, 
say from a = .6 to a = 2.0, the projected area for the elliptical- 
bladed propeller of hub diameter .2 of the propeller diameter is 
given with close approximation by the formula, 

Projected Area = (0.4267 0.09160) nd?h. 

From the above we have the following addit onal ratios for 
values of a between .6 and 2.0: 

Projected Area -j- Developed Area = 1.067 ~~ .2290. 

Developed Area -f- Disc Area = .509 nh. 

Projected Area -j- Disc Area = (.543 .11660) nh. 

Fig. 162 shows contours of the ratio (Projected Area) -5- (Disc 
Area) for elliptical three-bladed propellers. 

While the above formulae and Fig. 162 apply strictly only to 
propellers with elliptical blades and hub diameter tV of propeller 


diameter, they are accurate enough for practical purposes for any 
other hub diameter likely to be found in practice and are rea- 
sonably good approximations for any blades of oval type. 

5. Twisted Blades. Propellers with detachable blades nearly 
always have them fitted so that they can be twisted slightly in the 
boss, thus increasing or decreasing the pitch. The blade flange 
holes are made oval, as shown in Fig. 156. The twist or rotation 
of the blade is about a line or axis through the center of the flange 
perpendicular to the shaft. 

All pitch angles on the axis are changed a uniform amount. 

For points of the blade away from the axis of twist the change 
is less, and for points of the helical surface a quarter of a revolu- 
tion from the axis, if the surface were so great, there would be no 
change of pitch due to twist. For usual width of blade, however, 
the change in pitch angle is practically uniform over the blade 
and equal to the angle of twist. Hence the change of pitch due 
to twist will be investigated on this assumption. 

Let y denote the diameter ratio, 6 the pitch angle at a given 
point of radius r and pitch p. Let 7 denote the angle of twist 
and y the new diameter ratio after twisting. 

Then tan 6 = -*- = - y = - cot 0, 

2 irr iry TT 

tan (e + 7) = -^7, 


y =-cot(8+y)=- cotgcot T- i = I rrycoty- I = ycot 7 -7r 
TT TT COt 6 + COt 7 TT Try + Cot 7 iry -\- COt 7 

From the above formula, given y and 7, we can readily calculate y'. 
For a positive twist or value of 7 the new diameter ratio is less 
than the old, the new pitch and pitch ratio being greater. For a 
negative twist the opposite holds. 

The results are shown graphically in Figs. 163 and 164. In 
Fig. 163 the results are plotted upon diameter ratio. For each 
value of 7 a curve is drawn showing the new values of diameter 
ratio plotted as ordinates over the old values as abscissae. Con- 


tours are shown for each degree of positive and negative twist up 
to 6. 

Fig. 164 gives the same information as Fig. 163, but the results 
are plotted upon pitch ratio. 

Figs. 163 and 164 illustrate the relative advantages and disad- 
vantages of pitch ratio and diameter ratio when used as primary 
variables. Fig. 163 using diameter ratio, once the conception 
of diameter ratio is firmly grasped mentally, is simpler and 
more readily understood. This is largely because the diameter 
ratio at the tip of the blade is the natural starting point, and for 
any point of less radius the diameter ratio decreases directly as the 
radius. The conception of pitch ratio is more readily formed, but 
starting with the pitch ratio of the tips the pitch ratio increases 
inversely as the radius and becomes infinite for zero radius. In 
either case the tip value is a simple quantity of numerical value 
ranging in practice from .5 to 2. When using diameter ratio for 
any one blade the field covered, neglecting the hub, is that between 
zero and the tip value. When using pitch ratio the field is that 
between infinity and the tip value. 

20. Theories of Propeller Action 

i. Principles of Action Common to all Theories. There have 
been a great many different theories of propeller action propounded, 
but none which has been generally accepted as agreeing fully with 
the facts of practical experience. 

The principles underlying the chief English theories of propeller 
action are comparatively simple. The resulting formulae are more or 
less complicated, but not difficult to apply. In any theory in con- 
nection with which mathematical methods are to be used it is almost 
necessary to regard the blade as having no thickness. Fig. 165, 
which partially reproduces Fig. 158, indicates the motion of a small 
elementary plane blade area of radius r, breadth dr, in a radial 
direction and circumferential length dl. Looking down we see 
this element with its center at 0. If w is the angular velocity of 
rotation of the shaft, the transverse velocity of the element is 
cor. AOB is the pitch angle 0, BC the slip and BOC the slip angle 


<. We know that tan 8 = Considering Fig. 165 as a 

2 TTf ' 

diagram of instantaneous velocities, the line OA or cor represents 
the transverse velocity of the element. If there were no slip, the 
actual velocity would be parallel to OB since BOA = 6. Then 
AB would denote the axial velocity. 

AB = OA tan 6 = cor tan = cor 

2 TTT 2 7T 

When there is slip the transverse velocity of the element is un- 
changed, but the axial velocity is the speed of advance AC, which 
is denoted by V A . BC is the slip and AB, the speed of the screw, 
is the same as the speed of advance when the slip is zero. 

Denote the slip ratio by s. 

Then s = BC = AB ~ AC _2^ _ '__<P- 2vV A_i_y **. 
BA AB cop up 

2 7T 

Whence the speed of advance VA*= (i s) BC = s ^*- 

2 7T 2 7T 

If we take w as angular velocity per second and r in feet, then OA 
or the transverse velocity is in feet per second, and hence all other 
velocities are in the same units. 
Then we have 

Velocity of blade element in the direction of the perpendicular to 

its plane = CD = BC cos = s ^- cos 6. 

2 7T 

Axial or rearward component of above velocity = CE = CD 
cos 6 = s ^- cos 2 0. 

2 7T 

Transverse component of above velocity = DE = CD sin 6 = 

(j)p , 
s - 11 - sin cos 6. 

2 7T 

2. Three English Theories of Propeller Action. There are 
three theories of propeller action whose detailed consideration 
will be of value. They are all contained in papers before the 
Institution of Naval Architects. The first was by Professor Ran- 


kine in 1865, the second by Mr. Wm. Froude in 1878 and the 
third by Professor Greenhill in 1888. 

Rankine's fundamental assumption was that, as the propeller 
advanced with slip BC, all the water in an annular ring of radius 
r was given the velocity CD in a direction perpendicular to the 
face of the blade at that radius. Then, from the principle of 
momentum, the thrust from the elementary annular ring is pro- 
portional to the quantity of water acted upon in one second and 
to the sternward velocity EC communicated to it. 

Froude considers the element as a small plane moving through 
the water along a line OC which makes a small angle < with OB, 
the direction of the plane. Then Froude takes the normal pressure 
upon the elementary area which gives propulsive effect to vary as 
the area, as the square of its speed OC, and as the sine of <j> the slip 

Greenhill makes a somewhat artificial assumption. He assumes 
that the propeller is working in a fixed closed end tube. The 
result is that the motion communicated to the water is wholly 
transverse and would be represented by CF in Fig. 165. The 
blade is first assumed smooth, so that the pressure produced by 
the reaction of the water is normal to the blade and has of course 
a fore and aft component which gives thrust. In all three theories 
the loss by friction is taken as that due to the friction of the 
propelling surface moving edgewise or nearly so through the 

3. Relation between Direction of Pressure and Efficiency. - 
Neglecting friction for the present it is evident that all three 
theories start with a certain normal pressure. It follows that if 
this normal pressure be resolved into its axial and transverse com- 
ponents, say dT and dQ, we have 

41 = n = OA = 2L = 2jrr 
dQ ~ ~ AB ~ co = p 

2 TT 

Hence pdT = 2 -n-rdQ. 

Now 2 TrrdQ = total work done during one revolution and hence, 
neglecting friction, pdT = total work done during one revolu- 


tion. Now the useful work = dTp (i s), hence the efficiency 


It follows that, neglecting friction, if the reaction pressure from the 
water is normal to the face at all points of a screw of uniform pitch 
working with a slip s, the efficiency of each element and of the 
whole screw will be i s. Since the friction must reduce efficiency 
in all cases, it follows that upon the above supposition the efficiency 
of a screw cannot ever exceed i s. It is often thought that it 
is mechanically impossible for the efficiency of a screw to exceed 
i s. This, however, is not necessarily so. This limitation is 
associated with and dependent upon the assumption that the 
resultant pressure at each point of a screw surface is perpendicular 
to the surface. If the water can be made to move in such a man- 
ner that the resultant reaction is at an angle with the normal to 
the blade surface, we may have an efficiency, neglecting friction, 
greater than i s. This is an important point and worthy of 
careful investigation. 

Referring to Fig. 166, suppose we have acting on a point two 
forces OA and OB whose resultant OC makes an angle a with the 
axis of x, as indicated. Let the point O be moving with the 
velocity OE at the angle P with the axis of x as indicated. Then 
the work done by the reaction against the force OA = OA X OD. 
The work done by the force OB = OB X ED = AC X ED. 

Draw OF perpendicular to OC and denote EOF by 7. The 
ratio between the work done by the force OB and the work done 
by the reaction against OA is 


Now j3 = 90 a j. 

The above is readily applied to the propeller problem. Refer- 
ring to Fig. 167, which partially reproduces Fig. 166, consider an 
element at O whose pitch angle DOP is denoted by 6. Suppose 
OC is the resultant reaction upon the element O. Draw OF per- 
pendicular to OC. Then AO is the transverse force upon the 
element denoted by q, say, while AC is the thrust denoted by /. 


OD is transverse velocity V t and DE is velocity of advance V&. 
POD being the pitch angle 6, POE is the slip angle $. Then the 
efficiency is the ratio between the useful work done by t and the 

T) 7? 
gross input or work done by q and as before is - - Now if DE 


is speed of advance DP is speed of screw and -= = slip ratio = s. 

The efficiency of the element depends upon the directions of the 
resultant OC, and OF the perpendicular to it. Suppose the re- 
sultant OC is perpendicular to OP, then 7 = <, F goes to P and 

the efficiency is = i s. It appears, then, to be rigidly 

demonstrable that if the resultant reaction at every point of a 
true screw is perpendicular to the face the efficiency of every ele- 
ment, and hence of the screw as a whole, is i s. As the direc- 
tion of the resultant OC approaches the fore and aft line, or the 
perpendicular to AD, the efficiency of the element increases and 
would become unity if the resultant could become perpendicular 
to AD. As the direction of the resultant OC swings out from the 
fore and aft line beyond the perpendicular to the element, the effi- 
ciency becomes less than 1 5. Friction and head resistance 
always tend to swing the resultant in this direction, and the smaller 
the slip the smaller the values of AO and OC and the greater the 
relative effect of the force due to friction and head resistance. 

I will now, neglecting friction at first, develop the formulae for 
thrust and torque of a screw, following the three theories already 
referred to. For convenient comparison a uniform notation will 
be used, so far as practicable, differing slightly from the several 
notations of the original authors. 

4. Rankine's Theory of Propeller Action. Referring to Fig. 
165 by Rankine's theory, considering the annular ring of mean 
radius r, 

Annular area = 2 irrdr. 
Volume of water acted on per second = 

2 irrdr X AE = 2 irrdr X ^ (i - 5 sin 2 6). 



Stern ward velocity communicated = EC = s -*- cos 2 6 = s^ cos 2 6. 

2 IT DO 

Hence elementary thrust = mass of water per second X stern- 

ward velocity imparted = dT =-2 -n-rdr^ (is sin 2 6} s *- cos 2 

60 60 

= * s (i s sin 2 0) cos 2 2 

g 3 6o 

, 2 

Let q = cot = Then 2 Trrdr = " dq. sin 2 = - 

p 2 TT i + q 

cos 2 6 = - 


w p 2 R 2 ( 

dT = ~ A 5 

600 \i 

Q 2 __\ 

A 2 M 

g 3600 \i + q 2 (i + q 2 ) 2 / 2ir 

^L qdq 

At the axis q = o. Then, neglecting the hub, which a very slight 
investigation shows to have very little effect; if q denote now cotan- 
gent of the pitch angle of the blade tips, we have on integrating 
the expression for dT: 

_ _ loge(i + <7 2 ) _ 

g 3600 2 7T _2 2 \ 2 21 

W /> 2 /? 2 ^ r iog.(i + ^ 2 ) j /io gc (i + 9 2 ) i 

g 3600 47rL ^ \^ 2 i+ q 


Now pq = 2 irr p 2 q 2 = 4 7i 2 r 2 *-*- = irr 2 = if d is extreme 

47r 4 

diameter. Whence 


. j /log, i 

g 3600 4 L q 2 \ q 2 i + q 2 

Whence finally 

T = _jrw_ p2d 2 R2 J I _ log e (i + q 2 ) _ s /log e (i + q 2 } i_Yl 

14400 g L q 2 \ q 2 i + q 2 /] 


And the torque Q = 

2 7T 



5. W. Froude's Theory of Propeller Action. Consider now 

Fronde's theory. 

If / is the total blade length of all blades at radius r, then the 
total elementary plane area at this radius is Idr. This area ad- 
vances at the angle (j> (Fig. 165), with velocity OC, and from 
Froude's experiments if a is a thrust coefficient, we have a result- 
ing pressure normal to the blade = Idr aOC sin <. The ele- 
mentary thrust is equal to this pressure X cos 6. 
Then dT = Idr aOC' 2 sin < cos 6. 



cos 6. 

Also cos 2 6 = 

-afcx)' *: ddi + f' 

Whence, neglecting the hub as before, 

3600 /oOI+^ 27T 

The quantity under the integral sign is evidently dependent only 
on shape and proportions of the propeller and independent of its 
dimensions. It can be determined in any case by graphic integra- 
tion. For the present, let us denote it by the symbol X. Then 

from Froude's theory T= - -p*R*dsX, and as before O = - 

3000 2 TT 


6. Greenhill's Theory of Propeller Action. Coming finally to 
Greenhill's theory, we have (Fig. 165) 

Elementary area = 2 irrdr. 
Velocity of feed of the water = AC = *- (i s) = *- (i s). 

2 7T 6O 

Transverse velocity = s *- cot 6 = su>r = s * r. 

2 7T 60 

Transverse momentum per second = - 2 irrdr ^ (i s) s ^ r 

g 60 60 

W R 2 , x , 2 

= p s(i s) 4 irrdr. 
g 3600 

Torque = transverse momentum X r. 

7j TV 

Whence dO = - p s (i s) 4 ir^dr. 
g 3600 

_,_, 2 irdO W R 2 , \ofij 

dT = *- = - s (i s) 8 Tr 3 rdr. 

p g 3600 

Integrating from r = o to r = - we have 


IV R / \ TT iV / \ 

g 3600 28800 g 

And as before Q = ^ 

2 7T 

In connection with Greenhill's theory, it should be pointed out 
that the excess pressure at any radius is very simply expressed. 

w R 2 

We have above dT = - s(i s) S-n^rdr. 
g 3600 

But if AP be the excess pressure per unit area, dT = 2 wrdrAP. 

w R 2 

Whence dividing through AP = - s (i s) 4 w 2 r 2 . 

g 3600 

In other words, the excess of pressure varies as the square of the 
radial distance from the axis. 

7. Comparison of Theories with Each Other. Now, com- 
paring the three formulas for thrust and torque, it is seen that 
each one is composed of a coefficient, of a term involving the 


dimensions and revolutions or speed, and of a term varying with 
shape, proportions and slip but independent of the dimensions. 
Assuming, as is evidently possible, that we can expand X in the 
formula from Froude's theory in the form a fts + negligible 
terms, we can write for each formula T = <j> (pdR) (as 13s-). 

For Froude's theory < (pdR) = p z dR- and for Rankine's theory 
< (pdR) is p~(PR z . For Greenhill's theory <f> (pdR) is d 4 R 2 , of the 
same dimensions as before but independent of the pitch. Now, 
considering a and /3, it is evident that by the formula for Froude's 
theory /3 will be very small indeed compared with a. In the 
Rankine theory formula /3 will be smaller than a, but relatively 
larger than in the Froude theory formula. In the Greenhill theory 
formula /3 = a always. 

Still neglecting friction, we would have on the theory of all 
motion communicated to the water perpendicular to the blade 

Q = PI = - <f>(pdR) (as - /3s 2 ). 

2 IT 2 7T 

As a matter of fact, a very brief examination of experimental 
results shows that this cannot hold. If it were true, we could 
never have an efficiency greater than i s, and even when fric- 
tion is considered we get experimental efficiencies greater than 
i s. So it appears well to adopt tentatively as the general 

expression for the torque Q = - t $ (pdR) (ys 8s 2 ). 

2 7T 

8. Friction and Head Resistance. Now consider friction and 
head resistance. Referring to Fig. 165, if / denote the' coefficient 
of friction and dA an elementary area, we have with close approxi- 
mation frictional resistance = fdAOB?. In practice < is a much 
smaller angle than indicated in Fig. 165. where it is exaggerated 
for clearness. Suppose / is large enough to cover all edgewise 
resistance skin friction and head resistance together. 

Then dA = Idr, OB* = p*R* cosec 2 P = p 2 R 2 (i+q 2 ), q = > 

2 IT 

Then F = f% -*- dqfR 2 (i + q>) = ffdR* -*- \ l - (i + ? 2 ) dq I 
a 2 IT 2 IT (a } 


Fore and aft component = Deduction from thrust 

= F sin 6 = fp 2 dR 2 -- - Vi+q*dq=dT f . 
2 TT a 

Transverse component = F cos 6 = fp z dR 2 -'-q\/i-}-q 2 dq. 

2 TT d 

Difference of torque = F cos 6 X r = F cos 6 " 

2 7T 

X =^2/. 


Deduction from thrust for friction 

2 ird 

Addition to torque for friction = Q,= 

= --fp*dR 2 Z, where Z = f - ^ 2 V : T+~fdq. 

2 T J 2 ird 

2 7T / 2 TTtt 

Since for the working portions of actual propellers <? is greater 
than i, we will have in practice Z much greater than Y, and it is 
reasonable to ascribe the total friction loss to increase of torque. 

If we assume - constant = mean width ratio X number of blades, 

we can readily determine a curve of Z on q by plotting a curve of 

^2^/j i 02 

*- and integrating graphically. 

2 7T 

For actual propellers Y and Z can be determined without diffi- 

7 __ . 7 

culty by plotting on q curves of - Vi + o 2 and -q 2 Vi + o 2 

2 Tra 2 Trtt 

and integrating graphically. 

Fig. 1 68 shows curves of Y and Z and of for elliptical blades 


with hub diameter .2 the extreme diameter, plotted upon pitch 
ratio, and Fig. 169 shows curves of X for various values of s, 
namely, s = o, .20, and .40. 


9. Final Formulae on Theories of Rankine, Froude and Green- 
hill. Then the final formulae for thrust and torque including the 
friction term can be expressed in the forms below: 

Rankine's Theory: T = pWR 2 (as - /3s 2 ) - fdj?R 2 Y, 

Q = - [pWR 2 (ys - 5s 2 ) + fdp*R 2 Z], 

2 7T 

Froude's Theory : T = p 3 dR 2 (as - /3s 2 ) - fdf&Y, 

Q = -- [p 3 dR 2 (ys - 5s 2 ) +fdp 3 R 2 Z]. 

2 7T 

Greenhill's Theory: T = d*R 2 (as - /3s 2 ) -fdp 3 R 2 Y, 

Q = ^~ [d*R 2 (ys - 3s 2 ) +fdp 3 R 2 Z]. 

2 7T 

The above equations are simply to show the form of the ex- 
pressions. They do not imply that a and /3 in the Rankine Theory 
equation will be the same as in the Froude or Greenhill Theory 
equation, but simply that in each case a and /3 will be constant for 
a given propeller. The actual values of the constants will vary 
with the theory used. 

The formulae on Froude's theory are expressed in the above 
form, as previously noted, by assuming that X can be expanded 
with sufficient approximation in the form C sD, where C and 
D are independent of s. It is evident from Fig. 169 that this can 
be done and that D is much smaller than C. 

In all the theories, as has already been pointed out, it is assumed 
that the net reaction at each point is perpendicular to the blade 
surface. If this were true, we would always have a = y, /3 = <5, 
and the efficiency could never exceed i s even if there were no- 
friction. Since experience shows this is not the case, and as from 
considering the probable motion of a particle of water it is evi- 
dently not necessary that the net momentum impressed upon it 
shall be perpendicular to the blade surface, I have, while follow- 
ing the same form, used different coefficients for the torque 
expression, expecting that these coefficients y and 5 need not 
necessarily be the same as a and /3 used for thrust. 

It seems difficult at first sight to conceive of any fluid action 


upon a frictionless surface that is not at right angles to it, but if 
we consider the matter from the point of view of the velocity im- 
pressed upon the water the difficulty disappears. The suction of 
the propeller upon the water ahead of it causes a velocity which is 
practically all axial, or in the direction perpendicular to the plane 
of the propeller disc. Hence, the reaction upon the water is partly 
axial before the water reaches the propeller disc and partly normal 
or nearly so as the water passes through the disc, the final result- 
ant being at an angle with the normal in the direction which we 
have seen tends to make the efficiency greater than i s. 

10. Comparison of Theories with Facts of Experience. It 
does not require much reflection to render it evident that none of 
the three theories considered correctly represents the physical 
phenomena. This conclusion is very strongly confirmed by the 
results of model experiment and general experience. 

On Rankine's theory the water while passing through the screw 
disc is given the stern ward velocity EC (Fig. 165). This can 
occur only if the stream contracts materially while passing through 
the propeller or if a material quantity of water from abreast the 
disc is always flowing into it. Neither motion seems reasonable. 
Furthermore, on Rankine's theory, the thrust and torque are 
independent of the blade surface, one assumption of Rankine's 
theory being that "the length of the screw and number of its 
blades are supposed to be adjusted by the rules deduced from 
practical experience, so that the whole cylinder of water in which 
the screw revolves shall form a stream flowing aft." 

Practical experience with model propellers shows clearly that the 
result assumed by Rankine is unattainable. Rankine's theory 
further ignores variations of pressure which must occur in pro- 
peller action. 

Froude's theory goes to the opposite extreme of Rankine's. It 
assumes that the thrust increases always in direct ratio to the 
area. Model experiments show conclusively that, while within 
practicable limits thrust does increase as long as area increases, 
the increase in thrust is by no means proportional to the area 
increase, the rate of increase with area diminishing steadily as 
area increases. 


Greenhill's theory has the same obvious defect as Rankine's, in 
that it neglects the effect of area of blade. The portion I have 
used ignores the sternward velocity, deducing thrust entirely from 
the pressure set up by rotating the water in the disc, but it should 
be pointed out that his 1888 paper gives some consideration to 
other possible motions involving axial velocity of slip in the 

As, then, it seems that no theory we have considered can exactly 
represent the action of propellers, it would be necessary, in case we 
wished to adhere to formulae, to compare each formula with experi- 
mental results and select that one which seemed to agree most 
closely. Then using this as a semi-empirical formula, with coeffi- 
cients and constants deduced from experiments or experience, 
problems could be satisfactorily dealt with. But it will be ob- 
served that each formula is of the proper dimensions to satisfy 
the Law of Comparison. Hence if either formula holds, the Law 
of Comparison will hold, and experimental results, instead of 
being utilized to supply coefficients and constants for use with a 
formula, can be reduced to a form to be utilized directly by graphic 
methods. Per contra, if the Law of Comparison does not hold, the 
formulae on all of the three theories will fail. In either case there 
is obviously no advantage from a practical point of view in attempt- 
ing to reduce the formulae to forms for use in practice. A serious 
practical disadvantage is the fact that the formulas use a true slip, 
based upon true pitch, or a blade of no thickness. The face pitch 
of a blade with thickness, or its nominal pitch as it may con- 
veniently be called, is very different from the virtual or effec- 
tive pitch, and this fact causes material complications in using 

ii. Slip Angle Values. In connection with theories of pro- 
peller action it is desired to invite particular attention to the fact 
that propellers in practice operate with slip angles that are very 
small indeed. A slip of 20 per cent somehow seems to imply a 
large angle, but as a matter of fact it usually means in practice an 
angle of from i\ to 5 degrees only, and most propellers show their 
maximum efficiency at slips below 20 per cent. 


Referring to Fig. 165, where <p denotes the slip angle, we have 

CD BCcose 

sin <> = 

CO \/ r\ /( 2 i A/^~ 


cos 6 

2 7T 

2 , 

2 r 2 + 

47T 2 

Let y denote diameter ratio = - = Then r = "- 

p p 2 

Substituting, clearing and reducing, we have finally 

sin =5 

+ 7T 2 / VVy 2 + (l - S) 2 

Hence given s and y the value of </> is fixed. 

Fig. 170 shows graphically the relation between slip angle (j>, 
slip 5 and diameter ratio. Also at the top of the figure is a scale 
for pitch ratio, but reference to diameter ratio is more illuminating. 
Considering a screw of uniform face pitch it is seen that for a given 
slip per cent the slip angle is a minimum where the diameter ratio 
is greatest at the blade tip. As we go in from the tip the slip 
angle increases, reaching a maximum when diameter ratio = .3 
about, and then rapidly decreasing to zero at the axis. But on 
account of the hub the falling off of slip angle below diameter ratio 
of .3 is immaterial, and to all intents and purposes slip angle in- 
creases from tip to hub. The actual values for the diameter 
ratios and slips found in practice say below diameter ratio of i.i 
and slip ratio of .30 are quite small. 

The maximum efficiency of most propellers corresponds to a 
nominal slip in the neighborhood of 15 per cent, and for this the 
maximum slip angle at the hub is less than 5 and for the most 
important part of the blade it is in the vicinity of 3. These are 
small angles, and the fact that slip angles are so small should never 
be lost sight of in considering operation of propellers. 



21. Law of Comparison Applied to Propellers 

i. Formulae for Applying Law of Comparison to Propellers. - 

In connection with the Law of Comparison the formulae for the 
application of the law to propellers have been already indicated, 
but they are recapitulated below. 

Suppose we have a propeller and a smaller similar propeller or 
model. Let us use symbols as in the table following : 

For Large 

For small 
or Model. 

Diameter in feet - 



Revolutions per minute 


Speed of advance in knots 


Thrust in pounds 



Torque in pound-feet 


Pressure on propeller, pounds' per square inch 



Power absorbed 




Then if X denote the ratio of linear dimensions of model and full- 
sized screw we have the following relation: 

D = \d, 

V = 

T = \ 3 t, 

2. Conditions Governing Application of Law of Comparison. - 

Note that for the complete applicability of the Law of Comparison 
p l= x^i, or all pressures should be in the ratio of the linear dimen- 
sions. Now the pressure under which a model propeller works is 
made up of two components the water pressure due to its sub- 
mersion and the constant pressure of the atmosphere exerted upon 
the surface and transmitted through the water. 

When we consider the full-sized propeller we find the pressure 
due to submersion is or readily can be increased to scale; but the 
atmospheric pressure is not increased, and hence this component 
of the total pressure upon the full-sized propeller is only i -T- X of 
the value needed to have the Law of Comparison exactly appli- 
cable. Hence it might be inferred that, as the conditions required 
by the Law of Comparison are not present, model experiments are 


of little value in the investigation of propellers. But upon con- 
sideration it is evident that in each case the atmospheric pressure 
is transmitted through the water, appearing both in front of and 
behind model and propeller; and, since the forces upon model 
and propeller are due to reactions caused by the motions impressed 
upon the water, the Law of Comparison will apply provided the 
motions of the water around model and propeller are similar. 
The pressure relation fails in precisely the same way in passing from 
models to ships, but in this case the motions produced are not 
affected by the surface pressure and the Law of Comparison holds. 
Hence we may rely upon the Law of Comparison and design pro- 
pellers upon the basis of model results if we can but be sure that 
the motions of the water around model and propeller will be 

Now, we are reasonably certain that until we reach speeds and 
thrusts at which the phenomenon known as cavitation makes its 
appearance the motions of the water around model and propeller 
are so nearly similar that the Law of Comparison is applicable. 
When cavitation is present the Law of Comparison fails, because, 
as will be seen when discussing cavitation, the model does not cavi- 
tate as a rule, and hence results from it are an unsafe guide when 
dealing with the full-sized screw. But the majority of propellers 
as fitted are not very seriously, if at all, interfered with by cavi- 
tation, and for such propellers model experiments are of great 
value, since the Law of Comparison may be somewhat confidently 
relied upon in connection with them. Exact comparison of experi- 
mental data from a model and a full-sized propeller of large dimen- 
sions has never been made, but experiments at the United States 
Model Basin showed that for small or model propellers ranging 
from 8 inches to 24 inches in diameter the Law of Comparison 
applies reasonably well. (See paper entitled "Model Basin Glean- 
ings," Transactions Society of Naval Architects and Marine Engi- 
neers for 1906.) 

We have seen that theoretical formulas for propeller action all 
give the result that for a given propeller form advancing with a 
given slip the thrust and torque vary as the square of the speed of 
advance and also, that the thrust varies as the square and the 
torque as the cube of the linear dimensions. 


If this is the case, the Law of Comparison necessarily holds. 
There are a number of reasons for thinking that thrust and torque 
for a given propeller advancing with given slip vary as the square 
of the speed of advance. If the lines of flow or paths followed by 
the particles of water are the same, whatever the speed, then 
thrust and torque must vary as the square of the speed. For then 
the quantity of water acted upon must vary directly as the speed, 
and the velocity communicated to each particle acted on must 
vary directly as the speed. Hence the momentum generated per 
second, to which thrust and torque are proportional, must vary as 
the square of the speed. 

Experiments made at the United States Model Basin in 1904 
with 1 6-inch model propellers between speeds of three and seven 
knots showed that within the limits of experimental error thrust 
and torque varied very approximately as the square of the speed. 
The propellers whose thrust varied as a greater power of the speed 
than the square were usually those with very narrow blades. 
Those whose thrust varied as a lesser power of the speed than the 
square were usually those with very broad blades. 

Finally, experience in analyzing accurate trial results shows that, 
broadly speaking, when cavitation is not present, at speeds where 
the resistance of the ship is varying as the square of the speed the 
slip is practically constant, which of course means that the thrust 
of the propeller advancing with this constant slip varies as the 
square of the speed. 

At speeds for which the resistance of the ship is varying as a 
less power of the speed than the square the slip is falling off, and 
at speeds for which the resistance is varying as a greater power 
of the speed than the square the slip is increasing. This is 
fairly strong evidence from accumulated experience that the 
thrust of full-sized propellers varies as the square of the speed of 

In the light of present knowledge we appear to be warranted in 
concluding that the Law of Comparison applies to propeller action 
sufficiently well for practical purposes until cavitation appears. 
There is reason to believe, however, that cases have occurred 
where cavitation has been present without being suspected. 


22. Ideal Propeller Efficiency 

i. Thrust, Power and Efficiency of Ideal Propelling Apparatus. 

In a paper before the Society of Naval Architects and Marine 
Engineers, in 1906, entitled "The Limit of Propeller Efficiency," 
Assistant Naval Constructor W. McEntee, without setting up 
any special theory of propeller action, has pointed out the limit of 
propeller efficiency beyond which we cannot go. 

Suppose we have a frictionless propelling apparatus discharg- 
ing a column of water of A square feet area directly aft with an 
absolute velocity u, while the speed of the ship is v, both v and u 
being measured in feet per second. Then if w denote the weight 
per cubic foot of the water, the weight acted on per second is 


wA (v + u} and the mass is - A ( + ). 



The reaction or thrust T = A (v + u) u being equal to the 


sternward momentum generated per second. 


Useful work = - A (v + u} vu. 

There being no friction, the lost work is simply the kinetic energy 
in the water discharged. Hence we have 

Lost work = - A (v + u) 
g 2 

fiat not J Jj/, 

Gross work = - A (v + u) vu -\ A (v + u) 

g g 2 

Useful work v 

Efficiency e = -77 r- = 

Gross work . u 

v H 


Also solving for u in the equation for thrust T, we get 

/v 2 . sT v 
u = V/-+- 

V 4 wA 2 

Substituting in the expression for efficiency, we have 

- - 

This expression for maximum efficiency must involve the assump- 


tion that the water is discharged without increase of pressure. 
The effect of an increase of pressure would be to decrease the 
efficiency, since work done against pressure would be done with 

efficiency - Hence we conclude that the value of e above 

v + u 

is the maximum that could be attained by a perfect propeller. 

Suppose, applying this to a screw propeller, we write for A , 


where d is the diameter of the propeller in feet. Now if U denote 
useful horse-power delivered by the propeller and P denote gross 
horse-power, or horse-power delivered to the propeller, we have 

rr TV ~ <55 eP 6080 T , , 

eP = U = , whence T = M Also v = - - V, where V 

55 v 3 6o 

is speed of advance in knots. And g = 32.16, w = 64 for sea 
water. Substituting and reducing, we have finally 


16 24 e + 8 e 2 _ 2 3 e + e 2 

d?V s 292.2 e 3 36.52 

2. Discussion of Ideal Efficiency Results. From the above, 
Figs. 171 and 172 were drawn, Fig. 171 showing contours of 


efficiency on values of V as abscissae and of as ordinates and 

d 2 

Fig. 172 showing contours of efficiency on values of d as abscissae 

and of 7- as ordinates. 
V 3 

These figures should not be mistaken as representing actual 
efficiencies that are attainable. They are purely ideal diagrams, 
and their indication that efficiency always increases with increase 
of diameter is misleading if followed too far as regards actual pro- 
pellers. They are interesting and instructive, however, as giving 
us in any particular case a limiting efficiency beyond which we 
could not possibly go and which we must fall short of in practice. 

In Fig. 171 there is shown a supplementary scale of , or power 


per square foot of disc area. This of course bears a constant 


ratio to 


A striking result of the formula for ideal propeller efficiency is 
the high efficiency attained with large slips. The expression for 


slip ratio s\. in terms of v and u, is Si= The formula for 

v + u 


efficiency is e = - , whence expressing e in terms of s\, we have 


Fig. 173 shows a curve of e plotted on Si, as deduced from the 
above formula. This efficiency is everywhere above the line i Si. 

In this connection it is interesting to recall that numerous ex- 
periments with model propellers at high slips show an efficiency 
greater than i s. It should be remembered, however, that in 
the case of these actual small propellers s is derived from the pitch 
of the driving face, while in the ideal formula $1 is based upon the 
assumed sternward velocity u of the water, and the water is not 
supposed to have any transverse velocity. The actual sternward 
velocity of the water in the operation of actual propellers is not easy 
to determine or estimate, and transverse velocity is always present. 

On Rankine's theory we can readily establish the relation be- 
tween s and sternward velocity. In Fig. 165 the sternward velocity 
is EC = s cos 2 6. This is much less than BC, the slip velocity. 
While we cannot say that in actual cases the sternward velocity is 
EC, there is no question that it is very much less than BC, the slip 
velocity. It could be equal to BC only if there were no trans- 
verse velocity communicated to the water, and there is no ques- 
tion that in practice transverse velocity is always communicated. 
A very common mistake is to consider the sternward velocity 
communicated to the water the same as the slip velocity, or BC in 
Fig. 165. 

23. Model Experiments Methods and Plotting Results 

i. Experimental Propeller Models and Testing Methods. - 

Having concluded that the Law of Comparison is applicable to 

many cases of propeller action so that experiments with model 

propellers may be expected to be of value, I will now go into this 


question. Numerous experiments with model propellers have 
been made at the United States Model Basin. The details of the 
apparatus and methods used will be found in the author's paper 
of 1904 before the Society of Naval Architects and Marine Engi- 
neers entitled "Some Recent Experiments at the United States 
Model Basin." The experimental gear described in that paper 
has been changed subsequently only in minor details as improve- 
ments suggested themselves. 

The model propellers are usually made of composition, accu- 
rately finished to scale. Most of them have been 16 inches in 
diameter. When being tested the model propeller is attached to 
a horizontal shaft projecting ahead of a small boat which is rigidly 
secured to the carriage traversing the basin. The shaft projects 
so far that the propeller is practically unaffected by the presence 
of the following boat. The propeller shaft center is 16 inches 
below the surface of the water, so that the blade tips of a 1 6-inch 
model are immersed 8 inches, or one-half of a diameter. The hub 
is fitted with fair- waters in front and behind. Fig. 174 shows the 
arrangement for a hub 3! inches in diameter, which was a stand- 
ard hub diameter adopted for all models which did not represent 
actual propellers. For models of actual propellers, the hubs rep- 
resent to scale the actual hubs, appropriate fair-waters being fitted. 

Dynamometric apparatus, described in detail in the paper 
above referred to, enabled the torque and thrust of the model 
propeller to be accurately determined. 

By making runs with dummy hubs having no blades attached 
the hub effect was eliminated as far as possible, the endeavor 
being to determine experimentally the torque and thrust of the 
blades alone. 

The greater number of experiments were made at a 5-knot speed 
of carriage, this speed of advance being kept constant as nearly 
as possible, and slip being varied by varying the revolutions of the 
propeller. In the early stages of the experiments, however, a 
number of propellers were tested at speeds of advance ranging 
from 3 knots to 7 knots, and between these speeds it was found that 
within the limits of error the thrust and torque at constant slip 
varied practically as the square of the speed. As has been already 


pointed out, this agrees with the formula of Rankine, Froude and 
Greenhill, which agree in making thrust and torque vary as R 2 , 
and when slip is constant the speed of advance varies as R. 

In making the 5-knot experiments speeds of individual runs of a 
series would differ slightly from 5 knots, and the thrust and torque 
were reduced to the 5-knot speed by taking them to vary as the 
square of the speed. 

2. Methods of Recording Experimental Results. As will be 
seen upon consulting the original paper, during a run the thrust 
and torque are recorded continuously, and after uniform condi- 
tions have been reached the time and revolutions are recorded 
every 32 feet. For convenience the thrust and torque at 5 knots 
speed are plotted initially upon the revolutions made by the pro- 
peller upon a 64-foot interval denoted by pi, which is one of 
the quantities observed. 

Fig. 176 shows curves of thrust and torque plotted thus for the 
model propeller whose developed blade outline and blade sections 
are shown in Fig. 175. This is a 1 6-inch three-bladed model 
propeller of the true screw, ordinary type, the pitch being 16 
inches pitch ratio i.oo the blades being elliptical, of .25 
mean width ratio, and the sections ogival. The hub diameter is 
.2 the propeller diameter. The curves of Fig. 176 are plotted upon 
Pi, or revolutions per 64-foot interval. Lines showing the values 
of pi for various values of the slip are shown on the figure, the 
slip being based upon the nominal pitch of 16.0 inches. These 
lines are not equally spaced, for, p denoting pitch in feet and s 

the slip ratio, we have ppi (i s) = 64, or pi= - For 

P (i ~ s) 

equal increments of s the interval between successive correspond- 
ing values of pi constantly increases. 

It will be observed that p\ is dependent upon the pitch and 
slip only and for a given slip is quite independent of the speed. 
Furthermore, the experimental apparatus was such that pi was 
determined with great accuracy. Thus it was a very suitable 
quantity to use as a primary variable upon which to plot the 
experimental values of thrust and torque for the purpose of de- 
ducing curves of the same. 


24. Model Propeller Experiments Analysis of Results 

1. Methods of Plotting Information Derived from Experiment. 
- The results of model experiments having been plotted as curves 

of thrust and torque upon the revolutions made upon a 64-foot 
length as shown in Fig. 176, the lines for various definite nominal 
slips being indicated upon the same diagram, the subsequent 
treatment depends upon the purpose in view. 

For purposes of analysis, comparison of efficiency, etc., the 
methods would naturally differ from those most convenient for 
use in design. 

When we consider the best method of plotting for purposes of 
analysis, etc., curves deduced from model propeller experiments, it 
soon becomes evident that we may with advantage record the 
data as curves of coefficients quantities that do not vary with 
dimensions. As abscissae for such curves the slip ratio is a de- 
sirable quantity to use. It is not dependent upon size or speed, 
and is one of the primary variables involved in screw action. 

2. Virtual and Nominal Pitch and Slip. The question at once 
arises, however, whether we should use nominal slip, namely, slip 
based upon the pitch of the screw face, or real slip, i.e., slip based 
upon the virtual pitch, or pitch of the ideal blade of no thickness 
which would act as the actual blade. 

This virtual pitch is a thing very different from the nominal 
pitch. The ignoring of this fact has had a great deal to do with 
the prevention of correct conclusions as to propeller performance. 
In the case of a true screw the pitch of the driving face is known, 
but every point of the back has a pitch, and the back has much to 
do with screw performance. One might think without looking 
into it that for ordinary cases the pitch of the back is nearly the 
same as that of the face. The truth is that the pitch of the back 
varies prodigiously from the pitch of the face. Fig. 175 shows 
blade sections of a screw of not unusual blade thickness and of 
face pitch equal to diameter, the sections being of the usual ogival 
type. Taking face pitch and diameter as 16 feet, Fig. 177 shows 
plotted on radius the pitch of the back at the leading edge and at 
the following edge. It is seen that the pitch of the leading por- 


tion of the back will average somewhere about 50 per cent less 
than the uniform pitch of the face or the nominal pitch. On the 
other hand, the pitch of the following edge of the back is on the 
average somewhat more than 50 per cent greater than the nomi- 
nal pitch. It is quite obvious that such a screw cannot act as a 
theoretical screw, having blades of no thickness and of the uniform 
pitch of the face. It is evidently desirable to find some method 
of determining for a known screw its virtual pitch, or equivalent 
uniform pitch. Now, for all formulae we have, neglecting fric- 
tion, no thrust or torque at zero slip. Experimental results with 
screws of uniform nominal pitch and ogival type of blade section 
always show as in Fig. 176 both thrust and torque when the slip 
calculated on the nominal pitch is zero. It follows that for such 
screws the virtual pitch is greater than the nominal pitch. This 
might be inferred, too, from the fact that at the rear of the blade 
the pitch of the back is always greater than the nominal, and, if 
the back has any influence at all, it must increase the virtual 
pitch over the nominal pitch. Suppose, now, we consider some 
experimental results. Fig. 178 shows upon an enlarged scale the 
lower part of Fig. 176, being curves of thrust and torque as deter- 
mined experimentally for a 1 6-inch model of the propeller of 
Fig. 175 plotted upon p\, or revolutions required to traverse a 
distance of 64 feet, the speed of advance of the propeller being 
kept constant at 5 knots. Now on any theory we have at true 
zero slip a negative thrust T f and a positive torque Qf, both being 
due to the friction and head resistance only. From the formulae 
given when considering the theories of Rankine, Froude and 

T, = -fdp 3 R*Y, Qf=-- fdp*R 2 Z. 

2 7T 

Whence fdj?R*= ~^ = ^' 

pT f Y pT , 
Whence ^ ' = = -* ^ when s = o. 
2 irQf Z 2 -nQ 

Now is a fixed quantity for the propeller. For the propeller in 


question it is equal to .236, from Fig. 168. Fig. 178 shows the 

method to be followed. If s = o, we have p = - So we can 


plot a curve of p on pi as abscissa. Also we can plot the curve of 

i)T Y 

-^ ;, as shown. This has the value .236 = at PI= 42.11, 
2 irQ Z 

for which p = 1.520 feet. Then from the diagram the virtual 
pitch of the screw is 1.520 feet, or 18.24 inches, or 1.140 times the 
nominal pitch of 16 inches. Very frequently the virtual pitch is 
taken such that zero slip will give zero thrust. This is not quite 
correct, however, because at zero thrust there is a small negative 
thrust due to friction and an equal and opposite positive thrust 
due to slip. The error, however, is not great. In Fig. 178, at zero 
thrust PI = 42.70, p = 1.499 feet = 17.99 inches. The difference in 
virtual pitch is only about 1-3 per cent, and as it is very difficult 
to make model propeller experiments with minute accuracy, it is 
hardly worth while in practice to use the exact method. More- 
over, while we should always bear in mind that the nominal pitch 
is not the real pitch or virtual pitch, it is very desirable to use 
always the nominal pitch in practical cases. We shall see that 
this can be done, so that the question of virtual pitch, though of 
great scientific interest, is academic rather than practical. So, 
except for special applications, results for true screws of uniform 
face pitch will be plotted upon nominal slip corresponding to the 
face or nominal pitch. 

3. Determination of Efficiency. The ordinates for the curve of 
efficiency plotted upon nominal slip are readily and simply de- 
termined from the curves of thrust T in pounds and torque Q in 
pound-feet. For if p denote pitch in feet, R revolutions per min- 
ute and s the slip, speed of advance is p (i s) R, and useful work 
done in a minute = TpR (i s). The gross work, or work de- 
livered to the model, is Q X 2 irR. 

Now efficiency = (Useful Work) H- (Gross Work) = T P R ^ ~ s "> 

2 (JirK 

Tp(i -s} 

, Q ** 

Note that the quantity p (i s) is the advance of the screw 


for one revolution, and its value is the same whether nominal or 
virtual pitch is used, the slip in each case being that appropriate 
to the pitch. Since at the Model Basin the curves of T and Q are 
plotted upon pi, the revolutions per 64-foot interval, it is conven- 
ient to use this in the efficiency formula. 

We have pi = -. ' r Substituting and reducing, we have finally 
p (i - s) 

_ 10.186 T 

The values of T, Q and p being taken off for the values of pi for 
the various slips, as indicated in Fig. 176, the efficiencies are 
readily calculated and plotted on slip. 

4. Characteristic Coefficients. The next question is as to the 
curves of coefficients which will completely characterize the pro- 
peller. Various coefficients may be used. Papers by the author 
and Messrs. Curtis and Hewins of the Model Basin staff before 
the Society of Naval Architects and Marine Engineers give vari- 
ous forms of coefficients, but it is believed that those given below 
are simple and convenient. 

We have to deal with the power absorbed or propeller power P, 
the useful or net power E, the speed of advance in knots V, the 
revolutions per minute R, the slip and the size. 

Whatever formula we use we are led to the same type of ex- 
pression connecting power absorbed, speed of advance and diame- 
ter. Thus using Rankine's formula, 

2 7T 

Q = 

Gross power P = = - 1 (p z RW(ys - 6s 2 ) + fp*R 3 pdZ). 

33000 33000 

Now pR^ 1 ^^-. Letf-4. pd=- 

i s m m 

The P - 

J- I1CI1 f 

, \7v./ / \^vx 

(i-s) 3 X330oo m (i -s) 3 X 33000 J 
Using either Froude's or Greenhill's formula we are led to the 
same expression except that the Froude theory formula will have 
the first term in the parentheses divided by m and the Greenhill 


theory formula will have it multiplied by m 2 . In either case we 

d 2 V s A 

may write P = . where A is a coefficient independent of the 


size and speed of the screw but varying with the slip and depend- 
ent upon shape, proportions, etc. The divisor 1000 is introduced 
simply in order to give A a greater value than unity in practical 
cases. Otherwise A would be inconveniently small. Evidently 
then a curve of A plotted on slip will completely characterize a 
screw as regards the important question of its capacity to absorb 

If E denote the useful or effective horse-power delivered by the 


screw, we have E = eP = (PV 3 -- 


Let us denote eA by B. Then curves of e, A and B plotted 
upon slip will completely characterize the action of a propeller of 
given features independently of size and speed. 

We have already seen how to determine e from the curves of 
Q and T. These curves are for a fixed diameter and speed of 

advance and at any given point P - * 




33OOO p (l S) IOOO 

Whence A = 2^X101.33 

33000 p(i -s) 

From the experimental results for a model propeller for a given 
value of s we know everything on the right-hand side of the equa- 
tion and hence can determine A without difficulty. Similarly, it 
will be found that we may derive B from the thrust 7\ 

,, TV X 101.33 <PV 3 B 
n, = > 

33000 1000 

looo T X 101.33 
whence B = - ^in ' 

33000 tP V 

Then curves of A, B and e completely characterize a propeller. 
As a matter of fact any one of them can be derived from the 


other two. They are all functions of slip and proportions and 
characteristics of the propeller and independent of size and speed. 
Table XI shows the calculations necessary to determine curves of 
A, B and e from the experimental data recorded as in Fig. 176. 

Fig. 179 shows four curves of A, B and e as deduced from the 
results of model experiments for four propellers of the same nomi- 
nal pitch ratio 1.2 and mean width ratio .2 and of the different 
blade thickness fractions indicated. The curves are plotted upon 
nominal slip and show that for this blade width and pitch ratio 
efficiency increases as the blade thickness is reduced, but the power 
absorption coefficient A and the thrust coefficient B decrease as 
thickness decreases. 

5. Application of Curves of Coefficients from Model Propellers. 
- Curves of B are particularly valuable in estimating from model 
results the probable performance of propellers of ships. If there 
were no reactions between ship and propeller, that is, if the ship 
were a "phantom ship" as Froude calls it, which offers resistance 
the same as the actual resistance without disturbing the water or 
modifying the action of the propeller, the case would be very 

For the ship we would know from model experiments the E.H.P. 
at any speed V and would also know the diameter d of the pro- 

peller. Then B = 3 is known for any speed from consid- 

eration of the ship. But from the propeller model experiments 
we have a curve of B plotted on slip. So having determined B 

IOI "^ ^ if 

for a speed V we know what the slip must be. But R = ' oc> ; 

p (i - s) 

hence we know what the revolutions must be. Finally from the 
slip determined by B we may determine e and A corresponding. 

We can then determine the power P absorbed by the screw by 
either one of the two formulae. We have 


We shall see later that the case of the actual ship is not so simple 
as that of the phantom ship, but curves of revolutions and horse- 


power deduced entirely from model experiments for phantom ships 
agree surprisingly well in many instances with the actual curves 
determined by trial of the full-sized ships. 

The author has encountered cases where curves of revolutions 
and speed obtained from trials of full-sized ships presented fea- 
tures which appeared at first sight abnormal but were found to be 
duplicated almost exactly by the estimated curves of revolutions 
and speed deduced entirely from experiments made independently 
with models of ship and propeller. On the other hand, when the 
full-sized propeller shows cavitation, the curves deduced from 
model results differ materially from the actual curves, a fact 
which in some cases permits of the determination with a good 
deal of accuracy of the point where cavitation becomes serious. 

6. Methods of Plotting Information for Design Work. The 
preceding analysis and method of plotting results of model experi- 
ments is not very convenient when we come to design work. The 
designer of a propeller knows in advance or can estimate with 
reasonable accuracy the power P which the propeller is to absorb 
and the speed of advance of the propeller through the water VA. 
He either knows the revolutions R which are to be used, or, sup- 
posing the revolutions may be varied through a certain range, 
wishes to ascertain the effect of such variation upon his design. 
He then has to determine diameter, pitch, blade area, blade thick- 
ness and blade shape. 

It is evident, then, that in plotting model experiments for use in 
design it would not be advisable to plot them upon slip, because 
this is not a quantity that is known or can be closely approximated 
in advance. It is desirable to use variables independent of size 
but involving power, speed and revolutions, etc. There are many 
such expressions. For practical applications the following will be 
found convenient: 

p = 

where d is diameter in feet. The quantity p is practically the 
same as an expression suggested by Mr. R. E. Froude in discussing 
a paper by Barnaby before the Institution of Civil Engineers, 


May 6, 1890 (Vol. CII, p. 101). In discussing the same paper 

R Z P 

Mr. C. Humphrey Wingfield suggested the use of r^r 

The quantities p and 8 may be readily connected with the 
coefficients already used in analysis of propeller experiments or 

can be deduced directly from the model propeller results. 


Thus we have seen that P = d?V 3 and we know that 


-s) 2 = (ioii) 2 F 2 . 
Multiplying the two together, 


Whence ^- = (- ) -^- ^^ = io. 2 68[- ' A 

\pl (i -5) 2 looo \pl (i -5) 2 


Whence p = R y/ =3.204 

p i s 

The right-hand expression for p is independent of size of propeller, 
and values of p are correctly calculated from a curve of A plotted 
on s. Usually, however, it is just as convenient to calculate them 
from the curves of torque, etc., of the model propeller. It will be 
found that we may write 


Similarly, we may write 

(PV A ) 

* (4)* V^(i - 

Table XII shows calculations of values of p and 8 for one of the 
model propellers whose results are plotted in Fig. 179. 

Figure 180 shows for the four propellers of Fig. 179 curves of 
efficiency and of 8 plotted on p. The calculations it is seen are 
made for various values of s, and on the curves of 8 the spots 
corresponding to various values of s are indicated. The scale 
used for p is a variable one, the abscissa values being proportional 
to Vp instead of p directly. This is a convenient device for spac- 


ing widely the values of p which are the most important without 
extending the p scale unduly. 

The application of curves such as those in Fig. 180 to design 
work is very simple. 

Thus, suppose a propeller is to be designed to absorb 10,000 
horse-power with a speed of advance of 20 knots and to have 200 



V A 3200000 320 


3200000 320 V Vjf 17.888 

From Fig. 1 80 for p = 11.18 the value of 5 for the various blade 
thickness ratios varies from 54.8 to 57.6, the corresponding values 
of diameter varying from 12.25 to 12.88. 

It is seen, however, that the efficiency is low, only about .66, and 
the slip high. Evidently the pitch ratio of 1.2 is not adapted to 
the case and should not be used. But suppose the revolutions 
desired had been 100. Then we would have 

p = 5.59, d = .3551 8. 

For this value of p we have good efficiency, and if the law of 
comparison holds we would get good results from a propeller of 
pitch ratio 1.2. For p = 5.59 the values of 8 range from 54.2 to 
58.2 and of d from 19.25 to 20.66. In practice we would choose 
a value of d corresponding to a blade thickness fraction, then 
determine the actual blade thickness necessary, and if the result- 
ing blade thickness fraction differed much from that first esti- 
mated, a second approximation would be made using the correct 
blade thickness fractions. 

25. Propeller Features Influencing Action and Efficiency 

A number of experiments have been made with 1 6-inch model 
propellers at the United States Model Basin. Many of the results 
obtained were published in the Transactions of the Society of 
Naval Architects and Marine Engineers for 1904, 1905 and 1906. 

These results and others not published enable some conclusions 
to be drawn positively as regards 1 6-inch propellers and with con- 


fidence as regards propellers of ordinary sizes within the limits 
where the Law of Comparison is applicable. 

i . Number of Blades. There were tried a number of pro- 
pellers with blades identical but differing in number from two 
to six. It was found that efficiency was inversely as the number 
of blades; that is, a propeller with two blades was more efficient 
than a propeller with three identical blades, that one with three 
blades was more efficient than one with four identical blades and 
that one with four blades was more efficient than one with six 
identical blades. 

Also while total thrust and torque increase as number of blades 
is increased, the thrust and torque per blade fall off. A three- 
bladed propeller at a given slip does not show 50 per cent more 
thrust and torque than a two-bladed propeller with identical 
blades. Fig. 181 shows approximately for working slips the rela- 
tive efficiencies and coefficients for 2-, 3- and 4-bladed propellers 
identical except as to the number of blades. The curves are 
curves of ratios of the quantities concerned, those for 3 blades 
being taken as unity in each case. As we have seen : 

. _ loco P R _ i OOP E 


where d is diameter in feet, V is speed of advance in knots and P 
and E power absorbed and effective power. The subscripts refer 
to the number of blades, A^, for instance, denoting the value of 
A for 4-bladed propellers. It is seen that the power absorbed, 
depending upon the coefficient A varies more nearly as the number 
of blades than the useful horse-power depending upon the coeffi- 
cient B. The 2-bladed propeller shows slightly greater efficiency 
than the 3-bladed, and the 4-bladed distinctly less. It should be 
remembered that Fig. 181 refers to propellers working under 
identical conditions of slip, speed of advance, etc. This means 
that a 4-bladed propeller will absorb about 30 per cent more power 
than a 3-bladed and a 2-bladed propeller about 15 per cent less. 

In practice the question to be decided is whether to use a 
4-bladed or a 3-bladed propeller when the same power is to be 
absorbed. In this case the 4-bladed propeller would be smaller 


than the 3-bladed and hence might have a pitch ratio more favor- 
able to efficiency than the pitch ratio of the corresponding 3-bladed 
propeller. So the question of 3- or 4-bladed propellers would re- 
quire investigation in each case. The methods to be used will be 
considered later. 

2. Outline or Shape of Blades. The question of shape or 
outline of blade faces has been given much attention in connection 
with propeller designs and in some cases extravagant claims have 
been made for special shapes. 

Fig, 182 shows five blade shapes which were experimented with 
at the United States Model Basin. Blade thickness fraction was 
constant in each case, being .0575. Three pitch ratios were used, 
.8, i.o and 1.2. 

The results were quite consistent and showed that the blades 
with broad tips absorbed more power and gave more thrust but 
with slightly less efficiency. While the very pointed blades showed 
up slightly the best, there is some reason to doubt whether they 
would retain their superiority which was not very marked in 
full-sized propellers. The experiments justify us in looking with 
doubt upon claims for great gain of efficiency by reason of some 
special shape of blade, and appear to indicate that for all-round 
work the old well-known elliptical shape is probably as good as 
any, though it may be that some other oval shape may be found 
slightly better. On the other hand the conclusion seems warranted 
that if circumstances render some special shape desirable, it can 
be used without serious loss of efficiency provided it is not alto- 
gether abnormal. 

3. Rake of Blades. It is "a very common practice to rake or 
incline the blades of a propeller aft. Sometimes they are inclined 
forward. At the United States Model Basin, six propellers, all 
of .2 mean width ratio and .0425 blade thickness ratio, were tested. 
Three were of .6 pitch ratio and three of 1.2 pitch ratio. Of each 
trio, one had the blades inclined 10 aft, one had the blades set 
normal to the shaft and one had the blades inclined 10 forward. 
The diameter was 16 inches in each case. Fig. 183 shows radial 
sections of the blades. The experiments gave almost identical 
results, the difference of torque, thrust, and efficiency being 


slight. So far as efficiency goes, then, there seems no reason to 
rake the blades of propellers. The advantage sometimes claimed 
for blades raking aft is that they prevent a supposed centrifugal 
motion of the water. Careful investigation of 1 6-inch propellers 
on test failed to show any evidences of centrifugal action except 
for some models of very thick blades and coarse pitch tested at 
3 knots speed of advance with a slip of 75 or 80 per cent. These 
models were practically standing still and seemed to throw the 
water out under the conditions described. Numerous experiments 
with 1 6-inch propellers under normal conditions showed the propeller 
race to be practically cylindrical and that so far from there being 
centrifugal motion, there is a slight convergence abaft the propeller. 

There is little doubt that the advantages of rake as regards pre- 
vention of centrifugal motion are imaginary. 

A real advantage of rake in practice is that the blade tips of 
side screws are thereby given greater clearance from hulls of usual 
form than if the blades were radial or with the same blade clear- 
ance strut arms are shorter. A very real disadvantage is the 
increase of stresses in the blades because of centrifugal action. 
This will be discussed later. It is a serious matter for quick run- 
ning screws, and for such screws at least blades should never rake. 

4. Size of Hub. One of the features of the Griffith screw 
introduced some fifty years ago was a large hub sometimes 
with diameter a third that of the propeller. These screws were 
often very successful, and as a result of practical experience there 
have for many years been advocates of large hubs. Experiments 
with model propellers at the United States Model Basin have 
shown that large hubs are distinctly prejudicial to efficiency. 

Full-scale experiments with turbine vessels seem to have shown 
the same thing, material gains in speed having been reported 
after substituting solid propellers with small hubs for propellers 
with large hubs and detachable blades. The argument against 
the large hub is very simple. When a large spherical hub is mov- 
ing through the water there must be a strong stream line action 
abreast its center, the water flowing aft. Hence the inner por- 
tion of the blades must be working in a negative wake produced 
by the hub a condition prejudicial to efficiency. 


It is sometimes argued that with a small hub the inner portion 
of the blades offer more resistance than if they were suppressed 
and a large hub fitted. 

This is probably not true, especially when we consider that the 
large hub appreciably increases the vessel's resistance. But even 
if it were true, the prejudicial effect of the large hub upon the 
blade outside of it would be enough to turn the scale against it. 

With slow-running screws of coarse pitch the large hub, while 
prejudicial to efficiency, will not affect it seriously; but for screws 
of such fine pitch as usually fitted in turbine work the inner parts 
of the blades do relatively more work and are relatively more 
efficient than in the coarse screws. Hence, reduction of the work 
done by them and of their efficiency through a negative wake set 
up by a large hub is likely seriously to reduce the efficiency of the 
screw as a whole. 

5. Standard Series of Model Propellers. We have now con- 
sidered the minor factors affecting propeller operation and effi- 
ciency and will pass to major factors. These are pitch ratio, blade 
area, blade thickness and slip. In considering resistance of ships 
the major factors of residuary resistance were investigated by 
means of a standard series of models whose variations covered 
the useful range of the major factors concerned. Similarly, the 
field has been covered for propellers by a standard series of models 
of varying pitch ratio, mean width ratio, and blade thickness frac- 
tion. They were all 3-bladed propellers 16 inches in diameter, 
with blades that were elliptical in developed outline. The hubs 
were cylindrical and 35 inches in diameter, being practically .2 of 
the propeller diameter. Six pitch ratios were used namely, .6, 
.8, i.o, 1.2, 1.5 and 2.0. For each pitch ratio five blade areas were 
used. Fig. 184 shows the developed areas of the five blade faces. 
Their mean width ratios, as shown, were .15, .20, .25, .30 and .35. 
Six pitch ratios and five mean width ratios resulted in 30 propellers. 
These were made true screws with ogival blade sections, the backs 
being circular arcs, and with extra thick blades. 

After being tested, the thickness was reduced by taking metal 
off the back to form new ogival sections, the face being untouched, 
and thus new propellers with the same faces as before, but thinner 


blades, were made. These were tested as before. This process 
was repeated twice, so that each blade was tested in four thick- 
nesses, being finally unusually thin. This made 120 propellers 
tested in all. Table XIII gives their data. The original pro- 
pellers are numbered i to 30 and the successive cuts denoted by 
the letters A, B and C. Great care was taken when reducing 
thickness not to change the face, and toward the edges the recut 
blades were probably a shade thicker than true ogival sections. 

It is difficult to make model propeller experiments with minute 
accuracy, but in this case, owing to the number of propellers tried 
and the number of independent variables involved, irregular ex- 
perimental errors could be practically eliminated by cross fairing 
on pitch ratio, mean width ratio and blade thickness fraction. 

Figures 185 to 208 show the experimental results after this was 
done in the form of curves of thrust in pounds, torque in pound 
feet and efficiency. All refer to a 5-knot speed of advance. The 
results are plotted upon nominal slip as being most convenient for 
practical applications. 

The results of trials of these 120 propellers are worthy of the 
most careful study. We will now consider them briefly in con- 
nection with the influence of pitch ratio, blade area, blade thick- 
ness and slip upon thrust, torque and efficiency. 

6. Pitch Ratio. The effect of variation of pitch ratio is illus- 
trated in Fig. 209, which shows for propellers of .25 mean width 
ratio and .04 blade thickness fraction curves of maximum effi- 
ciency and of thrust and torque for 20 per cent slip. This 
figure is typical. It is seen that for constant slip and speed of 
advance, torque and thrust increase as pitch ratio decreases, the 
increase becoming more and more rapid as pitch ratio becomes 

The efficiency remains nearly constant over a fairly wide range 
of pitch ratio having its greatest value at a pitch ratio of about 
1.5. As pitch ratio decreases, however, efficiency begins to fall 
off, and below the value of unity the falling off is rapid. In prac- 
tice screws of fine pitch have frequently shown very low efficiency 
as a result of cavitation, but apart from this, screws of fine pitch, 
say below a pitch ratio of unity, are essentially less efficient than 


screws of pitch ratio 1.5 or so, and the smaller the pitch ratio the 
less the efficiency. 

7. Blade Thickness. When we study the influence of blade 
thickness we find that the thicker the blade the greater the thrust 
and torque for a given slip. This is perfectly natural when we 
reflect that the results are plotted upon nominal slip and that the 
thicker the blade the greater the virtual pitch. The effect of blade 
thickness upon efficiency is summarized in Fig. 210. It was found 
that for a given blade area the relative variations of efficiency with 
blade thickness were nearly the same for slips used in practice 
regardless of pitch ratio. Hence Fig. 210 shows for each blade 
width an average curve of relative efficiency plotted on thickness 
only; for each curve, unity corresponds to a different blade thick- 
ness fraction, the broad blades being thinner than the narrow 
blades. This is generally in accordance with what considerations 
of strength necessitate in practice. 

Figure 210 indicates that the efficiency of narrow blades increases 
rapidly as they are thinned, while for the broad blades thickness 
has little effect upon efficiency, and in fact the thicker blades 
seem slightly more efficient. When we remember that on ac- 
count of strength a narrow blade must be thicker than a broad 
blade the deduction from Fig. 210 is that practicable variations 
of blade thickness will have comparatively little effect upon 
efficiency. This conclusion, however, is from results of experi- 
ments where cavitation was not present, and it is generally agreed 
that to avoid cavitation propeller blades should be as thin as 

It is probable that in many cases if the blades are made too 
thick cavitation would reduce efficiency without the propeller 
actually breaking down, while it will be avoided altogether with 
thin blades. Hence we should make propeller blades reasonably 
thin in practice, in spite of Fig. 210. Where cavitation is likely 
they must be made thin. It may be remarked, however, that 
Fig. 210 appears to be in general accordance with facts of ex- 
perience with slow-running propellers. Coarse, heavy propellers of 
this type often give very good results in service in spite of thick 


8. Blade Areas. In the experiments with the standard series 
of propellers it was not practicable to investigate the question of 
blade area entirely apart from that of blade thickness. The broad 
blades were made thinner than the narrow ones, as would be the 
case with actual propellers in practice when it is a question be- 
tween a narrow-bladed propeller and a broad-bladed propeller to 
absorb the same power at the same revolutions and speed. 

It is owing to the greater thickness of the narrow blades, and 
hence their greater virtual pitch for a given nominal pitch, that in 
the fine pitches the narrow blades actually absorb more power and 
deliver more thrust for a given nominal slip than the broad blades. 
In the coarse pitches this is not the case for slips such as occur in 
practice, but the broad blades do very little more than the narrow 

Even after making allowances for the thickness effect it is evi- 
dent that the broad blades by no means absorb torque and deliver 
thrust in proportion to their areas. In fact the influence of blade 
area upon thrust and torque is surprisingly small. 

Considering efficiency it is seen that for propellers of pitch ratio 
usually found in practice the broad blades and the narrow blades 
are both less efficient than blades of medium width, say with a 
mean width ratio of .25 to .30. The differences are not great, 
however. It is interesting to note the superior efficiency of the 
narrow blades for the propellers of abnormally fine pitch. This, 
however, is not due to the fact that the blades are narrow, but to 
the fact that the narrow blades have greater virtual pitch ratio, 
and for the propellers of very fine pitch gain in virtual pitch ratio 
means gain in efficiency. 

The experiments with the standard series of model propellers 
warrant fully the broad conclusion that, when cavitation is absent, 
propellers may vary quite widely in pitch ratio (above 1.2 or so) 
and in area with little change in efficiency, provided diameter is 
such that they work at slips at or near that of maximum efficiency. 
This conclusion is fully borne out by experience, which has led 
many people to conclude that there was so little difference be- 
tween propellers that any propeller which allowed the engine to 
develop its power at the desired revolutions and showed a good 


slip was a good enough propeller. For low-speed work this is 
reasonably correct; for high-speed work, even leaving out of ques- 
tion cavitation, propellers which absorb the power at the desired 
revolutions are liable to vary seriously in efficiency, particularly if, 
as is usually the case, they must be of the fine pitch type. 

9. Slip. Figures 185 to 208 show that all curves of efficiency 
plotted upon slip present the same general appearance. Con- 
sidering nominal slip the efficiency is zero at a certain negative 
slip. The thicker and narrower the blade the greater in general 
the increase of virtual over nominal pitch, and the greater the 
numerical value of the negative slip corresponding to zero effi- 
ciency. It will be noted, however, that for the narrow blades of 
pitch above unity there seems to be a slight falling of! of virtual 
pitch with thickness beyond the A cut. This is probably due to 
the fact that as the thickness of these narrow blades is increased 
a point is reached where the water breaks away from the back, the 
latter losing its grip, as it were. The process is analogous to cavi- 
tation, though cavities are not formed. As the slip increases from 
that corresponding to zero efficiency, the efficiency rises very 
rapidly at first, then reaches a maximum and thereafter falls off. 
The nominal slip corresponding to maximum efficiency is nearly 
always between 15 and 20 per cent for blade thickness that 
would be used in practice, but slip can be increased to 25 per cent, 
and in some cases to 30 per cent, without serious loss of efficiency. 

But such an increase means an eno mous increase in thrust and 
torque. Hence a given propeller will vary widely its power and 
thrust without material change of efficiency. So it is not neces- 
sary in practice with propellers of coarse pitch, to aim very closely 
at some exact slip provided the propeller is so designed that under 
conditions of service its slip is not too small. A propeller which 
is too large, showing slip much below that for maximum efficiency, 
will be very inefficient. On the other hand, a propeller may be 
too small and work with slip a good deal greater than for maximum 
efficiency without much loss of efficiency. It should be remem- 
bered that the slips of Figs. 185 to 208 refer to propellers operating 
in undisturbed water, and the apparent slip of propellers attached 
to ships is usually less than the true slip. 


When dealing with propellers of fine pitch ratio, say in the 
neighborhood of unity, the question of efficiency as affected by 
slip is complicated by the question of efficiency as affected by 
pitch ratio. Thus in Fig. 190 we see that propeller No. 8, A cut, 
of .25 mean width ratio and .8 pitch ratio has a maximum efficiency 
of .632 at 15 per cent slip. From Fig. 194, propeller No. 13, A 
cut, of .25 mean width ratio and i.o pitch ratio has a maximum 
efficiency of .684 at 14 per cent slip and an efficiency of .632 at 
about 31 per cent slip. In a given case, then, where we could fit a 
propeller of the proportions of No. 8 working at maximum effi- 
ciency, we could make an improvement if we could fit a propeller 
of the proportions of No. 13 working below its maximum efficiency 
provided its slip did not exceed 30 per cent. This is a question of 
very considerable practical importance. In the next section will 
be given methods for determining the best combinations of pitch 
ratio and slip for given conditions. 

26. Practical Coefficients and Constants for Full-sized Pro- 
pellers Derived from Model Experiments. 

i. General Line to be Followed in Reducing Model Results. - 
The results of the model experiments for the standard elliptical 
3-bladed series will of course be of value in the case of any pro- 
peller design. It should be carefully remembered, however, that 
they cannot be applied blindly. We have determined experi- 
mentally the thrust and torque and deduced the efficiency of a 
number of small propellers at a 5-knot speed of advance through- 
out the range of slip likely to be found in practice. These small 
propellers covered for 3-bladed elliptical propellers the range of 
pitch ratio, mean width ratio, and blade thickness fraction likely 
to be found in practice. We know that so long as cavitation does 
not appear the Law of Comparison will apply satisfactorily and 
that the results of the model experiments will apply to full-sized 
propellers working under the same conditions as the models. But 
in applying the results we must remember that they do not 
hold for cavitating conditions, which will presently be considered 


The models were tested in such a manner as to be practically 
free from hull influence, and we know that for full-sized propellers 
driving ships there are material mutual reactions between pro- 
peller and ship. The question arises whether we shall attempt to 
take account of these reactions in reducing the model results or 
consider them separately. 

It is much better, and even simpler in the end, to attack the 
problem in detail. 

2. Reduction of Model Results. We have seen that by means 
of a p8 diagram, as in Fig. 180, the experimental model results may 
be reduced to a form convenient for practical applications. But 
if we simply construct a p8 diagram for each model tested it will 
be a very laborious process to locate and utilize the particular 
diagram adapted to a particular case. So it is necessary to de- 
velop diagrams, by interpolation if necessary, such that the pri- 
mary factors involved are readily determined. We have to deal 
with efficiency, diameter, pitch ratio, mean width ratio and blade 
thickness fraction. 

These are too many variables to be covered directly on a single 
diagram. The first three are the most important. Width and 
blade thickness are not independent in practice. To do a given 
work at given revolutions the narrow blade must be thicker than 
the wide blade. So four p5 diagrams, Figs. 211 to 214, have been 
constructed from the model results of Figs. 185 to 208. Figure 211 
refers to blades having a mean width ratio of .20 and a blade 
thickness fraction of .06. Similarly Figs. 212, 213 and 214 refer 
respectively to mean width ratios of .25, .30 and .35 and blade 
thickness fractions of .05, .04 and .03. We shall see later how to 
make slight changes involved by other blade thickness fractions. 

The application of the p8 diagrams is very simple : 


' = 

where P is the power absorbed by the propeller of diameter d 
feet at R revolutions per minute when advancing at a speed of VA 

Then p is the primary variable fixed by the conditions of the 
problem. Contours of 5 are plotted above p for equal intervals 


of pitch ratio and curves of efficiency for the same intervals. 
When p is known we can determine very promptly for any 
value of 8 the pitch ratio and efficiency. In addition to the con- 
tours of d above p contours of slip are plotted in dotted lines. 

3. Maximum Efficiency. The efficiency curves show many in- 
teresting and significant features. For a short interval each pitch 
ratio shows an efficiency greater than any other, and evidently if 
our choice is free we should for a given value of p use the pitch 
ratio corresponding to optimum efficiency. Hence, there is drawn 
an enveloping curve of maximum efficiency touching the suc- 
cessive efficiency lines for the various pitch ratios which has upon 
it a scale of the pitch ratios for maximum efficiency. 

In this connection attention may be called to the fact that the 
portion of each efficiency curve which gives the best efficiency for 
a given p is in general of an efficiency below the maximum efficiency 
attainable with the pitch ratio. This is particularly noticeable for 
the largest values of p. For all values of p above very small ones 
it is better to use a propeller of relatively coarse pitch and work 
it at a fairly high slip greater than that corresponding to its 
maximum efficiency than to use a propeller of finer pitch and 
work it at its maximum efficiency. This for the reason that for 
propellers of pitch usual in practice decrease of pitch means fall- 
ing off in efficiency. 

The p8 diagrams bring out clearly some of the basic conditions 
affecting propeller design. 

Once we fix for a propeller the power, P, it is to absorb, its revo- 
lutions per minute, R, and its speed of advance, VA, the value of p 
is fixed. Now it is apparent from the diagrams that for a given 
value of p there is a maximum efficiency beyond which we cannot 
go. We may very easily fall short of it, but even if we adopt the 
very best combination of diameter, pitch and blade area possible, 
we cannot get beyond a limiting efficiency. The p8 diagrams of 
Figs. 211 to 214 were deduced from experiments with models of 
3-bladed propellers with elliptical blades having ogival sections. 
Hence the limiting efficiencies shown in them are not exactly the 
same as for all types of propellers, though they are about as high 
as for any known type. But there is no doubt that they indi- 


cate well the general variation of efficiency with p for all types of 
propellers in present use. While there is a maximum efficiency, 
about p = 3, and the efficiency falls off on either side, the values of 
p that are found in practice are almost never materially below 3, so 
that in practice the larger the p the smaller the limiting efficiency. 
It is the high value of p produced, if we give low-speed vessels high 
revolutions, that has hitherto prevented the application to cargo 
vessels of turbines directly connected to the propeller. Thus, sup- 
pose we had a destroyer propeller absorbing 5000 shaft horse- 
power at 800 revolutions with a speed of advance of 30 knots. 
For this case 

The limiting efficiency for this value of p is about .65 which though 
low is not impossible. If now we had a large single-screw cargo 
and passenger vessel which required 5000 shaft horse-power to 
make 15 knots speed of advance and adhered to 800 revolutions 
per minute the value of p would be 

800 V cooo 

-- - 64.9. 

For this value of p the limiting efficiency would be inadmissibly 
low. To hold p at 11.5 the revolutions would have to be reduced 
to 142 which would make an inefficient turbine. An alternative is 
to hold revolutions at 800 and use multiple shafts. But in order to 
make the p value for each propeller 11.5 only, it would be neces- 
sary to divide the 5000 shaft horse-power between 32 shafts, which 
is of course impossible. 

Another fact of serious practical importance which the p8 dia- 
grams bring out is that there is practically a lower limit to the 
pitch ratio which can be used to advantage. At first the best 
pitch ratio falls off rapidly with increase of p, but for large values 
of p the pitch ratio falls off more and more slowly, and for no value 
of p which it would be advisable to use in practice is it desirable to 
go below a pitch ratio of .9 or a little less. 

The slip for the best all-round efficiency which is below .15 for 
small values of p increases steadily, until it is seen that propellers 


of a pitch ratio of .9 should be worked at over .30 slip. This is 
real slip, not apparent slip. 

It is interesting to note in this connection that the model ex- 
periments indicate that the broader the blades the greater the 
slip for the best results. Thus for a pitch ratio of i.o and the four 
blade width ratios of .20, .25, .30 and .35 the best slips are respec- 
tively .255, .265, .280 and .320. This is in accord with theoretical 

4. Methods of Calculations. In order to facilitate the calcula- 
tion of p in a given case there are given in Table XIV values of 
V A . 

It should be carefully borne in mind that VA is not the speed of 
the ship through the water but the speed of advance of the pro- 
peller through the disturbed water in which it works. The differ- 
ence between VA and V, the speed of the ship, will be considered 
in connection with the wake factor. 

The formula for 5 is 

or, when 8 has been determined, 

With a table of squares and cubes we can readily determine (PV A ) k 
by taking the square root of the cube root of PV 'A', R* is simply 
the square of the cube root of R. Hence the calculations re- 
quired in connection with the use of the p8 diagrams are readily 

5. Blade Thickness Correction. The four p5 diagrams for the 
standard series refer to a definite blade thickness fraction for each 
mean width ratio. We have seen in Fig. 210 the effect upon the 
efficiency of the standard series of variations of the blade thick- 
ness. This effect is not large enough to be of practical importance 
in most cases. But variation of blade thickness will also neces- 
sarily affect pitch ratio and diameter. Investigation shows, how- 
ever, that the effect is not large, and for blade width ratios from 
.25 to .35, and for propellers of about the proportions for maxi- 
mum efficiency, the average corrections required are shown in 


Fig. 215. The curves of this figure give for various values of p the 
percentages by which diameters and pitches determined from the 
p8 diagrams must be modified when the standard blade thickness 
fractions to which the pd diagrams correspond are departed from. 

The corrections are small and in practice may often be ignored. 
The standard p8 diagrams already take some account of thickness, 
the widest blades being only half as thick as the narrowest, but of 
course the actual blade thickness fraction in a given case is fixed 
mainly by considerations of strength. 

6. Four-bladed Propellers. The standard pd diagrams, Figs. 211 
to 214, refer to three-bladed propellers. It would be desirable to 
have similar diagrams from full experiments with four-bladed pro- 
pellers, but lacking such they can be used with fair approximation 
for four-bladed propellers. We have in Fig. 181 the relation be- 
tween power absorbed, thrust and efficiency of three and four- 
bladed propellers as deduced by analysis of experiments at the 
model basin with propellers having quite thin blades of rather 
broad tips. These may be taken as applying with reasonable 
approximation to the elliptical blades. 

Then the steps in a given case will be as follows: 

1. Determine p in the ordinary way and then divide it by the 
square root of the ratio between the coefficient A for a four-bladed 
screw and for a three-bladed screw these ratios are given in 
Fig. 181. Call the quotient P4. 

2. Using p 4 , determine by the use of the proper p8 diagram the 
proper diameter, pitch, etc., for a three-bladed propeller. 

Then upon adding a fourth identical blade to the three-bladed 
propeller we shall have a four-bladed propeller which will meet 
the conditions. 

For let P, R, and V A denote power to be absorbed, revolutions 
to be made and speed of advance. 

We have 

P = 

then P4 = 


where r. is the ratio of the A coefficients from Fig. 181. Then 
a three-bladed propeller based upon p 4 will, at revolutions R and 


speed of advance VA, absorb a power - But from Fig. 181 again 


a four-bladed propeller identical as to diameter, pitch and blades 


will absorb r times the power of the three-bladed one, or X r = P. 


Hence the four-bladed propeller will absorb the power P at revo- 
lutions R and speed of advance VA- The relative efficiencies 
may be obtained from Fig. 181. 

Since once we know p, we can determine the relative diameters 
of the three and four-bladed propellers; we can from each p8 dia- 
gram for three-bladed propellers determine, by using Fig. 181 as 
explained above, a figure giving ratios of diameter, pitch and 
efficiency for three and four-bladed propellers. It is found, how- 
ever, that as regards diameter and pitch the ratios are so nearly 
the same for all widths that the results may be averaged in a simple 
diagram (Fig. 216). 

This gives curves of coefficients by which the diameter and 
pitch of -a three-bladed propeller must be multiplied to determine 
the diameter and pitch of a four-bladed propeller of the same type 
of blades and mean width ratio that at the same revolutions and 
speed of advance will absorb the same power. 

Efficiency coefficients are also given. These are seen to be all 
less than unity, indicating a loss of efficiency by adopting four- 
bladed instead of three-bladed screws. 

The pitch coefficient is less than unity throughout, so the 
pitch of the four-bladed screw will be slightly less than that of 
the three-bladed screw, but the diameter is reduced more than the 
pitch, so that the pitch ratio of the four-bladed screw will be the 
greater. The diameter coefficient in Fig. 216 should be regarded 
as an upper limit. It will be feasible in practice to reduce the 
diameter of the four-bladed screw four or five per cent more with- 
out material loss of efficiency. 

7. Two-bladed Propellers. It is evident that the methods above 
may be utilized in order to apply the p8 diagrams for the three- 
bladed propellers to two-bladed propellers. 


In this case, however, the artificial value of p will be greater than 
the original value. 

Fig. 217 gives curves of coefficients, etc. It is seen that diame- 
ter, pitch and efficiency are all increased. The gain in efficiency is 
small, however, and there are practical objections to two-bladed 
propellers, so that their use is seldom expedient. This point will 
be discussed further in considering design of propellers. 

27. Cavitation 

i. Nature of Cavitation. The phenomenon known as cavita- 
tion has been given much attention of late years in connection with 
quick-running turbine-driven propellers. It appears to have been 
first identified upon the trials in 1894 of the torpedo boat destroyer 
Daring which had reciprocating engines. When driven at full 
power with the original screws this vessel showed very serious 
vibration evidently due to some irregular screw action. The pro- 
pulsive efficiency was poor, the maximum speed obtained being 
24 knots for 3700 I.H.P. and 384 revolutions per minute. 

Mr. Sidney W. Barnaby, the engineer of the Thorneycrofts, who 
built the Daring, came to the conclusion that at the high thrust 
per square inch at which the screws were working the water was 
unable to follow up the screw blades and that " the bad perform- 
ance of the screws was due to the formation of cavities in the 
water forward of the screw, which cavities would probably be 
filled with air and water vapor." So Mr. Barnaby gave the 
phenomenon the name of cavitation. The screws which gave the 
poor results had 6 feet 2 inches diameter. 8 feet 7! inches pitch and 
8.9 square feet blade area. Various alternative screws were tried, 
and the trouble was cured by the use of screws of 6 feet 2 inches 
diameter, 8 feet n inches pitch and 12.9 square feet blade area. 
With these screws 24 knots was attained with 3050 I.H.P. and the 
maximum speed rose from 24 knots to over 29 knots. 

For the Daring cavitation appeared to begin when the screw 
area was such that the thrust per square inch of projected area was 
a little over u pounds per square inch. For a time it was thought 
that the thrust per square inch of projected area was a satisfactory 
criterion in connection with cavitation and that the limiting 


thrust per square inch of projected area found on the Daring was 
generally applicable. 

This, however, is not the case. Greater thrusts have been suc- 
cessfully used and cavitation is liable to appear at much lower 
thrusts. In one case within the author's experience cavitation 
appeared when the thrust was about 5 pounds per square inch of 
projected area, the tip speed being about 5000 feet per minute, 
and in another when it was about 7.5 pounds, the tip speed being 
about 650x3 feet per minute. There is little doubt that the prime 
factors involved in cavitation are: (i) the speed of the blade 
through the water, which is conveniently measured by the tip 
speed, and (2) the shape of the blade section. 

2. Accepted Theory of Cavitation Inadequate. When we at- 
tempt to explain just how or why vacuous cavities at the backs of 
screw blades cause the serious loss of efficiency associated with cavi- 
tation we encounter insuperable difficulties. Suppose, for instance, 
the cavity is a vacuum and covers the whole blade back. Then the 
thrust per square inch of projected area due to the vacuum on the 
blade back would be between 14 and 15 pounds and the thrust due 
to the face would be added to that. As cavitation will appear in 
some cases at thrusts per square inch of projected area as low as 4 
pounds, it is evident that in such cases there cannot be a vacuum 
over the whole blade back and thrust in addition on the face. 

But suppose the blade had a vacuum over a portion of the back 
only. There would be no increase of thrust from additional suction 
of that portion of the blade back, but neither would there be any 
increase of torque due to that portion of the blade back. The only 
loss of efficiency would be a small amount due to the propeller 
working with a slightly higher slip, while the loss of efficiency 
accompanying cavitation is very much greater than this. 

Fig. 218 shows a propeller blade section advancing through the 
water at an angle of slip of 3 degrees not an unusual angle. 
There are three regions indicated: 

1. The leading portion of the back, denoted by A. 

2. The following portion of the back, denoted by B. 

3. The face, denoted by C. 

It does not appear possible that cavities can form at A. This 


portion of the section contributes negative thrust, and although the 
point of demarcation between the portion of the back contributing 
negative thrust and the portion contributing positive thrust (suc- 
tion) probably varies in position with speed through the water and 
slip angle, it appears reasonably certain that A always contributes 
negative thrust and quite probable that this negative thrust in- 
creases indefinitely with the speed. 

Qver B a cavity will form when the speed is high enough. It 
will probably be small at first, and as the speed is increased, cover 
a greater and greater portion of the section back. It cannot cover 
the whole back, however, because it cannot extend over A to the 
leading edge. 

As regards C it has been generally assumed that the thrust from 
the face always increases with increase of speed of the section 
through the water. 

3. Possible Theories of Cavitation. Now how is it conceivable 
that cavitation can cause a rather sudden loss of efficiency when 
the section is pushed to a sufficiently high speed ? 

A. It is possible that when a vacuum is formed at B this portion 
of the blade contributes no more suction or thrust while the nega- 
tive thrust at A continues to increase with resulting loss of efficiency. 
This explanation would seem to involve the further assumption that 
by far the major portion of the thrust of a propeller is due to the 
suction of the blade back. 

B. It is possible that when a vacuum is formed at B it is spoiled 
by air obtained from the surrounding water and the suction of the 
blade back is decreased. This explanation is possible only if, when 
the water still hugs the blade back, it sweeps away any air which 
is sucked out of the water, so that while the water is in contact with 
the back it is possible for the latter to exert a suction approaching 
that of a perfect vacuum. But when the water breaks away from 
the back, air leaking into the space is carried away by entrainment 
only from the rear of the cavity, where the water comes together 
again; and when the rate of entrainment is equal to the rate of leak- 
ing into the cavity there is a balance of pressure, and though there 
is a partial vacuum in the cavity the pressure is much greater than 
a complete vacuum. 


C. It is possible that when cavitation sets in the thrust from the 
blade face falls off absolutely or relatively. 

A, B and C above appear to cover the possible theories of the 
phenomena associated with cavitation. Whether cavitation is due 
to one or more of these explanations or to something different still, 
can be satisfactorily determined by experiment only, either on 
models or on full-sized propellers. 

4. Experimental Investigation of Cavitation. Experiments 
with cavitation using full-sized propellers have not hitherto been 
made, except inadvertently. While no theory of cavitation should 
be fully accepted until confirmed by full-sized experiments the ex- 
pense of a general investigation with large propellers has been 
hitherto prohibitive, to say nothing of the time required and the 
practical difficulties in the way. Small scale or model experiments 
on cavitation present special difficulties. For the law of com- 
parison to apply in spite of cavitation it would be necessary to have 
the pressure around the model in the ratio of the size to the pressure 
around the full-sized propeller. 

This requires the model to work in water whose surface is covered 
by a partial vacuum, or in hot water which has a vapor pressure 
partially neutralizing that of the air. 

The Hon. C. A. Parsons has done some work using the latter 
method, but little has been published of the results. There are 
great practical difficulties in making experiments along this line, 
except with very small models. 

A second possible method of investigating cavitation experimen- 
tally by means of models is to test the model, not at the corre- 
sponding speed, but at the actual speed of advance of the full-sized 
propeller. When this is done, the pressures per square inch at cor- 
responding points of propeller and model are the same, and if one 
shows cavitation so will the other. This method is hardly prac- 
ticable for the model of the propeller of a 33-knot destroyer, but 
for propellers of slow and moderate-speed vessels experiments 
could be made without serious difficulty or great expense, either 
in a model basin or from a special testing platform in front 
of a vessel. This method, however, has not been used in practice. 
For model propellers of any size, say 15 inches to 18 inches in 


diameter, it would require very powerful driving and measuring 

A third method is to use the propeller testing gear already in- 
stalled in a model basin with small propellers of such abnormal 
proportions and shape that they will show cavitation within the 
limits of speed and revolutions available. 

Some experiments along this line have been made at the United 
States Model Basin. 

To obtain pronounced cavitation from small propellers 12 inches 
to 1 6 inches in diameter, tested at speeds of advance not over 7 
knots or so, it is necessary to make the pitch ratio much smaller 
and the ratio of thickness to width of blade much larger than for 
the propellers used in practice. Sixteen-inch models representing 
propellers of ordinary proportions will not cavitate satisfactorily 
at low speeds of advance, and the experimental gear available was 
not powerful enough to drive them at high speeds. 

The results obtained with the fine pitch propellers appear, how- 
ever, to throw some light upon the subject under consideration. 

Figure 219 shows expanded blade outline and blade sections for a 
1 6-inch model propeller of 6.4-inch pitch. Figure 220 shows curves 
of thrust and torque for this propeller plotted upon slip for speeds 
of advance of 5, 6 and 7 knots. The major portion of Fig. 220 is 
from Fig. 10 of a paper by the author before the Society of Naval 
Architects and Marine Engineers in 1904, but the curves for the 
5-knot speed have been extended, and the curves for the propeller 
reversed have been added from the results of subsequent experi- 
ments. For the propeller reversed the nominal slip is figured from 
the nominal pitch of the back as tested (the face before reversal). 

Figure 220 shows conclusively that, so far as this propeller is con- 
cerned, the thrust per square inch of projected area has little to do 
with the cavitating point. At a nominal slip of 15 per cent there 
is evidently cavitation at the 7-knot speed. At this point the thrust 
is 80 pounds, or almost 4.3 pounds per square inch of projected area. 
At 5 knots, however, the thrust per square inch of projected area 
at which cavitation begins is about 9 pounds. 

Other conclusions might be drawn from Fig. 220, but more illu- 
mination can be obtained from the results of trials of a small pro- 


peller especially designed to show cavitation. This propeller was 
14 inches in diameter and of 4.2 inches pitch. Its developed blade 
outline and blade sections are shown in Fig. 221. At the points 
A, B, C, D and E small holes were made on each blade connecting 
to the shaft, which was hollow. The hole in the shaft communicated 
in turn with a pipe forward of the hub, which led finally to a tank 
under air pressure, there being a pressure gauge on the line and 
valves for turning on or cutting off the air pressure as desired. 
When making trials one hole only was left open in each blade. 
This apparatus measured suction or partial vacua with great facility 
but had to be handled carefully to measure pressure. When mea- 
suring suction, the air pressure was cut off, when the propeller itself 
would quickly exhaust the air and the amount of vacuum was read 
on the gauge. When measuring pressure, the air valve was barely 
cracked, so that a small quantity of air was dribbling out all the 
time through the hole where pressure was to be measured. 

In this way the passages in the propeller were kept clear of water, 
whose presence would have prevented obtaining the pressure at the 

A gauge pressure of a pound and a half or so was sufficient to keep 
the air passing out when the propeller was at rest or turning over 
very slowly, and the difference between this initial pressure and the 
gauge pressure shown while running was taken as pressure at the 

In the early part of a run for pressure the air would stop coming 
out of the propeller; it would accumulate in the pipe and the gauge 
pressure rise until air again began to come out and the gauge became 
steady. At the end of a run the instant the propeller began to slow 
down the air would burst forth. 

While the apparatus and methods described above for measuring 
pressure and suction are certainly not of minute accuracy, they gave 
consistent results which are believed to be reasonably accurate. 

For looking at the propeller under the test there was fitted a 
fixed disc with a small slot, and immediately behind it a revolving 
disc with a similar slot, which was driven at the same speed as the 
propeller. The propeller was illuminated by a searchlight and when 
looking through the slot in the fixed disc the propeller was seen once 


during each revolution always in the same position. The discs and 
searchlight could be shifted so that either back or face of the pro- 
peller could be observed. 

Figure 222 shows for the propeller of Fig. 221 and three knots 
speed of advance curves of thrust, torque and of pressure or suc- 
tion at the points indicated. The curves are plotted upon nominal 
slip and pressure and suction are measured in pounds per square 
inch. A scale showing tip speed is also given. 

Figure 223 gives the same data as Fig. 222 for five knots speed 
of advance. 

When watching the operation through the slotted discs any 
cavities present were plainly visible and it was easy to trace the 
development of cavitation. 

At about 3000 feet tip speed cavities appeared at the following 
portions of the back and the leading portions of the face. The 
cavities appeared first on the face, as might be expected from Figs. 
222 and 223, which show that the suction at A is always more 
intense than at D. 

The cavities first show themselves near the blade tips and creep 
in toward the center as speed is increased. 

In Figs. 222 and 223 the thrust has returned to zero, when the 
tip speed is between 5000 and 6000 feet per minute. When this 
is the case the cavities at the back of the blade extend in from the 
tip about two-thirds of the blade length and near the tip cover 
nearly two-thirds of the blade back. 

On the face under the same conditions the cavities extend along 
the leading edge practically in to the hub and near the tip from the 
leading edge to the following edge. 

5. Theory and Cause of Cavitation. From the experimental 
curves of Figs. 222 and 223 and observation of the cavities it is 
obvious that the cavities at the rear of the blade do no harm. It 
is the cavities on the driving face which grow rapidly as tip speed 
is increased, combined with the negative thrust of the leading por- 
tions of the blade back that stop the increase of thrust and then 
actually cause it to decrease to zero and below. 

These conclusions apply strictly to the 14-inch model propeller 
of somewhat abnormal type shown in Fig. 221, but it seems reason- . 


ably certain that they apply more generally, and that harmful cavi- 
tation is due not to cavities at the backs of propeller blades, but to 
cavities at their driving faces. 

When we seek a cause for these cavities, it seems fairly obvious. 
Fig. 218 shows the section of a propeller blade advancing with a 
slip angle of 3 degrees, which is not an exceptionally small angle, 
as is evident from Fig. 170. But the face C, advancing through 
the water at an angle of 3 degrees, is associated with the leading 
portions of the back, whose direction is such that it is advancing 
through the water at an angle of over 20 degrees. Fig. 63 shows 
diagrammatically the nature of the motion of water past a plane 
with a sharp edge. In the case of the propeller we have virtually 
two planes in association ; namely, the face and the leading portions 
of the back. Considering the face alone, the water tends to cascade 
around the leading edge from front to back. Considering the back 
alone, the water tends to cascade around the leading edge from 
back to front. Actin? In association, the back of the blade with 
an inclination of ;^er 20 degrees overpowers the face with an incli- 
nation of 3' degrees, and as a result the water cascades from the back 
of the blade to the face around the leading edge, causing first eddies 
and then cavities on the face of the blade. 

In regarding the leading portion of the propeller blade as made 
up of two planes, we should remember that the motion at each point 
is circular, not linear. A plane in linear motion can drag a good 
deal of dead water behind it, and water brought to rest relatively 
to the plane passes aft again without any motion across the plane. 
The propeller blade is moving in a circle and cannot carry water 
with it in the shape of " dead " water for any distance. Centrifu- 
gal action would rapidly throw it out, and no doubt strong centrif- 
ugal force acts upon the water which is brought nearly or entirely 
to rest relatively to the blade by impinging upon the leading edge. 
It is possible that this strongly localized centrifugal force plays 
a part in causing cavitation. 

It is evidently necessary to consider separately the cavitation 
which appears over the backs of propeller blades and the cavitation 
which appears over the faces. 

The former is not seriously objectionable. If the cavities at the 


blade backs were perfect vacua they would be helpful rather than 
harmful. It is seen from Figs. 222 and 223 that for model propel- 
lers in the fresh water of the model basin these cavities do approach 
perfect vacua. Sea water contains a good deal of occluded air, and 
it may be that for full-sized propellers in sea water the cavities are 
more or less filled with air. But, even so, the air could be pumped 
out without serious difficulty. Hence we may conclude that cavities 
at the rear of a blade are not an insuperable bar to efficiency. This 
is fortunate, for there is no question that when a curved surface, 
such as the back of a propeller blade, is driven through the water 
at a sufficiently high speed, cavities are necessarily formed over its 
rear portions. 

The case of the cavities over the blade faces is different. These 
have no redeeming feature. In the first place, they are due to an 
edge angle so large as to produce large negative thrust from the 
leading portion of the back of the blade. In the second place, they 
nullify the thrust which the blade" face would otherwise contribute, 
and, all things considered, are obviously fatal to efficiency. 

Hence, it is essential to efficiency to minimize or avoid entirely 
face cavitation. The method which has been most used with satis- 
faction in practice consists in fitting very broad blades so that the 
thrust per square inch of projected area is kept below a limit found 
to be safe by experience. But the thrust per square inch of pro- 
jected area is not the primary feature causing cavitation. Tip 
speed and blade section are without doubt the main factors. Still, 
for a given type of propeller the thrust is a function of tip speed 
and blade section, and hence might be used as a gauge of cavitating 
conditions. Thus Barnaby, for the type of propeller used on the 
Daring, found that with a tip immersion of one foot, cavitation 
showed up when the thrust per square inch of projected area was 
above 1 1 pounds. The trouble with this method is that the limit- 
ing thrust permissible would have to be determined for each type 
of propeller. 

6. Reduction of Cavitation by Broad Blades. From the theory 
of cavitation set forth above the advantages of a wide, thin blade 
are obvious. It has a smaller edge angle, so that it can be driven 
to a much higher tip speed than a narrow blade without causing 


face cavitation. Also after face cavitation begins it spreads slowly 
with increase of tip speed so that the wider the blade the greater 
the area of the face whose thrust is not nullified by cavitation. 

In fact, if the blade is so wide that the manner of the water leav- 
ing it is not materially modified by cavitation, the thrust will not 
be materially modified even if there is a cavity over the leading 
portion of the face. This result is readily explicable. Thus, sup- 
pose we have a cavity at the leading portion of a blade face. The 
vacuum results in the water being impelled toward the face, forward 
momentum being communicated to it. If the face is sufficiently 
wide, the water will impinge upon it again. Through the loss of 
its momentum it will communicate a corresponding thrust to the 
blade, and then will pass from the blade, if it is wide enough, in 
nearly the same manner as if there were no cavitation over the 
forward portion of the face. Hence, the net change of velocity and 
resulting thrust will not be much affected by the cavitation. But 
if the blade is so narrow that the face cavity extends nearly to the 
following edge there will not be enough blade beyond the cavity to 
absorb the forward momentum of the water and direct it again in 
the way it should go. With the wide blade the loss of pressure 
on the leading portion of the face due to cavitation is nearly made 
up by additional pressure on the following portion of the face. 
With the narrow blade there is virtually no following portion. 

Figures 224 and 225 show experimental results which indicate the 
advantages of breadth of blade in preventing harmful effects from 
cavitation. Two 1 6-inch model propellers of the same pitch ratio 
0.4 and blade thickness fraction, but of mean width ratios of 
.125 and .275, were tested with smooth backs and with strips secured 
to the backs, as indicated in the figures. The sections shown were 
taken in each case at two-thirds the radius. The curves in each 
case refer to a 5-knot speed of advance. Neither propeller showed 
harmful effects of cavitation with a smooth back. With the strip 
attached the narrow-bladed propeller showed pronounced cavitation, 
while the broad-bladed propeller showed none, though its strip was 
materially larger than that of the narrow-bladed propeller. As 
might be expected, the torque is much increased by the presence of 
the strip. But until cavitation appears the thrust of the narrow- 


bladed propeller is but little reduced by the strip, and for the broad- 
bladed propeller the thrust is actually increased by the presence of 
the strip. Upon the theory of cavitation which has been set forth 
a reasonable explanation of the peculiar features of Figs. 224 and 
225 is as follows: 

The strips increase the negative thrust on the leading portion 
of the blade back in each case, increase the suction or cavitation 
of the following portion of the blade back, thus increasing thrust, 
and cause face cavitation over the leading portion of the blade face. 
The net result of the two former actions is small or even results in 
an increased thrust. But when face cavitation is set up strongly, 
the narrow blade breaks down, while the broad blade holds its own, 
because the face cavitation over the leading portion of the face is 
neutralized by the action of the following portion of the face. 

7. Cure for Cavitation. We have seen that the wide blade of 
usual type has two advantages from the point of view of cavitation. 
Its smaller edge angle will allow high tip speeds to be reached with- 
out cavitation, and when cavities do appear the tip speed can be 
still further increased without the harmful effects due to the face 
cavities, which are usually characterized by the term "cavitation.'* 
Now we do not mind cavities on the back of the blade, so the ques- 
tion whether it is possible fully to cure harmful cavitation depends 
entirely upon whether it is possible to avoid entirely face cavitation. 

The difficulties in the way of this are practical difficulties of con- 
struction. Thus, if we could make propeller blades without thickness, 
there would be no face cavitation. The water would cascade around 
the leading edge from front to back. There would be back cavita- 
tion only, and solid water over the face. But we cannot make pro- 
peller blades of no thickness. The best we can do in practice is to 
approximate to the ideal plane along the leading edge, making the 
face straight, or very slightly convex, and the leading portions of 
the back hollow, as indicated in Fig. 226, and keeping the edge 
angle down as close as possible to the slip angle. 

It might seem that the edge angle could be made double the slip 
angle without danger of face cavitation, since when so made the edge 
would part the water evenly. But the slip angle is an average angle, 
and usually at some part of its revolution the blade of an actual 


propeller will have a slip angle but little if any greater than half 
the average value. Another reason for making the edge angle as 
small as practicable is the fact that no matter how sharp the edge 
is made it is not a mathematical edge, and when advancing at enor- 
mous speed through the water will show slight cavitation if it is 
attempted to split the water evenly on each side. Hence, the en- 
deavor should be to have the water naturally tend to cascade around 
the edge from face to back. 

It might seem that this could be accomplished without extreme 
sharpening of the leading edge by making the leading portion of the 
face convex, as indicated in Fig. 227. This is true, and a propeller 
so shaped would not show face cavitation near the leading edge, 
but with even a moderate convexity of the face it would show 
severe cavitation over the following portion of the face. There 
was a case of a United States battleship whose propeller did not 
differ materially in dimensions, etc., from those of her sister vessels, 
but had sections which were abnormally curved at the leading por- 
tion of the face, as indicated in Fig. 228. 

This vessel showed over a knot less speed than her sister vessels 
for the same power, and although her tip speed was only about 
6000 feet per minute, there is little question that she showed very 
serious face cavitation. It is not possible to say what convexity is 
permissible in a given case without cavitation, but it is certain that 
the higher the tip speed the smaller the permissible convexity, and 
for tip speeds of 10,000 feet and over it probably should be very 
small indeed. Pending careful full-scale experiments on this point, 
the safest plan is to avoid convex blade faces for propellers of high 
tip speed. 

It need hardly be said that it is not easy to make hollow-backed 
propellers with leading edges as sharp as a knife. It is advisable 
to use cylindrical ribs on the back, extending from the leading edge 
to the thicker portion of the blade. If the leading edge is serrated 
with a rib extending to the point of each tooth, the blade edge need 
not be quite so sharp. Such a form of edge seems to get through 
the water with less tendency to face cavitaiion, and when this does 
set in it seems to confine itself to rather narrow rings, starting from 
the angles where the roots of the serrations join. 


The ribs on the back must of course be well sharpened where 
they cut the water. They increase back cavitation, but that is not 
a very serious matter. 

While the prevention of face cavitation is essentially a question 
of the extreme leading portion of the blade back, the blade should 
not thicken so rapidly as we pass aft from the hollow portion that 
owing to its angle of action there is large negative thrust. 

This is of course always objectionable, but particularly so when 
there is pronounced back cavitation. After this has set up, the 
suction of the back does not grow so rapidly as before with increase 
of speed, and hence negative thrust, which continues to increase 
indefinitely with speed, should be avoided with peculiar care. 

The practical conclusion in this connection is that blades made 
hollow-backed to avoid cavitation should not be of narrow type 
but fairly wide say from .30 to .35 mean width ratio in 
order that they may be made fairly thin in the center. 

Such blades should avoid cavitation without the excessive widths 
which are necessary with blades of ogival section and which involve 
material loss of efficiency through large blade friction. 

8. Pressure Due to Blade Edge Speed. In connection with the 
question of cavitation it is interesting to note that at the tip veloci- 
ties of modern high speed propellers enormous pressures are liable 
to be set up upon the leading blade edges. Suppose we have a 
small plane advancing through water perpendicular to itself. The 
maximum pressure upon it is that due to a head equivalent to the 
velocity, the formula being 

wv 2 

where p is pressure in pounds per square foot, v is velocity of advance 
in feet per second, w is weight of a cubic foot of water and g is the 
acceleration due to gravity. If we assume that at a blade edge 
there is always a small portion which is virtually a plane surface, 
it follows that the motion of the blade through the water will cause 
at its edge the pressure given by the above formula. 

Table XV shows for various blade edge velocities in feet per 
minute, g being taken as 32.16, the corresponding pressures in salt 


water weighing 64 pounds to the cubic foot. The pressures are 
expressed in pounds per square inch. 

When we consider in Table XV the very rapid growth of blade 
edge pressures with velocity and the very high pressures reached 
when the velocity is 10,000 feet per minute and over, it is obvious 
that for high-speed propellers the area of blade edge over which 
such pressures are set up must be reduced to a minimum. In former 
days propeller blades were often made of elliptical section, and even 
now, for fairly high-speed propellers ogival blades are frequently 
finished with a quarter round. Such blades will certainly break 
down by cavitation at high-speeds and quick running propellers 
should by all .means have sharp leading edges. It is difficult to make 
an edge which is mathematically a sharp edge, but the more nearly 
this is approached the better. 

28. Wake Factor, Thrust Deduction, and Propeller Suction 

Hitherto the ship and the propeller have been considered apart. 
It is necessary now to take up their very important reactions upon 
one another when the ship is being driven by its propeller or pro- 

i. Components of Wake. Owing to its frictional drag upon the 
surrounding water there is found aft in the vicinity of the ship a 
following current or wake, called the frictional wake, which is in most 
cases greatest at the surface and in the central longitudinal plane 
of the ship and decreases downward and outward on each side. 
Superposed upon the frictional wake there is a stream line wake, 
caused by the forward velocity of the water closing in around the 
stern. This also will be greatest at the surface and center and 
decrease downward and outward, though its law of decrease will be 
different from that of the frict onal wake. 

Superposed upon the two wakes above we have the wave wake. 
If there is a wave crest under the stern, the water is moving forward 
with velocity which decreases downward from the surface and, prob- 
ably in practical cases, decreases slightly outward from the center. 

Under a wave hollow the velocity is sternward, the wave wake 
velocity in this case may be said to be negative, the wake being 
regarded as positive when its velocity is forward. 


There is a final factor, often ignored, which will be considered in 
more detail later in connection with shaft obliquity. The water 
aft is not flowing exactly parallel to the shaft. It rises up behind 
the stern and closes in horizontally, thus causing the slip of a pro- 
peller blade to be greater than the average over one portion of its 
revolution and less than the average over another. This condition 
of affairs does not materially affect the wake action, except in certain 
cases that will be considered later. For the present we will consider 
the wake proper made up of the three components enumerated 

2. Effects of Wake. The propeller of an actual ship does not 
work in undisturbed water, but in water which has a very confused 
motion. The wake velocity varies over the propeller disc at a given 
speed, and at a given point of the disc varies with the speed. It 
is necessary to assume a uniform velocity of wake over the screw 
disc. This velocity of wake may conveniently be expressed as a frac- 
tion of the velocity of the ship, the ratio being called the "wake 
fraction " and denoted by w. The wake was first explored by R. E. 
Froude, who published some methods and results as long ago as 1883 
in a paper before the Institution of Naval Architects. Froude used 
model propellers behind ships' models. Suppose the speed of the 
ship model is V. If the model screw is tested at given revolutions 
separate from the model at a speed of advance V into still water, 
we get a certain thrust and torque. 

Suppose, now, keeping the revolutions constant, the model screw 
is tested behind the ship model. The thrust and torque are changed 
and are the same as would be found at the constant revolutions at 
a speed of advance Vi, say, into still water. V\ is nearly always 
less than V. So the wake behind the model at the speed V is equiv- 
alent, so far as the screw is concerned, to a uniform following cur- 
rent of velocity V V\ or wV. The thrust and torque of the 
screw are then those appropriate to a speed of advance of V\. The 
power absorbed is the same as if the screw were working in undis- 
turbed water with speed of advance V\. But if T denotes the 
thrust, the useful work as far as the ship is concerned is not TV\ 
but T V. Hence the efficiency or ratio between the useful work and 
power absorbed is, if V is greater than FI, greater than in undis- 



turbed water, the ratio being - The fact is that the following 


wake assists in pushing the ship ahead, using the propeller as the 

3. Thrust Deduction and Hull Efficiency. While the ship acts 
upon the screw through its wake, the screw acts upon the ship 
through its suction. 

Through its suction, the resistance of the ship is virtually in- 
creased beyond what it is without the screw. This is a cause of 
increase of power absorbed in propulsion. If R is the resistance 
of the ship at speed V, and T the screw thrust required to drive 
the ship at speed V, we have T greater than R. The quantity 
T - R is called the thrust deduction, being the difference be- 
tween the actual thrust and the net thrust or tow-rope resistance. 
It is usually denoted by tT, so that R = T(i t) and i / is called 
the thrust deduction factor, t being called the thrust deduction 

Suppose, now, we have a propeller absorbing a certain power, P, 
at certain revolutions per minute and driving a ship at speed V. 
In undisturbed water the propeller when absorbing the same power 
at the same revolutions would have a speed of advance Vi, and its 
efficiency would be a definite quantity, e say. Its thrust is T. De- 
note the effective horse-power necessary to propel the ship by E 
and its resistance by R. Then E is not equal to eP, as it would 

R V 

be if there were no wake or thrust deduction, but to eP X X 

T V\ 

R V 

The expression X is called the hull efficiency, and its two factors 
T V\ 

R V 

and are called respectively the thrust deduction factor and 
T V\ 

the wake factor. Since R = T (i /) and Fi= V (i w) we have 

the hull efficiency = ~ X rr = ~ 

T Vi i w 

Froude expressed the wake as a fraction of Vi, the speed of ad-' 
vance, not V, the speed of the ship. Calling this w p , Froude denoted 


the wake factor by i -f- w p where w p is the "wake percentage." 


There are some advantages in using the " wake fraction " as already 
denned, but care must be exercised not to confuse it with Froude's 
"wake percentage." The relation connecting them is 

In most cases the hull efficiency does not depart greatly from 
unity, the thrust deduction factor i t being less than unity, and 

the wake factor greater than unity. 

i w 

This is readily understood when we reflect that the more favorably 
a screw is situated to catch the wake the more direct its suction as 
a rule upon the after part of the ship. Single screws, for example, 
may be expected to show larger thrust deductions and wake factors 
than twin screws. Also the stream line wake is increased by full 
lines aft, but the fuller the after part the stronger the propeller 
suction upon it and the larger the thrust deduction factor. 

4. Variations of Wake Fraction and Thrust Deduction. The 
wake fraction and thrust deduction are affected by many considera- 
tions, and in the present state of our knowledge the actual values 
in a given case can seldom be estimated accurately without special 
model experiments. 

The most comprehensive information in this connection available 
at present is contained in a paper read at the 1910 Spring Meeting 
of the Institution of Naval Architects by W. J. Luke, Esq. This 
paper contains data as to the wakes and thrust deductions of models 
of various vessels that had been previously published, mainly by 
Mr. R. E. Froude, and gives a great deal of valuable new infor- 
mation obtained at the John Brown and Company's experimental 
tank at Clydebank, Scotland. These experiments were made with 
a single model 204 inches long, 30 inches broad, of 9 inches mean 
draught, displacement 1296 pounds in fresh water and having .65 
block coefficient. All variations of propellers, etc., were tried with 
the bare hull and many with propeller bosses or brackets inclined 
225 degrees from the horizontal. In addition some special experi- 
ments were made with bosses at other angles, ranging from horizontal 
to vertical. 


In what may be termed the standard conditions, two three-bladed 
model propellers 6 inches in diameter, of 7.2-inch pitch with straight 
elliptical blades were used with centers i inches forward of the 
after perpendicular and 5 inches from the center line. 

Experiments were made varying separately speed of vessel, pitch 
ratio and diameter of propellers, fore and aft and transverse position 
of propellers, number and area of blades, etc. 

Briefly summarizing the main results of the twin screw experi- 
ments, which were always made with both outward and inward 
turning screws, Luke found that variation of number and area of 
blades had no appreciable effect upon wake factor and thrust 

Change of pitch ratio produced changes of secondary importance 
for the bare hull, both wake and thrust deduction increasing slightly. 
With the 22\ degrees bossing the changes were slight and much as 
before with outward turning screws, but with inward turning screws 
the wake fell off with increase of pitch. 

Changes of diameter caused material changes in wake and thrust 
deduction, but Luke concluded that they were due as much to 
changes in clearance between hull and propeller as to the changes 
in diameter per se. 

Change of speed of vessel resulted in practically no change in 
thrust deduction, but whether with bare' hull or bossing the wake 
fell off steadily with increase of speed, the wake fraction decreasing 
with the bare hull and propellers in standard location from about 
.19 for speed-length ratio of .6 to .1452 for speed-length ratio of i.o. 
In the paper the wake is characterized by the wake percentage 
values following Froude. These have been converted to wake frac- 
tions as already defined. For a speed-length ratio of .8, about what 
such a vessel would usually be driven at in service, the wake fraction 
was .167 for inturning screws and .173 for outturning screws, the 
thrust deduction / being about .155 in each case. 

With the bossing the thrust deduction was still practically the 
same with out- and inturning screws and varied little from .16. 
The wake fraction fell off with the speed as with the bare hull, but 
the wake was materially greater for outturning than for inturn- 
ing screws. For the .8 speed length ratio it was .191 instead of 


.173 for outturning screws and .146 instead of .167 for inturning 

Luke's experiments show clearly that for the model tried the most 
important factor affecting wake and thrust deduction is the location 
of the propeller with reference to the hull. Thus with the bare hull 
and the 6-inch propeller the results were as follows : 

Center of propeller from center of model 




Wake fraction outturning screws 



. no 

Wake fraction inturning screws 




Thrust deduction, t, both cases 


. I ^O 


It is seen that in this case a transverse change of \ the diameter 
caused wake arid thrust deduction to vary a great deal, both being 
larger the closer the propeller was to the hull. When the distance 
of the propeller from the hull was varied by shifting it fore and aft 
the effect was not so great, but still material. 

A few experiments were made with a single screw behind the 
model, diameter being varied. The wake was found markedly 
greater for this propeller location, the thrust deduction being also 
increased, but not nearly so much as the wake. Curiously enough 
the smaller the propeller the larger the wake. Thus, the wake frac- 
tion varied from about .275 for a 5-inch screw to .226 for an 8-inch 
screw. The corresponding thrust deduction values were .155 and 
.185, the smaller screw thus profiting not only by the larger wake 
but by the smaller thrust deduction. 

Luke's paper makes it clear that location with respect to the hull 
is a very important factor in connection with wake and thrust 
deduction. Experiments such as described in his paper made with 
models of varying fineness are much needed. 

5. Approximate Wake Fractions and Thrust Deductions. Since 
the speed of advance of a propeller, a vital factor in design, depends 
upon the wake fraction, it is important to be able to approximate 
to it in a given case. In Luke's paper, as already stated, are given 
a number of wake factors for single and twin screw ships and of 
thrust deduction coefficients, t, for twin screw vessels. For twin 
screw vessels, Froude laid down the dictum many years ago that, 


broadly speaking, wake factor - - and thrust deduction factor 

i w 

i t were reciprocals or w = t. The data given by Luke confirms 
this, and shows also that we may, so far as present knowledge 
goes, reasonably assume wake fraction to vary linearly with block 

Then from the data published by Luke we may say with reason- 
able approximation iv = .2 + .55 b = I, where w is wake frac- 
tion, / is thrust deduction coefficient and b is block coefficient. 
This formula ignores the matter of screw location, but may be taken 
as applying to screws about abreast the after perpendicular and 
with centers about 1.2 the radius from the center line. 

For lesser clearance w will be greater and t will also increase 
somewhat, but the formula is and can be, from the available data, 
only a rough approximation. 

For center screws in the usual position the approximate formula 
indicated is" w = .05 + -5 b. 

Data is not available for a formula for t for center screws, but 
Luke's experiments would appear to indicate that for them t would 
be increased but little over its value for twin screws. It follows 
that if the hull efficiency is unity for twin screws it is somewhat 
over unity for single screws, particularly for full vessels. 

The formulae above apply to the bare hull or to vessels fitted with 
struts or bossing which does not interfere with the natural water 

It should be remembered that they are deduced from model 
experiments and will nearly always exaggerate the wake of the full- 
sized ship. It is desirable, however, if we cannot determine the 
wake accurately, to overestimate it rather than underestimate it. 
If it is overestimated, the engines on trial will turn somewhat faster 
than estimated, which is generally allowable. If it is underesti- 
mated, it may be impossible to run the engines up to the designed 
speed without decreasing propeller pitch or reducing propeller 

6. Approximate Determination of Wake Fraction. Since the 
wake is explored by trial of model screws working behind models of 
ships the question naturally arises whether we cannot gain some 


light upon the subject from trials of full-sized ships. Analysis soon 
makes it evident that the apparent slip of propellers on trial is often 
very much below what must have been the real slip. We know 
that in any case the power absorbed by a given propeller advancing 
through undisturbed water depends only upon the revolutions and 
the speed of advance. For an actual propeller advancing through 
the water disturbed by the ship we can reasonably reduce the actual 
disturbance to an equivalent uniform motion. Throughout the 
range where the Law of Comparison holds we can determine for 
any propeller for which we have model experiments the relations 
between power absorbed, revolutions and speed of advance. Hence, 
if we know any two of these quantities, we can determine the third. 
Now from the results of trial of a vessel we know corresponding 
values of indicated or shaft horse-power, revolutions and speed of 
vessel. The shaft horse-power is practically the power, P, absorbed 
by the propeller, and from the indicated horse-power P can be 
estimated with reasonable accuracy. Hence, although we do not 
measure V A directly, we can estimate it from the power and revolu- 
tions if we have reliable model experiments with the propeller and 
the Law of Comparison holds, and knowing VA and V we can deter- 
mine the wake fraction. The reduction of the results of model 
experiment to a form convenient for this application is simple. We 

have seen that we may write P = A - where P is power absorbed 


by the screw, d is diameter in feet and V A is speed of advance in 
knots. A is a coefficient independent of size and speed and de- 
pending only upon the slip and the proportions and shape of the 

/ PR \ 3 d? 

So let us write P = S{ ) where p is pitch in feet and R 
Viooo/ p 

/IOOO\ 't) 

denotes revolutions per minute. Then S = I ) jj P and is like 

\ pR I a 6 

A, a coefficient independent of size and speed and depending only 
on the slip and the proportions, etc., of the propeller. 

From experimental results with models we can readily determine 
a curve of 5 plotted on the slip. Thus, for a 1 6-inch model with a 
speed of advance of 5 knots we have 5 =.3129^ (i s) 2 where Q 


is torque in pound-feet. Or we may determine 5 from a curve 


101.33 1000 (ioi.33) 3 \ioooj p 

Whence S =.9610% A (i s) 3 . 


Fig. 229 shows curves of S plotted on s for the four propellers of 
Fig. 179. Now suppose we have a full-sized propeller similar to 
the model of .0448 blade thickness fraction and 18 feet in diameter, 
making 120 revolutions per minute and absorbing 12000 horse- 
power. Its pitch will be 21.6 feet. Then from the data of the 

/ 1 OOO\ & 

full-sized screw S = ( ) ^P = 2.552. From Fig. 229, for the 
\ pR I a 6 

propeller in question, when S 2.552, s = .2340. So the true slip 
of this propeller would be .2340, and its true speed of advance, 

V A = iQ-593- Suppose the speed of the ship V is 


21 so that the apparent slip, 5', is .1790. 
Then V = ^ R ^ ~^ = 21 knots. 


The wake = V V A = - (s - s') = 1.407 knots. 


ITT- i r 4.- V VA I0l S s 

Wake fraction = - = -7 - = - -. = .0670. 


It is very easy to derive curves of S from the Standard Series 
results of Figs. 185 to 208. 

Figures 230 to 233 show contours of slip plotted on S and pitch 
ratio for four blade widths and the blade thickness fractions indi- 
cated. For propellers closely resembling the Standard Series these 
figures may be used in connection with accurate trial data to obtain 
a reasonable approximation to the wake so long as there is no cavi- 
tation. The propeller power, P, however, must for reciprocating 
engines be estimated from the I.H.P. Methods for this will be con- 
sidered under analysis of trials. 


These figures may be used, however, to obtain rough approxima- 
tions to the wake for propellers very different from the Standard 

For three-bladed propellers with oval blades and extra wide tips 
the correct values of 5 will be somewhat less than in the figures, but 
the difference for practical propellers will not be great. In order 
to use Figs. 230 to 233 for four-bladed propellers we need only 
divide the actual propeller power, P, by the proper power ratio for 
four blades, obtained from Fig. 181. We thus obtain approximately 
the power absorbed by a three-bladed propeller having blades iden- 
tical with the four-bladed propeller and working with the same 
revolutions and speed of advance. 

From this we determine 5 and use Figs. 230 to 233 as before. 
It will be found in practice that the methods above for estimating 
the wake from full-sized trials will generally give values that seem 
too low. We know that the wake values for a full-sized ship should 
be less than for its model, but another factor present at times and 
tending to lower the wake deduced from the S value is a slight 
failure of the Law of Comparison connecting model and full-sized 
propeller. We know that the Law of Comparison fails when a pro- 
peller breaks down by cavitation, but it is probable, particularly 
with blunt-edged blades, that there is more often than might be 
supposed a certain amount of eddying in the operation of the full- 
sized propeller not found in the operation of the model. This might 
not seriously reduce efficiency and would manifest itself mainly by 
a slip of the full-sized propeller somewhat larger than would be 
inferred from the model results. The wake deduced from the S 
values would be correspondingly reduced. 

The 5 value method should not be used when the wake can be 
investigated by model experiments. Lacking model experiments, 
we can roughly approximate to the wake by the formulae already 

There is great need for a systematic and thorough experimental 
investigation of the question of wake, following the lines of Luke's 
experiment, which will enable it to be closely estimated in any prac- 
tical case likely to arise. But there is a mass of accummulated 
trial data extant for vessels whose models never have been, and 


probably never will be, tested, and it is worth while for those possess- 
ing it to investigate the wake fraction even by a method which is 
only roughly approximate. For practical purposes the wake frac- 
tion of a vessel seldom requires to be determined with minute accu- 
racy. It is principally of use for settling the diameter and pitch 
of the screw, and neither these nor the efficiency will often be much 
affected by a moderate error in the wake fraction. 

If by use of Figs. 230 to 233 we find a certain wake for a vessel 
of a given type, we can use this for a vessel of the same type with 
similar propeller location, and for the purpose of determining diam- 
eter and pitch of screw it will make little difference whether the 
nominal wake from Figs. 230 to 233 is the real wake or departs 
materially from it. Whatever the departure, it will be practically 
the same in the two cases. 

7. Effect of Shaft Brackets upon Wake. Reference has already 
been made to the apparent effect upon the wake of the direction 
of flow of the water aft. 

This has a marked effect when large shaft brackets are fitted 
which modify the natural flow of the water. 

Thus, if a shaft bracket is fitted with a wide horizontal web, it 
interferes seriously with upward flow aft and the water closes in 
with a much stronger horizontal motion or current inwards than 
otherwise. The conditions over the lower half of the propeller disc 
are somewhat, but not very seriously, modified from bare hull con- 
ditions, much greater modifications occurring over the upper half 
of the disc. Considering the upper blades, the effect of the inward 
flow of the water is materially to increase the slip angle for outward 
turning propellers where the upper blades are moving against the 
current, while for inward turning screws with the upper blades mov- 
ing in the same direction as the current the slip angle would be 
decreased. Hence, we may expect a large horizontal shaft bracket 
materially to increase the apparent wake for outward turning screws 
and to decrease it for inward turning screws. 

A case in point is that of the Niagara II, a steam yacht 247' 6" 
X 36' X i6'4^" draught and 2000 tons displacement. 

This vessel had a Lundborg stern, involving wide horizontal 
shaft brackets, and her deadwood aft was not cut up. 


She had two six-hour trials under similar conditions, except that 
the screws were interchanged, being inward turning on the first trial 
and outward turning on the second. While the horse-power was 
not accurately determined, it was closely estimated at 2100 with 
inward turning screws and 1950 with outward turning screws. 

Nevertheless, with inward turning screws the average speed was 
12.8 knots with an apparent slip of 26.4 per cent, while with out- 
ward turning screws the average speed was 14.12 knots with an 
apparent slip of but 13.3 per cent. 

This marked difference in apparent slip can be due only to the 
fact that the horizontal shaft webs force a strong inward motion 
of the water above them along horizontal lines, and while this motion 
is not a wake, being transverse or perpendicular to the line of 
advance of the ship, its effect upon the upper blades of the pro- 
peller is equivalent to a positive wake for outturning screws and a 
negative wake for inturning screws. 

It would seem that the lower blades are not much affected, such 
action as there may be upon them being much less than that upon 
the upper blades. 

Luke's paper already referred to, gives most interesting and in- 
structive results of a model investigation of shaft bracket angles 
and direction of screw rotation. The model was the same as already 
described, 204 inches long, 30 inches broad, 9 inches draught, 1296 
pounds displacement in fresh water, having a block coefficient of .65. 
The model screws were three-bladed, 6 inches in diameter, 7.2 inches 
pitch, having straight elliptical blades. Their centers were 5 inches 
out from the center line of the model and ij inches forward of the 
A.P. Brackets were fitted at angles ranging from horizontal to 
vertical and the model tested with inturning and outturning screws, 
the screws and their positions remaining unchanged as the shaft 
bracket angles were varied. The results are summarized on the 
following page. 

These results show relatively enormous variations of wake with 
variation of bracket angle and direction of turning and make it 
clear that under some conditions the virtual wake due to obliquity 
of water motion may overshadow the real wake or forward motion. 
It is obvious that for a given real wake outturning and inturning 



screws should give practically the same derived wake. We see, how- 
ever, that with horizontal brackets the wake fraction is about 2\ 
times as great with outturning screws as with inturning screws while 
with vertical brackets the wake fraction with inturning screws is 
nearly four times as great as with outturning screws. 



Angle of bracket with horizontal 


4 c 



^ 7 ake fraction outward turning screws 

. 2dl 

. TOO 

. I4.S 


^^ake fraction inward turning screws 


. 14? 




Thrust deduction outward turning screws. . . 
Thrust deduction inward turning screws . . . 
Hull efficiency outward turning screws 

. 164 


I . IO< 

. 169 

. 160 

I .026 

. 164 



Hull efficiency inward turning screws 

9 7 .8 


I .02 ? 

I 064 

I 087 

Model resistance in terms of bare hull resistance 


I .040 



I . I 2O 

These differences can be due only to the fact that transverse 
motion of the water affects inturning and outturning screws very 
differently. Horizontal motion inward is equivalent to a positive 
wake for outturning screws and a negative wake for inturning 
screws. Vertical motion upward is equivalent to a negative wake 
for outturning screws and a positive for inturning screws. 

In the light of Luke's experiments the remarkable trial results of 
the Niagara II are readily explicable. 

While horizontal shaft brackets in his experiments resulted with 
outturning screws in a high hull efficiency this was accompanied 
by an increased hull resistance, so there was no appreciable net gain. 

It would seem that in practice from the point of view of resistance 
and propulsion, shaft brackets should be arranged to parallel if pos- 
sible the lines of flow. It is generally, however, more convenient 
to arrange them more nearly horizontal and when this is done the 
screws should obviously be outturning. 

8. Propeller Suction. The thrust deduction is due to the suc- 
tion of the propeller upon the ship's hull. It is well to consider in 
connection with it the question of propeller suction generally and 
its effect upon the water. An experimental investigation in this 
connection has been made at the United States Model Basin, and is 
described in a paper entitled " Model Basin Gleanings " read before 
the Society of Naval Architects and Marine Engineers in 1906. The 


suction of 1 6-inch model propellers was measured over the surface 
of a vertical plane, parallel to the propeller axis, which could be set 
at various distances from the propellers. Figure 234, showing the 
variation of pressure along various horizontal lines of the plane when 
set f inch from the tips of a propeller of 1 6-inch diameter and 16- 
inch pitch which was working at a nominal slip of 30 per cent, is 
typical of all the results. 

Now a necessary result of this suction is that it draws the water 
inward toward the propeller axis and aft toward the disc. An impor- 
tant fact, which seems to have been generally ignored, should be 
pointed out. When a propeller works with sternward slip velocity 
of the water, the supply of water necessary to allow slip velocity 
comes ultimately from the free surface. For referring to Fig. 235, 
which indicates a submerged propeller, consider an imaginary plane 
X Y perpendicular to the shaft axis and just forward of the screw 
disc, as indicated by the dotted line. But for the screw action all 
the water in that plane would be at rest. Owing to the screw action 
the water is flowing aft through the screw disc and forward is 
flowing from all directions toward the disc. Now the water flowing 
through the plane does not leave a vacuum behind it; and particles 
of waler flowing toward the disc from points forward of the plane 
cannot leave vacua behind them. Their places must be taken by 
other particles of water. Where can these particles come from ? 
The water being practically incompressible, there are only two pos- 
sible sources of supply. It is possible to conceive that the water 
flowing aft through the plane just forward of the screw disc spreads 
out astern, and finally to an equal amount flows forward again 
through the plane. In other words, the suction draws a certain 
amount of water through the plane and the thrust behind the pro- 
peller forces an equal amount across the plane in the opposite direc- 
tion at points some distance from the disc. This action goes on 
when a screw is operated with no speed of advance as in dock trials. 
Careful study of the action of advancing screws, however, indicates 
clearly that in this case the water to take the place of that sucked 
to the piopeller disc simply flows downward from the surface, pro- 
ducing a depression of the surface, which advances with the speed 
of the propeller. Figures 235, 236 and 237 show results of an experi- 


mental investigation of this question made at the United States 
Model Basin. Two 1 6-inch propellers of identical blade profiles, as 
indicated, one with 12. 8-inch nominal pitch, the other with 19.2- 
inch nominal pitch, were operated as indicated with their tips 
8 inches below the surface and the resulting surface depressions for 
5-knots speed of advance and various slips observed. It is seen 
that contour lines in the depression over the propeller are approxi- 
mately circular. The point of maximum depression is in each case 
a little astern of the propeller, and as to be expected, the greater 
the slip the greater the depression ; also the finer the pitch the greater 
the depression. This too is to be expected. The propeller of fine 
pitch exerts much the greater thrust for a given slip and speed of 
advance. Hence the actual sternward velocity communicated to the 
water is greater for the propeller of fine pitch than for the propeller 
of coarse pitch, and the surface depression greater accordingly. 

As a result of the fact that sternward velocity of water entering 
the screw disc is obtained ultimately by sucking water from the 
surface, it follows that if a screw is so arranged that it cannot draw 
water from the surface, the sternward velocity of the water entering 
the screw disc is reduced. The suction in such cases, not being 
absorbed by giving velocity to the water, is likely to be exerted 
upon the ship and cause abnormal thrust deduction. Once the 
water has reached the screw disc it is difficult to conceive, as pointed 
out in discussing Rankine's theory, how it can be given much addi- 
tional sternward velocity. We must conclude that while in the disc 
the change of velocity is nearly all rotary, as in Greenhill's theory. 
It is true that this involves changes in pressure, and Greenhill, on 
account of the increase of pressure involved in his theory, considered 
it necessary to confine the screw disc and race by a cylinder. Green- 
hill has pointed out, however, that it is conceivable to have a defect 
of pressure behind the screw at the center, the pressure increasing 
as the circumference is approached until at the outside of the screw 
race it is normal. It should be pointed out that, since there is 
quite a defect of pressure in all the water passing into the screw 
disc, its pressure while in the disc can be materially increased by 
the action conceived by Greenhill without exceeding the normal 
pressure of the surrounding water. 


To sum up, it appears that a reasonable theory of what happens to 
a particle of water which is acted on by a propeller is about as follows: 
When some distance forward of the screw, it is sucked aft and in 
toward the shaft axis, its pressure being reduced at the same time. 
Hence, it enters the screw disc with a certain sternward velocity 
and reduction of pressure. As it passes through the disc its stern- 
ward velocity is changed but little. It has impressed upon it a 
rotary velocity and an increase of pressure, so that its pressure on 
passing out of the screw disc is probably very close to normal pres- 
sure again for particles near the circumference of the screw race and 
still below normal for particles in the interior of the race. 

9. Effect of Immersion upon Suction and Efficiency. The stern- 
ward velocity into the screw disc is affected by the situation of the 
screw. Probably immersion alone does not affect it much. The 
more deeply immersed screw is, it is true, farther from the surface 
from which its water supply must come, but it is in a position to 
draw upon a larger surface area. Still from this point of view there 
is nothing favorable to efficiency in deep immersion, the reasons 
rendering it desirable in most cases and necessary in some having 
to do not with efficiency but with prevention of racing in a seaway. 

If vessels worked always in smooth water, there is little doubt 
that screws could be located with their tips quite close to the sur- 
face, provided they did not suck air in operation, without loss of 
efficiency. In fact, in a paper by W. J. Harding, read March 13, 
1905, before the Institute of Marine Engineers, on " The Develop- 
ment of the Torpedo Boat Destroyer," we find the statement when 
discussing the question of propellers of destroyers: 

" The least immersion of the propellers gave the best results, both 
in speed and coal bill." This conclusion was deduced from con- 
sideration of a number of trial results of destroyers in smooth water. 

A screw propeller placed under a wide flat stern, or with the flow 
of water to it obstructed in any way by the hull to which it is 
attached, must evidently work more after the Greenhill theory than 
a screw with a free flow of water to it. 

Apart from the increased thrust deduction this must involve a 
reduction of propeller efficiency. It is, of course, necessary at times 
to fit screws in tunnels, or so that they are hampered by the hull, but 


when this must be done allowance should be made for the loss of 
efficiency involved. 

29. Obliquity of Shafts and of Water Flow 

x. Shaft Deviations, Actual and Virtual. Propeller designs 
and calculations are usually based explicitly or implicitly upon the 
assumption that the propeller advances in the line of the shaft axis. 
As a matter of fact, it is unusual to find a shaft which is exactly 
horizontal when the propeller is working. Shafts of center screws 
are in a fore and aft line, but side screw shafts generally depart in 
plan from the fore and aft line. 

The divergence of propeller shafts from a horizontal fore and aft 
line is seldom so great that the resolved horizontal fore and aft 
thrust differs materially from the axial thrust. But there is a very 
serious departure from ideal conditions as regards slip of blade dur- 
ing revolution. The slip angle is a small angle, as a rule, and if the 
shaft axis is changed from the line of advance of the screw, the slip 
angle at one part of the revolution is increased by the amount of 
angular change and at another part is decreased by an equal amount. 
The slip angle is a function of the slip ratio and the pitch ratio or 
diameter ratio. Fig. 170 shows slip angles for the range of pitch 
ratio and slip ratio found in practice. 

The small size of these slip angles renders it evident that shaft 
deviations occurring in practice must cause the slip of a blade to 
vary materially during a revolution. 

2. Wake and Obliquity of Water. The variation of wake is 
another perturbing factor. The slip of the blade will be greatest 
where the wake is strongest. Evidently a virtual deviation of shaft 
axis can be imagined which would give practically the same effect 
as the variation of wake. Finally, the water itself has a motion 
across the shaft axis. 

3. Variation of Slip. The net result is that, in practice, instead 
of the thrust, torque and efficiency of a blade remaining constant 
during a revolution, they vary throughout the whole revolution. 
To fix our ideas, suppose we consider a starboard side propeller turn- 
ing outward. In considering shaft inclination we will always take 
it as we proceed forward from the propeller. 



If the shaft inclines upward from the propeller, the slip angle will 
be decreased by the amount of shaft inclination for a blade in a 
horizontal position inboard and increased by the same amount for 
the blade in a horizontal position outboard. For the blade at the 
top and bottom of its path there will be no appreciable change. 
Similarly, for a shaft inclined inboard, as we go forward there will 
be no effect for the horizontal position of the blades, a maximum 
increase of slip for the top position of the blade and a maximum 
decrease for the lower position of the blade. 

If the wake is strongest next the hull on a horizontal line, the 
result is equivalent to a downward inclination of the shaft, hence 
we may say that such a wake causes a virtual downward inclina- 
tion. Similarly, a wake strongest nearest the surface gives a virtual 
inclination inward. Water rising up gives a virtual upward inclina- 
tion, and water closing in gives a virtual inward inclination. 

The table below gives the positions for maximum and minimum 
slip of blades due to shaft inclination. Of course, when the shaft 
has both horizontal and vertical inclination, the positions of maxi- 
mum and minimum slip are neither horizontal nor vertical. In all 
cases, the plane of zero effect is that including the shaft axis and 
the line of advance of the center of the propeller. The plane of 
maximum effect is that through the shaft axis perpendicular to the 



Right-handed Screws. 

Left-handed Screws. 













E d 

p e. 



E o. 


3 . 

E . 

' E 


E d 



E .5- 



e d 


3 . 

F a 



E d 


3 & 


c w 



c en 




'E 53 



S to 


a to 

' 53 















,1 UP 







Down .... 
Inboard. . 



In the above, " in " means that the blade is in the horizontal 
position next the ship. " Out " means that the blade is in the hori- 
zontal position away from the ship. P means horizontal position 
to port for center screw and 6" the horizontal position to starboard. 
" Up " means blade vertical upward, " down " means blade vertical 

The following table gives virtual inclinations of shaft correspond- 
ing to wake and transverse motions of the water: 



Motion of Water. 

Right-handed Screws. 

Left-handed Screws. 




. Port. 



Wake increasing inward 





Wake increasing upward 

Water rising vertically 



Water closing in horizontally 

4. Virtual Inclinations in Practice. In practice, in most cases 
of twin screws, the wake increases inward and upward and the water 
rises vertically and closes in horizontally, the latter motion being 
strongest over the upper half of the disc. Then for inturning screws 
(port right-handed and starboard left-handed) we have a positive 
virtual upward inclination, since both water motions give a virtual 
upward inclination. As regards horizontal angle the wake gives 
a virtual outward inclination and the horizontal water motion a 
virtual inward inclination. So the net virtual inclination may be 
either in or out, being the difference of the two components. 

For outturning screws we have a positive virtual inward inclina- 
tion, and vertically the inclination is the difference of two virtual 

If we wish to secure uniform turning force on each blade, we must 
neutralize the virtual shaft inclination due to water motion by actual 
inclination in the opposite direction. 

While we have not data for exact quantitative results, it is evi- 


dent from what has been said that with inturning screws the shafts 
should incline downward from the screws and for outturning screws 
the shafts should incline outwards. 

Outturning screws, with shafts inclining outward, are desirable 
for maneuvering purposes. 

5. Effect upon Efficiency. This question of desirable shaft 
angles is of importance in practice, and it is to be hoped that some 
day it will be given accurate experimental investigation. At present 
we can deal with it in quantitative fashion only. As regards effi- 
ciency, a moderate variation of slip during the revolution of a blade 
will not seriously reduce efficiency so long as the average slip is that 
corresponding to good efficiency and the variation of slip is not 
extreme. But it is difficult to see how a shaft inclination as great 
as ten degrees, which has been often fitted on motor boats, can fail 
to be accompanied by a loss of efficiency. With such an angle of 
inclination it is evident, from Fig. 170, that each blade will work 
with negative nominal slip at one portion of its revolution and with 
excessive nominal slip at another portion even if the average 
slip is that corresponding to good efficiency. 

If the thrust of a propeller were due solely to the action of the 
face such a variation of slip would be wholly inadmissible. Irregu- 
lar turning forces and thrust would cause serious vibration and there 
would be great loss of efficiency. But the back of the blade through 
its suction is always an important and often a dominant factor in 
the production of thrust. The slip angle for the following portion 
of the blade is greater than the slip angle for the face by the value 
of the edge angle at the following edge. This edge angle is seldom 
less than twelve or fifteen degrees and is often twenty-five or thirty. 
Hence a shaft inclination of two or three degrees will affect com- 
paratively slightly the action of the blade back, and even the large 
inclination of ten degrees will seldom cause the suction of the back 
to be reversed into negative thrust at any portion of the revolution. 
Such a large deflection, however, is liable to produce very irregular 

6. Vibration. An important consideration in this connection 
is that of vibration. With turbine propelled vessels, practically 
all vibration which is quite strong in some turbine steamers 


is due to pounding of the water against the hull as the blades pass 
or to unbalanced propeller action. There can be no doubt that the 
latter cause of vibration, which is practically the only cause if the 
propeller tips are not too close to the hull, is affected by the shaft 
angles, and it is particularly advisable with turbine steamers to 
choose shaft angle.5 which will tend to uniformity of propeller action. 
Suppose, for instance, we have a propeller shaft carried by a nearly 
horizontal web. We have seen that there will be a very strong wake 
above the web and vertical motion of the water will be interfered 
with. In such a case, for inturning screws, the shaft should incline 
down and out, and for outturning screws, up and out. 

7. Obliquity of Flow. While the wake through its variation 
of strength over the propeller disc produces a virtual shaft devia- 
tion, it is evident from consideration of Figs. 50 to 59, showing lines 
of flow over models, that the water closing in and rising up aft fol- 
lows lines which will in many cases make material angles with the 
shafts. The effect of the obliquity of the water flow will vary a good 
deal with the position of the propeller. 

For vessels of usual type it would seem that the farther aft the 
propeller the less the obliquity of the water flow. But experiments 
with the model of a four-screw battleship indicated that at the for- 
ward screws the water was rising at an average angle of about 10 
and closing in at an average angle of about 5. For the after screws 
these angles were 1 1 and 4 respectively. The after screws, however, 
were not very far aft. These angles seem large when we compare 
them with the slip angles to be expected in practice. The obliq- 
uity of horizontal water flow will usually be greater over the upper 
portion of the propeller disc than over the lower, so that the virtual 
wake to which the obliquity of motion is equivalent will be stronger 
over the upper portions of the screw disc. Now this virtual wake 
will, for outturning screws, be positive over the upper part of the 
disc and negative over the lower. Fcr inturning screws the virtual 
wake will be negative over the upper portion of the screw disc and 
positive over the lower. 

The strength of the virtual wake being in the upper part of the 
disc, where it is positive for outturning screws and negative for in- 
turning, it would seem that side screws well forward of the stern 


post should be outturning in order to make the most of the virtual 
wake due to the obliquity of the water motion. 

30. Strength of Propeller Blades 

In view of the importance of blade thickness in many cases it is 
advisable to make a careful inquiry into the matter and endeavor 
to reduce to rule the stresses upon propeller blades. This can be 
accomplished only by certain assumptions, which will be pointed 
out and justified as they are made. In order to apply the well- 
known formula for beam stress to a propeller blade, it will be assumed 
that the section of a blade by a cylinder at a given radius is devel- 
oped into a plane tangent to the cylinder. This section will then be 
treated as a beam section. This assumption probably errs on the 
safe side, since the actual strength as a beam of the curved blade 
would be greater than that of a developed cylindrical section of the 

i . Fore and Aft Forces and Moments. In considering the 
forces upon a blade it is convenient first to consider separately fore 
and aft forces, or thrust and transverse forces, producing torque. It 
is convenient to use the disc theory or Rankine's theory, by which 
the thrust upon a blade may be taken to vary radially directly as 
the distance from the shaft center. For a ring of water one inch 
thick at ten feet radius, say, would contain twice as much water 
as a ring of the same thickness at five feet radius. If each ring be 
given the same sternward velocity, involving the same thrust per 
pound of water acted upon, then the thrust from the ring of ten 
feet radius would be double that from the ring of five feet radius. 
Put into symbols, if dT denote elementary thrust from a ring of 
thickness dr at radius r, we have dT= krdr where k is a constant 
coefficient over the blade depending upon the total thrust. Then 
integrating we have for thrust, 

T = \ kr\ 

Applying the limits > - , where di is diameter of hub and d is diam- 

2 2 

eter of propeller, we have, if T is total thrust of one blade, 

T k/d* dA 

1 o = 



This enables us to determine k, since from the above 

fa = "" 

Suppose, now, we wish to determine the thrust T\ from the tip to 
a radius r\, 

We have Ti**-\--rA 



4 ' 4 

From the above, if fi = , then J"i = T Q , and if ri = - , then 

2 2 

TI = o as it should. Now we need to know not only the thrust 
on the blade beyond any radius, but its moment at the radius. 
The moment at radius r\ of the elementary thrust dT at radius r 
is dT (r ri) = kr (r r\) dr. 

Call dMi the moment of this elementary thrust. Then 
, s 8 To , 

2 r , 
Upon reduction we have 

At the root section r\ Substituting and reducing, we have 
at the root section, 

A /J , 

6 (tf + 
hrust were 
Then we should have 

Suppose, now, the thrust were concentrated at a point ki - out. 



Whence, equating these two values of MI, we have 

, d di 2 d 2 ddi dj 2 

i ^ 73 I j \ 


Upon reduction this gives us 
, _ 2 

d(d + <f 

The value of k\ in the above formula depends only upon the ratio 
between di, the diameter of hub, and d, the diameter of propeller. 
Numerical values are given below: 

, d 2 d 3 d 4 J 
d\ = 

10 10 10 10 

.689 .713 .743 

These values of ki agree very well with values deduced by entirely 
different methods upon the blade theory or Froude's theory. Upon 
the blade theory ki is nearly constant at .7. 

2. Transverse Forces and Moments. Let us now take up the 
transverse moment, which denote by M 2 . Let dQ denote the ele- 
mentary transverse force in pounds at radius r. Let p denote pitch 
in feet, s the slip ratio and e the efficiency of the elementary portion 
of the blade at radius r. Then the gross work done by the element 
of the blade in one revolution is in foot-pounds dQ X 2 irr. 

The useful work is 

dQ X 2wr X e = dT X p (i - s) = krdrp (i - s). 

dQ kp (i - s} 

Whence -^ = -^ - - 

dr 2 ire 

Now over a blade the quantities on the right in the above equation 
are all constant except e. The variation in e over the part of the 
blade that does the most work is probably not great, so let us assume 

it constant and write -p = g, where g is a constant coefficient to be 



We have seen that dQ X 2 -n-r = element of work done in one revo- 

lution in foot-pounds. Then / d Q 2 ?rr = work per blade per revo- 


lution = 33 -> where PI is the power absorbed by one blade. 



- = / dQ 2 irr = I g 2 TIT dr = Trgr 2 

TT , , ,>. 4 X 33000 Pi I 

= -e (d?- di z ) or s. = ^ ~z rr 


" i 

= 42,017 a __^_ 

Then for Af 2 , the transverse moment at any radius, r\, due to the 
moments of the elementary transverse forces from the tip in to the 

radius, r\, we have 


(r rO dr = g\ 

Then, upon substituting its value for g and reducing, we obtain 

Now as to the radial position of the transverse center of effort we 
have the total transverse force equal to 

The arm of this force beyond r\ is obtained by dividing moment 
by total force and equals 


- - 


_x/rf y 

2\2 / 

The center of transverse effort, then, is by this method halfway be- 
tween the tip and the radius considered. So if k z - denote the dis- 



tance of the center of effort of the whole blade from the center of 
the propeller d\, denoting the root diameter, we have 

2 2 2 \2 2/ 

OF kz= ~ 

This gives us the values below of k 2 for the values of - indicated, 

-=.i .2 .3 4 

2 =-55 -6 .65 .7 

These compare fairly well with values of k z deduced by entirely 
different and more complicated methods upon the blade theory. 
These values of k% varied from .710 for a coarse pitch ratio of 2 
to .600 for a pitch ratio of i. 

Let us now recapitulate the results to this point. 
Let d denote the diameter of the propeller in feet, 

di the diameter of the hub or diameter to root section, 

ri the radius to the point at which we wish to determine thick- 


TO whole thrust of the single blade in pounds, 
PI horse-power absorbed by the single blade, 
R revolutions per minute, 

M i fore and aft bending moment at radius r\ in Ib.-ft. 
Mz transverse bending moment at radius r\ in Ib.-ft. 

Then we have deduced M 1= ^ (d + r % (d ~ 2 Tl} \ 

3 (P-di* 

3. Moments Parallel and Perpendicular to the Sections. These 
moments above are of fixed direction independent of the angle of 
the section. This angle varies with the radius of the section. The 
next step is then obviously, to resolve the above moments parallel 
and perpendicular to the section. For the ordinary screw whose 


driving face is a true helicoid this face develops into a straight line, 
and we will resolve the moments parallel and perpendicular to this 
line. For sections of varying pitch we will resolve parallel to the 
tangent at the center of the face. Figure 238 shows an ordinary 
ogival type of section expanded from its cylindrical shape. Let 
6 denote the pitch angle, or the angle which the face line makes 
with a plane perpendicular to the shaft. Then OB = Mi = fore 
and aft moment . and OA = Af 2 = transverse moment. If M c denote 
the resultant moment perpendicular to the face, we have from 
Fig. 238, 

M c = OC + OD = Mi cos + M z sin 0. 

Similarly, if MI denote the moment parallel to the face, we have 
MI= BD - AC = Mi sin - M z cos 0. 

Now 0, the pitch angle, depends upon the pitch and the radius. 
If p denote the pitch and r\ the radius, we have 

2 irr\ d 2 


Denote the pitch ratio proper, or ^ by a. 



Then tan = 


TTT1 . ad 2 irr\ 

Whence sm = ; cos = = 

4 Tr a 4 TT 

We have seen above that M c = MI cos + M% sin 0. 
Substituting their values obtained above for MI, M 2 , cos and 
sin and reducing the results, we obtain 

(d 2 ri) 2 f2 TT 


Let us next express r\ and d\ as fractions of the diameter d, the main 
dimension. Write r\= and d\= cd. Upon reducing we have 


,, (i m) z [IT , ( v . 5252 aPi~\ 

M c = - - ^-rr , - T md (2 + m) + - r 

(i - c 2 ) Va 2 + Tr 2 ^ 2 L6 R J 


Proceeding in practically the same manner we obtain 

^ad (2 + m) - 5 2 5 2^ 

(i - c 2 )Va 2 + A 2 6 R 

In any particular case of design we will know P\ and R, but 
generally will not know T . The equation connecting TO and PI is 

Since p = ad, this gives 


= er\. 

= 33000 eP l 
adR(i -s) 

Now in practical cases e approximates but is generally somewhat 
less than i j for practical slips, being greater than i s only for 
very high slips. So if we assume e = i s, the result will be to 
make the value of T generally greater than the truth. In other 
words, we shall generally introduce a moderate error on the safe side 

and simplify our expressions enormously. So write T = "4 ^ . 


Also introduce the factor 12 in the expressions for M c and MI, so 
that these moments, heretofore expressed in pound-feet, will be 
expressed in inch-pound units. This is desirable because it is con- 
venient to measure dimensions of the propeller sections in inches. 

o ^ooo P\ 

Then substituting in the expression for M c and MI, T = ^ i , 


and multiplying by 1 2 we have after reduction 

= 63 ' 24 

(i - m} 2 f ~\Pi 

MI= 132,0007 - ^"7 i m 

(i -c*)Va* + ir*m*\- JR 


In the above expressions x is the factor depending upon the work 


done. The complicated fractions involve m the fraction of the 
radius; a, the extreme pitch ratio; and c, the ratio between diameter 


of propeller and diameter of hub. Hence these complicated frac- 
tions can be calculated and plotted once for all. So write 
(i - m)* I" m(2+m) , "I 

c = 63,024 7- ; 2 . /-rr-T-i 3 ' 29 " + a ' 

(I c*-) \/a? _(- ^2^2 L a J 

(i - m 2 } f 

L = 132,000 \ = (i - ). 

(i - c 2 ) Va 2 + 7r 2 w 2 

Then finally, M C =C^, M^L^- 
K K 

Figure 239 shows curves of C and L plotted upon m for various values 
of a. For these curves c was taken uniformly at f . Even if the 
hub has a different diameter, this is generally an amply close approx- 
imation for practical purposes. Since, however, for very large hubs 
a correction may be needed, there is given in Fig. 240 a " Curve of 
Correction Factors " for hub diameters. It is seen that for m = f 
the factor is unity. For smaller hubs the factor is less than unity, 
and for larger hubs greater than unity. Unless, however, the hub 
diameter is one-third of the propeller diameter or more, it is not 
worth while to undertake to correct the regular values of C and L in 
Fig. 239, namely, those for the hub diameter y the propeller diameter. 
4. Resisting Moments of Section. The above expressions en- 
able us, by the use of Fig. 239, to obtain very readily with sufficient 
approximation the longitudinal and transverse bending moments at 
any section of a given propeller of known power and revolutions. It 
is next in order to consider the resistance of the section, using, as 
already stated, the developed section. Referring to Fig. 241, let A B, 
the length of a section in inches, be denoted by /, and CD, its thick- 
ness at the center in inches, be denoted by /. The center of gravity 
will be found on CD at a point G, say. Denote DG by gt, where g 
is a coefficient. Let 7 C , or the moment of inertia about a horizontal 
axis through G, be denoted by k c lP and // or the moment of inertia 
about CD, by kj?t. Then for the type of section above we have 
due to M c : 

Tension at A and B = f-M c = f -j- 

K c lt K c It 

Compression at C = L ^ M c = -:-* 

K c trl 


Due to MI we have, if B is leading edge, 

Tension at A = compression at B= = - MI= 

k/t 2ki Pi 

These are general expressions. The coefficients g, k c and k t depend 
upon the type of section, and / and / are the dimensions. It will 
be well, then, to consider the range of values of the coefficients g, 
k c and k L for various possible types of section. The most usual type 
of section is the ogival, where AB is a straight line and the curve 
ABC the arc of a circle. This type of section, however, is difficult 
to reduce to rule, the coefficients varying with the proportions. The 
ogival section, however, is practically the same as a section with a 
parabolic back, so the latter may be considered. 

In addition to the parabolic back as representing the ordinary 
type of blade section we will consider two other types of blade of 
the same maximum thickness. In one the parabola is replaced by 
a curve of sines. In the other, thickness is equally distributed be- 
tween face and back, each being a curve of sines. Figure 242 shows 
the three types of blade section, and below each are given the equa- 
tion characterizing it, the expressions giving the area in terms of 
length and thickness, and the value of the coefficients for it. 
Then for the three types of blade section we have with sufficient 

No . i . 

No. 2. 

No. 3. 

Maximum tension at 




Expression for maximum ten- 

_g M c I MI 
k c it 2 2 ki fit 

g M c I Ml 
k c It 2 2 kl Pt 

g Me 

k, nt 

Maximum compression at 
Expression for maximum com- 


i - g M c 

I - g M c 

I - g M c 


k Ift 

t HI 

b ]ft 

5. Compressive Stresses. It is not obvious whether for types 
i and 2 the maximum tension is greater or less than the maximum 
compression. It is found, however, upon investigation of blades as 
they are found in practice, that the compression stress is the greatest 


and the only one that need be considered in the case of material 
that is as strong in tension as in compression. 

It would seem, then, to be the best plan in practice to design the 
blade thickness from considerations of compression and then deter- 
mine tension of the blade thus designed. In the rare cases where 
the tension is found too high it is easy to make the necessary 
changes. The formula for compression at the center of the back of 
the blade in pounds per square inch is, 

Compression = -~ ^ 
k c It 


Now M c = C - , and it is seen from Fig. 242 that for all these types 


of blade a safe value for ^ is 14. Then our final formula is, 

k c 

Maximum Compression at Center of Back in Pounds per Square 

P i 
Inch, = 14 C X ^ ) where C is obtained from Fig. 239. 

/V 1 1 

We are now in a position to investigate the stress, not only at a 
root section, but at any point along the radius, by the aid of the 
above formula and Fig. 239. The result for a blade of rather wide 
tips and a mean width ratio of .2 is shown in Fig. 243. This shows 
for various pitch ratios, and plotted on fractions of radius, curves of 
thickness in center for constant compressive stress, the thickness 
being expressed always as a fraction of the thickness at .2 the radius. 
Beyond .2 of the radius these curves are so close together for the 
various pitch ratios that it is impossible to plot them separately. 
Below .2 of the radius the curves separate. It is seen that the outer 
portion of the thickness curve in Fig. 243 is not quite straight, being 
slightly curved. The curvature is so slight, however, that if we 
follow the nearly universal practice of making the back of the blade 
straight radially, the thickness at the tip being not zero but the 
minimum that can be conveniently cast, the stress per square inch 
will be practically constant. Unfortunately it is clearly unsafe to 
make the line of the blade back concave as we go out, thus decreas- 
ing thickness and gaining efficiency for high speed propellers. It is 
true that sometimes the line of blade back is made concave when the 


blade has a small hub and is narrow close to the hub, but this is due 
to a thickening of the inner part of the blade not a thinning of 
the outer part. The only practicable method, then, of accom- 
plishing reduction of blade thickness is to use material capable of 
standing high stress. 

Figure 243 indicates that for propellers with small hubs less 
than .2 the diameter the thickness should be determined, not at 
the root, but at .2 the radius, the straight line of back being 
extended inward to the hub. 

For convenience in design work Fig. 244 has been prepared. This 
gives values of C, from .2 the radius to .4 the radius for pitch ratios, 
from .8 to 2.0, thus covering the practical field. 

6. Tensile Stresses. Coming back now to the question of ten- 
sion, it seems that sections of Type 3 are the simplest. The maxi- 
mum tension for it is the same as the maximum compression. But 
sections of Type 3 are not desirable for use. For sections of Types 
i and 2 the case is not so simple. Taking the maximum tension as 
that at A and the maximum compression as that at C, and denoting 
by /i the tension factor or value of maximum tension -i- maximum 
compression, we have 

g M c i MJ 

k if 2 k n 

Now ~ = , and with sufficient approximation we have from 
M c C 

Fig. 242 g =.4 and ^ = .7 1 k c . 

Whence, after simplifying, /i = .666 + 1.17 - - L and C are given 

c/ i> 

in Fig. 239, but to facilitate computation Fig. 245 gives curves of 
1.17 from m = .1 to m = .4, and for final pitch ratios from .6 to 2. 


This covers the practical ground. For narrow cast-iron blades with 
solid and hence small hubs it will generally be necessary to determine 
tensile stress with care. 

7. Stresses Due to Centrifugal Force. In addition to the 
stresses upon a propeller blade due to thrust and torque, there are 


stresses due to centrifugal force. These are appreciable. In any 
given case they can be calculated with sufficient approximation with- 
out serious difficulty. If W denote the weight of that portion of a 
propeller blade outside of the radius r\ of a given section, r 2 the 
radius of the center of gravity of the portion of blade and v the 
circumferential velocity of the center of gravity, while g, as usual, 
denotes the acceleration due to gravity, then the centrifugal force 
of the portion of blade may be taken as equivalent to a single 
force perpendicular to the shaft through the center of gravity of 
the portion of blade. The amount of the force in pounds will be 

W V 2 

equal to Knowing the force and its line of application, the 

g r z 

stresses upon the bounding section of the portion of blade can be 
determined by well-known methods of applied mechanics. 

It appears advisable, however, to make a general mathematical 
investigation of a case sufficiently simple to admit of such investi- 
gation and sufficiently resembling the cases of actual propellers to 
enable us to apply the results of the mathematical investigation, in 
a qualitative way at least, to actual propellers. It will be seen that 
we can thus learn a good deal about the laws governing the stresses 
of propeller blades caused by centrifugal action. 

Fig. 246 shows an elliptical expanded blade touching the axis at O. 
Consider the weight of each section such as CD concentrated at 
the blade center line at E. Let bd denote the minor axis BN, d 
being the propeller diameter and b a fraction. The equation of the 
ellipse referred to the point where it touches the axis, is 

y b V2 dr 4 r 2 , 

where r denotes radius and y the semibreadth at radius r, 
Then Breadth = 2 y = 2 b v/2 dr 4 r 2 . 

Now for r substitute m - where m is fraction of whole radius varying 

from o at to i at A . 

Then Breadth = 2 bd Vm - m z . 

When we come to thickness, the axial thickness is rd, where T is 
blade thickness fraction. The tip thickness is not fixed by consid- 


erations of strength, being from considerations of castings, etc. ? 
usually materially thicker than it need be for strength. We wish in 
considering centrifugal force to be sure we take the tip thick enough, 
so will assume it as .15 the axial thickness. It will usually be less 
in practice for large propellers. Then the back of the blade center 
being a straight line, the thickness at m is rd ( i .85 m). 

Assuming the section as parabolic, the area of a section = 
width X thickness = X 2 bd Vm m?X rd (i -.85 m} 

(i .85 m) v ' m m 2 . 

We are now able to formulate the elements of curves to be plotted 
upon m and integrated graphically to obtain the results needed. 
The element of blade volume = Area of section X dr. 

-NT m d j d j 

Now r = ? dr = - dm. 

2 2 


Element of volume = - (i .85 w) \/m m 2 dm. 


Let 8 denote weight per cubic foot of the material of the blade 
Element of weight = - - (i .85) m) \/m m 2 dm. 


n o T 

Element of centrifugal force = X weight = - - X weight 

g *g 

, N A / , j 
m (i .85 ?w) v w mr dm. 

\J O 

f*m _ 

Let / m(i .85 m) \/m m 2 dm ^>\(m). 

J \ 

Then total centrifugal force from the tip to the section m is 

u> 2 8rbd 4 , / x 


If there is no rake the effect of the centrifugal force is simply to 
cause a tension over the area. This tension = 



3 g 4 rbd 2 (i .85 m) "vm m 2 

in pounds per square foot. 

4 g (i .85 m) Vm m 


Expressed in pounds per square inch it is T ] of the stress in pounds 
per square foot. So we have Tension in pounds per square inch 
due to centrifugal force when there is no rake 

(m) u 2 8d? , . N 

. . 

576 g(i -.S$m)Vm - m z 57 

It appears, then, that for a blade without rake the tensile stress due 
to centrifugal force varies as the weight per cubic foot of the blade 
material, as (cod) 2 , or as the square of the tip velocity, and as </> 2 (m) 
where < 2 (#0 is a quantity depending upon radial position, blade 
shape, proportions, etc., but independent of size and pitch. 

Since it is usually more convenient to express angular velocity by 

the revolutions per minute, denoted by R, we may substitute 


for co. Also, in order to avoid small decimal factors, multiply numer- 
ator and denominator by 1,000,000. Then Tension in pounds per 
square inch due to centrifugal force when there is no rake 

22 8d 2 . , loooooo . f v 
X - - X - - 02 O), 

3600 57^ g loooooo 
8d 2 R 2 [4000000 TT 

ioooooo_ 3600 X 576 g 

8d?R 2 f 4000000?^ 0i (m) 

1000000 [3600 X 576 g ( x _ .85 m) Vm - m 

I 000000 

Figure 247 shows curves of <i(w) and (j> t . It is seen that <i(w), 
which is proportional to total centrifugal force, increases always from 
tip to axis, as might be expected. Since the assumed blade has no 
area at the axis, (f> t , which is proportional to the stress per square inch, 
is infinity at the axis but falls off very rapidly at first as we go out. 

We wish mainly, however, to investigate the effect of rake or 
inclination upon the stresses on propeller blades due to centrifugal 
action. Let id denote the total rake of the blade along its center 
line, where i is a comparatively small fraction, and assume the weight 
of the section concentrated at the center line. Then idm denotes 
the rake from the axis to the radius corresponding to m. 


Suppose we wish to determine the moment due to centrifugal 
force about the section corresponding to mi. 

The element of force at m beyond mi is, as before, 

m (i .85 m) vm m 2 dm. 

Its lever to radius mi is id (m mi). Hence element of moment 
Abid 5 

O o 

( N ( >. / o , 

(m mi) m(i .85 w) v w m- dm. 

,, a> 2 3rta/ 5 r r m ' ,, v / - = , 

Moment to mi = - m z (i .S^m)\ / m m* dm 

3 LJi 

r\m\ _ ~| 

mil m (i .85 m)Vm m 2 dm 

The second integral is <i(w), but we can denote the whole thing 
by </>a (mi) and after obtaining results by graphic integration use m 
instead of m\. Then we have 

Moment from tip to m = - </> 3 (m) = M f , say. 


O o 

Now the moment MI is in the plane through the axis and the 
center line of the blade. Its effect upon the section is best ascer- 
tained by resolving it parallel and perpendicular to the section. 

If 6 be the pitch angle at radius r, tan 6 = -* = *. = - , if 

2 irr Trmd irm 

we use a to denote the pitch ratio * 


Then sin 6 = cos 6 = 

If Me' and ML denote the moments resolved perpendicular and 
parallel to the blade face we have 

nf r *f a , / x 

Me = M cos 6 = - 7T03 (m) 

ML' = M ' sin 6 = - < 3 (m) _. 

Finally, by applying at the center of the section forces equal and 
opposite to the forces producing the moments, we have the section 


affected by a force and two couples. The force is the same as the 
outward force when there is no rake. The couples are perpendicu- 
lar and parallel to the section and their moments are given above. 
The result of the force and couples is as follows, reference being 
had to Fig. 241, where B is the leading edge: 

1. The force causes a certain tension over the whole section. 

2. The perpendicular couple causes compression at C and ten- 
sion at A and B. 

3. The parallel couple causes tension at A and compression at B. 
Now from consideration of thrust and torque only we have already 

found that the maximum compression is at C and the maximum 
tension at A . Centrifugal action evidently increases the tension at 
A more than at B. Hence, as regards tension we need consider the 
action at A only. 

As regards compression, when we neglect centrifugal action this 
is a maximum at C. The tension due to the force decreases com- 
pression at B and C equal amounts. Then the parallel moment 
increases compression at B and the perpendicular moment increases 
compression at C. We need to find which increase is the greater, 
and if C has greater compression from centrifugal action we need 
consider C only. 

The necessary coefficients for the parabolic sections are found in 
Fig. 242. Consider the tension increases at A first. We have three 

Due to force alone in pounds per square inch, fa (m). 

576 g 

Due to perpendicular moment, -^ -r-^- , where the factor 12 


has been introduced because we wish stresses per square inch and 
Me was calculated in pound-foot units. 

Now in feet / = 2 bd ^m m?= 24 bd \/m m z in inches. 

Also t = rd (i .85 m) in feet = 12 rd (i .85 m) in inches. 
So the tension per square inch at A due to the perpendicular mo- 
ment is 

105 Mr' 

24 X 144 b^d? Vm m 2 (i .85 w) 2 


Substituting the value of Me' 

Tension at A due to perpendicular moment 
__ 35 Tru?dTbid 5 <j> 3 (Tn) X m 

1152 X 3 gbrW Vm-m 2 X (i -.85 m) 2 V a 2 + 
35 TT i _ uPdcPfa (m) X m __ 
3456 g T V m - m 2 (i - .85 m) 2 V a 2 + 
Due to parallel moment 

rr 15 X 12 ML' 

Tension at A = -* - -- 

1 1 

Reducing this similarly we have 
Tension at A due to parallel moment 

576 g b ( m - W 2) ( z _ g 5 

Suppose now we denote by N the tension per square inch due 
to centrifugal force only and express these other tensions in terms 
of N: 
We have 

o w 
576 g 57 6 g ( J - -85 ) ( - 

Then Tension at A due to perpendicular moment 

6 r ^(w) (! -. 

= -^-^> 4 . 
3 * 

Tension at A due to parallel moment 

,, i 03 (m) _ a __ j\j- 
~ 5 b fa (m) ( m - ~ ' 

In the above ^ and 0s involve a as well as m and should be expressed 
by contour diagrams. 

Consider now the compression at C due to the perpendicular 

,, . 13.121; X 12 X Me ir i 

moment. This is -^ s2 -- As before in inches 


I = 24 bd (m w 2 )*, 
t = i2rd(i .85 m). 


Hence compression 

13.125 X 12 Me 

24 X 144 X br 2 d 3 (m - w 2 ) 4 (i -.85 m)' 2 

13.125 jr u?8rbid 5 <f)3 (m) m 

288 3 g br 2 d? (m - w 2 )* (i - .85 m)*Va*+v*ni< 
26.25 E2 i 03 (m) m 

576 g 3 r ( m - W 2)i (i _ .85 w) 2 vV+ ir 2 2 

And in terms of N 
Compression at C 


() (i -.{ 

We can now express the ratios between extra compression at C 
and compression at B due to parallel moment. The latter is the 
same as tension at A due to parallel moment 


(m) ( m - 
Extra Compression at C 

Parallel moment Compression at B 15 T a i.&$m 

Now we may safely say that in practice b is greater than 47. 
If we put b = 4 T, TT = V > we nave f r above ratio, 

m (m w 2 )* 
22 * -' 
a i .85 m 

The hub is such that m may be taken as .2 or more. Putting 


w =.2 we have ratio above = So for propellers in practice 


the extra compression at C due to centrifugal action will always be 
greater than that at B due to the parallel moment. When, too, we 
recollect that there is a large opposing tension at B due to the per- 
pendicular moment, it is obvious that the maximum compression is 
at C, and only that need be considered. 

Figures 248 and 249 show contours of </> 4 and < 5 plotted on a and 
m and curves of $i(w), <& and 4>z(m] which involve m only are 
shown in Fig. 247. 

We have finally for stresses due to centrifugal forces 


Tension in pounds per square inch neglecting rake = N 


Extra compression at center of blade back = AM- < 4 l] 

dd?R 2 ^ (i ^ \. 
= <p t [- 94 i ] m pounds per square inch. 

1000000 \T I 

(2 i i \ 
9^ + 7 95 + I ) 
3 T * I 

dd?R z /2 i , . i, \. . , 

= 9, 9<+ r 95+ 1 m pounds per square inch. 

1000000 \3 T b I 

In the above i is ratio between rake and diameter, T is ratio 
between axial thickness and diameter, and b is ratio between maxi- 
mum blade width and diameter and may be taken as 1.188 (mean 
width ratio). 

The above formulae and the accompanying figures apply strictly 
only to blades whose expansion is an ellipse touching the axis and 
whose tip thickness is .15 the axial thickness. 

The methods used can be followed to determine 9i(w), (f> t ^>z(m), 
4>4 and < 5 for blades of any type, but the results of Figs. 247, 248 
and 249 can be applied in practice with sufficient approximation 
to any oval blade that does not depart widely from the elliptical 

Since centrifugal stresses increase as the square of the tip speed, 
they evidently need to be given much more careful consideration 
for quick running propellers than for those of moderate speed. Thus, 
suppose we had a manganese bronze propeller for which dR = 4000, 
or the tip speed is over 12,000 feet per minute. For manganese 
bronze 5 = 525 about. Then N = 525 X 169^ = 84009^. For 
m = .3, <f>i= .135 about, soN = 1134. If the pitch ratio is about 


unity, 94 = 2 \ about, and if - has the value of 3 or the rake is three 


times the axial thickness, -941 = 6, or increase in compressive 


stress at .3 the radius is the large amount of 6700 Ibs. This is an 
extreme but not impossible case. As tip speed falls off, stresses due 
to centrifugal force decrease rapidly, but it would seem the part of 
wisdom to avoid them entirely by avoiding backward rake. More- 


aver, it seems advisable when tip speed is very high to give a moderate 
forward or negative rake, thus opposing the tensile and compressive 
stresses due to the work done by opposite stresses due to the cen- 
trifugal forces. When backing, centrifugal force would add to the 
natural stresses, but propellers are not worked backward at maxi- 
mum speed. 

In calculating stresses due to centrifugal force we need values of 
5 or weight per cubic foot of the various materials used for pro- 
peller blades. For manganese bronze or composition we may use 
525 for 5, for cast iron 450 and for cast steel 475. 

8. Stresses Allowable in Practice. While for quick-running 
propellers centrifugal stresses must be calculated separately, in the 
majority of cases they are not very serious and may be allowed for 
by using a low stress in our main strength formulae. 

P i 
Compressive stress in Ibs. per sq. in. = S c = 14 C 1 X %' 

1\ LI 

Tensile stress in Ibs. per sq. in. = ST = S c (.666 + 1.17) - - 

\^ L 

In applying these formulae to the root section of any blade we will 
know C, PI, R and /. Then we fix t by giving S c a suitable value 
and calculate ST to see if that has a suitable value. Now what 
are suitable values of S c for the various materials of which we make 
propeller blades ? They cannot be fixed arbitrarily from considera- 
tion of only the tensile and compressive strengths of the material. 
For one thing our formulae are approximations only. In order to 
apply the methods of Applied Mechanics we start by developing the 
cylindrical section of the blade into an ideal plane section. It is 
probable that this ideal section is materially weaker than the actual 
section, especially in the case of propellers of varying pitch. Hence, 
if this were the only perturbing factor, we could allow high stresses 
in the formulae, because the stresses per formulae would be greater 
than the true stresses. But when we consider the conditions of 
operation of propellers we find other very serious perturbing factors 
which we cannot reduce to rule. In the formula, PI is the average 
power absorbed by the blade. But even in still water the blade, 
owing to inequalities of wake, will absorb more power than the 


average at one portion of the revolution and less at another. And 
in disturbed water, what with the motion of the water and the 
pitching of the ship, the blade is liable to encounter stresses very 
much in excess of those due the average power which it absorbs. 
This is especially likely to be true of turbine driven propellers. 
With reciprocating engines, when a propeller encounters abnormal 
resistance the engine will soon slow down, the kinetic energy of the 
moving parts being rapidly absorbed. With turbines, however, we 
are likely to have the kinetic energy of the moving parts per square 
foot of disc area much greater than for reciprocating engines, and 
the flywheel action, so to speak, of the moving parts is then capable 
of causing a relatively greater extra stress. 

To determine with scientific accuracy allowable stresses for use 
in the formula we would probably have to test to destruction full- 
sized propellers which is impracticable. The next best thing is 
to find from the formula the stresses shown by actual propellers 
which have been successful in service, and also those of propellers 
which have shown weakness in service. We can thus establish, with 
sufficient accuracy for practical purposes, the maximum stresses that 
can be tolerated. The advantage in this connection of a formula 
upon a sound theoretical basis is that a stress found satisfactory for 
a fine-pitched, quick-running propeller, for instance, will be almost 
equally satisfactory for a coarse-pitched propeller, and vice versa, so 
that satisfactory allowable stresses can be deduced from less data 
than would be necessary for a formula partaking largely of the rule 
of thumb nature. 

There are advantages in the use of a simple semi-graphic method 
which will enable data from completed vessels to be recorded for 
use in design work. 

We have deduced as the final formula for S c the compressive 
stress in pounds per square inch for blades of the usual ogival section 

where C is a coefficient depending on radius and pitch ratio, PI is 
the power absorbed by the blade, R denotes revolutions per minute 
of the propeller and I and t are width and thickness respectively of 


the blade in inches. Also we should generally use in determining 
S c the values of C, I and t at about .2 the radius of the propeller. 
Let us now express I and t in terms of coefficients and ratios already 

Put / = 1 2 chd where d is diameter in feet, h is mean width ratio 
and c is a coefficient depending upon the shape of the blade. 

It is not such a simple matter to determine a rigorous expression 
for /, because the tip thickness is more or less independent of the 
root thickness. 

If rd denote axial thickness as usual, and krd the tip thickness, 
we have for .2 the radius 

t = 12 rd [k +.8 (i - k)]= 12 rd (.8 +.2 k}. 

In practice k is seldom much less than .1 or greater than .2 
Now k = o, t = 9.6 rd, k =.i, t = g.&4.rd, k =.2, t = 10.08 rd. 
So it is a sufficient approximation for practical purposes to put 

t = 10 rd. 
So, returning to the stress formula, we have 

C* /"* ^ v/ 

O c = 14 C X 

14 C PI i 

12 chd IOOr 2 J 2 I2OO 


Let Ci= - Figure 250 shows plotted upon pitch ratio a curve 

1 2OO 

of Ci for .2 the radius. 

Then S c = ^~ X ~^- 


Suppose, now, we put = #, chr z = y: 

then we have 5 C = - 


Figure 251 shows contours of values of S c plotted on x and y. In 
the case of a given propeller we know or can readily calculate chr 2 and 


Hence, we can locate a spot on Fig. 251 corresponding to 


the propeller which will show the root compression or value of S c in 
pounds per square inch. Figure 251 shows by crosses a number of 
spots each of which corresponds to an actual propeller. They are 


nearly all for vessels of war, and all for manganese bronze or other 
strong alloy. It is desirable, when using the method for design work, 
to reproduce Fig. 251 on a large scale. It is evident from Fig. 251 
that the designers of the propellers referred to differed widely as to 
the allowable stress. No. n refers to a destroyer which would 
very seldom develop maximum power, and then only in smooth 
water. But even for such vessels it is not advisable to go to such 
stresses. No. 14 was a vessel which much exceeded her designed 
power, on trial, and also sprung her propeller blades. With man- 
ganese bronze and similar alloys now available it is inadvisable to 
exceed 15,000 Ibs. even for destroyers. For other fast men-of-war 
which seldom develop full power, suitable stresses, based upon full 
power, are 10,000 to 12,000 pounds per square inch. For merchant 
vessels, always at nearly full speed, particularly passenger steamers 
that are driven hard in rough weather, it is not advisable to exceed 
5000 to 6000 Ibs. The above all refer to blades of manganese bronze 
and similar alloys. Good cast-steel propellers can be given the 
same stresses as those of manganese bronze. 

For cast iron it is advisable not to exceed 5000 Ibs. for compression 
and 2000 Ibs. for tension. 

As already stated, designers differ widely as to the proper stresses 
to allow for propeller blades. It is a simple matter for any designer 
with an accumulation of data for actual propellers to record it on 
a large diagram similar to Fig. 251 and form his own conclusions 
as to the stresses which he will allow in a particular case. 

While it is desirable for a designer fully to understand all de- 
tails involved in determining propeller blade thickness, it may be 
pointed out that when centrifugal forces are not serious, and the 
blade thickness is to be fixed from considerations of compressive 
stress only, Figs. 250 and 251 are all that need be consulted. 
For when number of blades, diameter and pitch have been deter- 
mined we can determine P\, R and d. C\ can be taken from Fig. 

C P 
250, so we will know From the blade outline we can deter- 

mine h and c, the latter usually falling between .6 and .8 in practice. 
Thus in a practical case, after having calculated ch we need only 
to determine r. 


C P 
So we enter Fig. 251 with the value of * * and from the stress 

chosen determine c^r 2 , and c/s being known r 2 and T are readily 

9. Connections of Detachable Blades. While somewhat apart 
from the question of strength of propeller blades it seems advisable 
to consider briefly the question of the strength of the connections 
of detachable blades. We have seen that the formulae for trans- 
verse and fore and aft moments in pound-feet are: 

ff n. To (d+R l )(d-2r l }* 

Fore and aft moment M\= . - 

3 d?-d? 

Transverse moment M 2 5252 - ^= "Tjj 

Also with a margin for safety we may write T Q = 


Making this substitution and multiplying by 1 2 to reduce moments 
to inch-pounds, we have: 

P l (d 

= 132,000 

(d - 2 

Now with sufficient approximation we may write d\ d. 
Also we may take r\ or the radius to hub flange to which the blade 
is bolted, as \ the propeller radius with a slight error on the safe 
side. Substituting and reducing, we have in round numbers 

,, 116000 PI ,, PI 

M!=- , M 2 = 52,400 

d K J\. 

These two moments may be compounded into a single moment 
whose direction makes with the direction of the shaft axis 

tan ~ l ^-^ - and whose amount in inch-pound units is 
1 1 60 

p i . It NO , /n6oV P! 

Y/ (524) 2 +(^-yj =H say. 


The amount and angle of the moment depend upon - and a only. 



Figure 252 shows plotted upon a, or extreme pitch ratio, curves 
of values of the angle of inclination of the moment and of the 
coefficient H. 

The moment above must be resisted by the bolts securing the 
blade flange to the hub and the flange itself. The bolts are, of 
course, disposed on each side of the direction of the moment, and 
it is good practice to use more bolts for the side where the bolts 
are in tension when going ahead. Thus, if there are nine bolts in 
all, five will be in tension when going ahead and four in tension 
when backing. 

Theoretically, the blade flange will pivot under stress about some 
point on its extreme circumference and the leverage of each bolt 
will be the length of a perpendicular from its center to a line drawn 
through the pivoting point tangent to the circumference. 

For a conventional assumption, however, which is an adequate 
approximation, v^e may take the effective leverage of each bolt in 
tension as the diameter of the circle through the center of the 

Investigation of actual propellers upon this basis indicates 3000 
pounds per square inch as a fair average of the stresses allowed on 
steel flange bolts by designers, the actual stresses varying from less 
than 2000 pounds to some 4000 pounds. 

Even after making all allowances for the conditions of service it 
would seem that 3000 pounds per square inch is a low stress for 
such bolts and that 4000 pounds or more might be used without 

For quick running propellers the stress taken account of should 
include that due to centrifugal force upon the blade. The expres- 
sion for force in pounds is 

and for moment in pound-feet, 

The moment may be taken as parallel to the shaft axis. It is seen 
from Fig. 247 that we may, with fair approximation, use .09 for 


<i(w) and .04 for ^>a(w). Substituting these values, and putting 

2 irR 

g = 32.16 and co = , we have: 

Force in pounds = 


Moment in pound-feet = 

31. Design of Propellers 

i. Number and Location. Nearly all the matters of detail 
involved in propeller design have been already considered, but it is 
proposed briefly to review the general considerations involved, and 
illustrate the methods already explained by working out a few ex- 
amples. The question of the number and location of propellers is 
not very often an open one at any stage of the design, being usually 
fixed by practical or other considerations which have little to do 
directly with propeller efficiency. From the point of view of pro- 
peller efficiency only, the best location for a propeller is in the center 
line, as far aft as possible. In the center line it gets the maximum 
benefit from the wake and the farther aft it is the less the thrust 
deduction. Practical considerations of protection from damage re- 
quire the screw to be forward of the rudder, but a suitable arrange- 
ment by which the screw was located abaft the rudder, so that its 
suction would not produce appreciable thrust deduction, would un- 
doubtedly increase efficiency of propulsion. Since, however, suc- 
tion will have no retarding effect upon a fore and aft plane, about 
the most that can be done in practice to reduce thrust deduction 
upon a single screw vessel is to make the after portion as fine as 
possible. In many cases there might be more done in this direction 
than is done. Fineness at the water surface is what is needed. 

As to vertical location, it is the usual practice to locate screws 
as low as possible. For seagoing ships this is desirable to reduce 
racing, and even for ships intended for smooth water service only, 
it is generally necessary, because such vessels are usually of shallow 
draft, and to get the propeller sufficiently beneath the water sur- 
face it must be placed low. But propellers are not placed so low 
that their tips project below the keel if this can be avoided. 


This is simply to reduce risk of damage in case of grounding, and 
in some cases it is necessary to ignore this risk and allow the 
propeller tips to go below the keel. 

There is little doubt, that contrary to what is generally supposed, 
a propeller for smooth water work is more efficient the closer it is 
to the surface, provided it is not so close that it draws air from the 
surface. This, for the reason that in this position it gets the greatest 
useful reaction from the wake. Frictional, wave, and stream line 
wakes are all strongest near the surface. 

One is apt to conceive of the frictional wake as a vertical belt of 
nearly uniform horizontal thickness. But an examination of Figs. 
50 to 59, and careful observations of actual ships, would seem to 
indicate that the frictional wake abreast the stern widens rapidly 
as we approach the surface, and in fact we may almost regard the 
wake as made up of a vertical layer close to the ship and a horizon- 
tal layer extending out some distance from the ship, but not extend- 
ing deeply into the water. The higher a center line propeller is the 
more it gains from the vertical layer, and if it is high enough to reach 
the horizontal layer it gains still more. But as already pointed out, 
it is necessary to give a good submergence to the screw of a sea- 
going vessel to avoid racing in a seaway. A broken shaft is too 
serious a matter to be risked in order to secure slightly greater pro- 
pulsive efficiency in smooth water. Furthermore, in rough water a 
deeply submerged screw which does not race will have much higher 
propulsive efficiency than one close to the surface that is racing con- 
stantly. So in practice we usually find screws of seagoing vessels 
immersed as deeply as practicable. 

The best location for a side propeller is probably the nearest loca- 
tion practicable to the best location for a center line propeller. 
Where twin screws are fitted they would, under this rule, be placed 
as far aft as possible and as close to the center line as possible. 

It must be said, however, that the fore and aft location of a side 
screw appears to have surprisingly little effect upon its efficiency. 
We saw in considering actual and virtual shaft deviations that for 
a four-screw vessel the after pair were about as badly off in this 
respect as the forward pair. We would expect, however, a priori, 
that a side screw well forward would usually have greater virtual 


shaft deviation than one well aft, and would also gain less from the 
wake and have a greater thrust deduction. 

It is undesirable to place screws so that their tips are too close 
to the surface of the hull. When a screw tip strikes the belt of 
eddying water adjacent to the hull, the virtual blows resulting are 
communicated to the ship, shaking rivets loose and causing vibra- 
tion. The irregular forces upon the propellers also cause vibration 
of the ship. 

In some twin-screw ships this trouble has been partially avoided 
by leaving an opening in the dead wood abreast the propellers. 
This saves the ship, and with large propellers of moderate speed of 
revolution the tips can be brought quite close to one another with- 
out giving trouble. For small, quick-turning propellers, such as those 
fitted with turbines, vibrations are very likely to be set up unless the 
blade tips are kept well clear of the hull, say 30 inches to 36 inches. 
It seems a pity to lose any of the beneficial action of the wake, and it 
is possible that if the hull abreast the propeller tip were made of cir- 
cular shape, with the shaft as a center, specially strengthened to stand 
the pounding, and the propeller tips fitted close to the hull so that 
they caught the dead water through a large arc, the beneficial effect 
of the wake might be had without very objectionable vibration, 
though such propellers would probably be noisy. That is a matter, 
however, which could be determined only by a full-sized trial. The 
only solution now known to be successful is to keep the blade tips 
well clear and accept the slightly reduced efficiency. 

When triple screws are fitted, it is obviously desirable that the 
races from the side screws should almost or entirely clear the disc 
of the center screw. This result is best attained when the side 
screws are forward of and above the center screw. 

For a side screw located well forward the question of virtual 
deviation due to the water rising up and closing in aft is frequently 
given less attention than it should receive, resulting in loss of 
efficiency and vibration from the screws. 

When four screws are fitted the after pair are located in the 
natural location of twin screws, and the forward pair are placed 
forward and higher so as to avoid interference as far as possible. 
These forward screws, if badly placed, are liable to serious virtual 


shaft deviations, and the questions of their location, shaft angles, 
etc., should receive most careful consideration. They may, from 
their high location, get a better reaction from the wake, and hence 
not lose in propulsive efficiency as compared with the after screws. 

The number of screws depends upon various considerations. If 
there is no limit to diameter and revolutions, there is no question 
that the single screw should be the most efficient. There is prob- 
ably not much to choose between twin and triple screws as regards 
propulsive efficiency. Quadruple screws are likely to be somewhat 
the least efficient as regards location. In practice, however, in a 
given case, diameter and revolutions are not unrestricted, and the 
number of screws is apt to be fixed from other considerations than 
those of slight differences of efficiency due to number of screws. 

Twin screws were adopted for men-of-war primarily to secure 
greater immunity from complete breakdown, greater protection of 
screws and engines on account of smaller size, and ability to do 
some maneuvering independent of the rudder. The same considera- 
tions influenced the adoption of twin screws for high-class passenger 
vessels, but another consideration came in here. With the very 
great powers used for such vessels the engines or shafts became 
too large with single screws. This consideration has also largely 
influenced the adoption of triple and quadruple screws. 

With the advent of the turbine the question of revolutions 
already of importance in fixing the number of screws for quick- 
running engines became a very important one. 

For steam economy and weight saving the turbine should use 
high revolutions. But a propeller which absorbs great power at 
high revolutions must be given so much diameter in proportion to its 
pitch that its efficiency becomes too small. Hence, with turbines 
we usually find three or four shafts. In the early days of turbines 
multiple screws were often fitted two or three on each shaft. 
This practice has now been abandoned, however, as a result of 
experience, the present practice being to fit but one screw on each 

While in many cases with turbines it is desirable for the best 
economy to use three screws, it is rather difficult with three screws 
to secure satisfactory arrangements for the rudder post and rudder. 


Still it is possible to do this, and three screws are used until ques- 
tions of economy or size of units drive us to the use of four screws. 

2. Direction of Rotation. Obviously, when we have a center 
line screw it will give the same efficiency whether it is right-handed 
or left-handed. Hence the direction of rotation of single screws and 
of the center screw of triple screws is immaterial. The desirable 
direction of rotation of side screws depends upon considerations of 
water flow and shaft obliquity already discussed in detail. 

For ships as they are, in the vast majority of cases, it seems 
probable that side screws would be slightly more efficient if outward 
turning. For side screws very far aft, with shafts supported by 
struts, so that the fittings for carrying the shafts do not interfere 
with the natural water flow, it matters little as regards efficiency 
whether the screws be in or out turning. With shaft webs approach- 
ing the horizontal, the side screws should be outturning for effi- 
ciency. With shaft webs approaching the vertical, they would be 
more efficient if inturning. Such shaft webs are, however, prac- 
tically unknown. Side screws materially forward of the stern, how- 
ever their shafts are supported, should turn outward for the best 

As regards efficiency, then, in about all practical cases side screws 
should be outturning. For maneuvering by means of the screws 
alone, when a vessel has not steerage way, outturning screws are 
distinctly preferable for practically all types of vessesl. For many 
vessels this consideration alone would outweigh minor difference 
of efficiency, but as outturning screws have the advantage as re- 
gards efficiency in nearly all practical cases, they should be adopted 
in the vast majority of cases. Cases may occur where it is a matter 
of indifference, and cases are conceivable where, as with vertical 
shaft webs, inturning screws are more efficient, but outturning 
screws should be the rule and inturning screws should be fitted 
only for good and sufficient reasons, which in practice will exist very 
seldom indeed. 

3. Number of Blades. When the number and location of pro- 
pellers are settled and it becomes necessary to get out finally the 
design of the propeller, we will know the power which it is expected 
to absorb and the revolutions it is to make. The speed of the ship 


will be known, and we can estimate the wake factor and thus deter- 
mine the speed of advance. About the first thing to be settled is the 
number of blades. Two-bladed propellers are hardly worth consid- 
ering for jobs of any size. Figure 217 indicates that appreciable gain 
in efficiency is not to be expected from them, and they are distinctly 
inferior as regards uniformity of turning moment and vibration. 

So, in practice, the choice will lie between three blades and 
four blades. Model experiments of a comparative nature appear to 
indicate that three-bladed propellers are essentially more efficient 
than four-bladed. 

It is seen from Fig. 216, however, which probably exaggerates, 
if anything, the inferiority of four-bladed propellers that this inferi- 
ority is small, and it may well happen in practice that a four-bladed 
propeller exactly adapted to the conditions will be superior to a 
three-bladed propeller not so well designed. 

Many designers are firm believers in the superiority of the four- 
bladed screw as well as many sea-going engineers. Probably in 
rough water the four-bladed screw will show a slightly more uniform 
turning moment and less tendency to produce vibration. But some 
of the fastest Atlantic liners that are driven at top speed in fair 
weather and foul have three-bladed screws. All things considered, 
there are probably few cases in practice where with equally good 
design the three-bladed propeller is not somewhat to be preferred. 
It should always be lighter and cheaper, and this is a matter worthy 
of consideration, especially when the propeller is to be made of an 
expensive composition. 

In some large four-screw turbine jobs, two of the screws have been 
made four-bladed and two three-bladed with satisfactory results. 

With this combination the chance of objectionable vibration due 
to synchronism is practically eliminated. Where special reasons such 
as this exist, or where strong prejudices exist, it may be advisable 
to use four-bladed propellers, but in the vast majority of cases three 
blades should be used. 

We have seen in Section 25 that propellers witn solid hubs are 
slightly more efficient than those with detachable blades. The dif- 
ference is small, however, except for quick-running propellers, which 
are usually of small diameter. There are great difficulties in the 


way of accurately casting and finishing large propellers with solid 
hubs say propellers over 12 feet in diameter. Hence, such pro- 
pellers should nearly always be made with detachable blades. 

4. Material cf Blades. For the material of propeller blades 
we have a choice between cast iron, cast steel, and some copper 
alloy, such as composition, manganese bronze or other special alloy. 
Forged steel blades. have been used, but are not found now. 

For such a vessel as a tugboat, with its wheel near the surface 
and liable to strike floating objects, cast iron is regarded as desirable. 
Its brittleness and weakness here become virtues, for when a blade 
strikes something it breaks without endangering the shaft or engine, 
and it is cheaper and shorter to renew the propeller than the shaft 
or portions of the engine. Cast steel is superior to cast iron in 
strength and is largely used for merchant work. 

Manganese bronze and other special alloys can now be had with 
strength equal or superior to that of cast steel. They can be given 
a better surface, and from the point of view of efficiency of propul- 
sion are decidedly the better materials. They have two drawbacks. 
The first cost is higher, and through galvanic action they are liable 
to cause excessive corrosion of the portion of the ship's structure 
adjacent to them. This damage can, however, be neutralized in 
practice by the use of zinc plates properly secured to the hull. 

A very serious objection to iron and steel blades is their tendency 
to corrode. The backs of the blades where there is eddying water 
probably mixed with air seem peculiarly subject to extensive and 
rapid corrosion. 

The practical conclusion is that noncorrosive blades should by 
all means be used, unless their first cost prohibits them for the job 
in hand or unless for special reasons cast iron is indicated. 

But in many cases cast iron or steel blades as a gift would be in 
the end more expensive than noncorrosive blades, owing to the loss 
of efficiency and greater coal consumption caused by their extra 
friction when corroded. This extra friction is the more objection- 
able the finer the pitch of the propeller. 

5. Width of Blades. The blade area of a propeller of given 
diameter and pitch varies directly as the mean width ratio. While 
it has sometimes been thought that comparatively small changes of 


blade area had large effects upon propeller action and efficiency, this 
view is hardly sustained by practical experience. When cavitation 
is not present, rather large changes in blade area produce quite 
small effects. It should be remembered, too, that in practice change 
of blade area involves change of blade section with attendant change 
of virtual pitch. 

The p8 diagrams indicate clearly that when cavitation is absent 
the best mean width ratio is between .25 and .30. For mean width 
ratio of .35 the efficiency is appreciably reduced, and for wider 
blades still it falls off quite rapidly. These conclusions are for very 
smooth blades. In practice blades become more or less roughened 
and foul, and when this is the case the wider blades will have the 
greater loss of efficiency. 

The conclusion indicated as a practical rule is that where cavita- 
tion is not to be feared the best all-round mean width ratio is about 
.25 or less. To avoid cavitation wider blades up to a mean width 
ratio of .35 or so should be used, even with thin blades of hollow- 
backed type. In extreme cases even wider blades may be required, 
in spite of their excessive friction loss. 

6. Examples of Design. The principles governing propeller 
design and the application of the methods that have been given 
will now be illustrated by some typical cases. 

First Case. Design the propeller for a turbine Atlantic liner 
which develops 80,000 shaft horse-power upon four screws making 
200 revolutions per minute each and has a speed of 28 knots. Here 
we may take the propeller power as 20,000. The first thing neces- 
sary is to estimate the wake factor. In the case of a job of such 
importance this would be done nowadays from model experiments. 
Let us suppose that we are considering the after screws and that 
the wake factor is 10 per cent. 

Then ^ = .9 X 28 = 25.2. 

2OO V / 20,OOO _ 

So p = r^~ = 8.87. 

(25. 2) 2 ' 5 

Al J * (20,000 X 25.2)* 

Also d = d - .. J ' =.2608 5. 


We are now prepared to enter the p8 diagrams (Figs. 211 to 214). 


Since, however, we know that this is a case where cavitation is to 
be carefully provided against, we would expect to use a blade of wide 
type, so we will use only Fig. 214 for a mean width ratio of .35. 
In Fig. 214 for p = 8.87 the best pitch ratio is 1.140 and the best 
value of 5 = 57.4. Then diameter d = .2608 X 57.4 = i4'-97 and 
pitch = 14.97 X 1.140 = if. 07, the real slip being 25.2 per cent. 
These for a blade thickness fraction of .03. Now the power PI 
absorbed by each blade is 6667. From Fig. 250 for a pitch ratio 
of 1.14, Ci= 910 and (i4-97) 3 = 3355. 

( v CiPi 910 X 6667 

Hence for Fig. 251 x = =-= = *- *- = 9.04. 

Rd* 200 X 3355 

Now it seems advisable in such a job to keep the stress down to 
moderate limits. So let us try for it 7500 Ibs. per square inch. 
From Fig. 251 where x = 9.04 and compressive stress is 7500, 
y = ch-r 2 = .ooi2 about. Now we know h =.35, and if c = , which 

will be somewhere near the truth, we have r 2 = - * = .00514; 


7 =.072, axial thickness = 12". 9. Now Fig. 214 being based upon 
a blade thickness fraction of .03, it is necessary to correct the 
results obtained by using Fig. 215. From this figure when 
p = 8.87 for each .01 increase of T the diameter should be de- 
creased i.i per cent and the pitch ratio increased 0.9 per cent. 
So the total decrease in diameter would be i.i X 4.2 = 4.62 per 
cent and increase of pitch 0.9 X 4.2 = 3.78 per cent. This would 
make the diameter 14.97 X .9538 = 14'. 28, pitch 17.07 X 1.0378 = 
i7'.72. If we allowed a stress of 10,000 Ibs. per sq. in. which might 
be admissible in such a high-class job as this we would have from 
Fig. 251 y = chr* = .ooo(). Whence, for c = h = .35, 

2 .0009 X 3 
T - J =.00386. 

T =.0621, axial thickness = n".2. 

The reduction of thickness is not very much, but we could probably 
stand an axial thickness of 12 inches. 


Now the tip speed will be over 9000 feet per minute and even 
with the best possible shape of blade section some cavitation is to 
be expected. So as much increase of slip would involve rapid fall- 
ing off of efficiency, it would seem advisable to make the propeller 
a little large in order to provide against this and adopt as the final 
dimensions: Diameter 15 feet, pitch 17 feet 6 inches, mean width 
ratio .35, axial blade thickness 12 inches. The propeller effi- 
ciency to be expected, barring cavitation, is about 67 per cent. 

Second Case. Design the propeller for a large twin-screw tur- 
bine destroyer to make 34 knots with 25,000 shaft horse-power at 
800 revolutions per minute, the wake fraction being .03. 

Then V A = 34 X-97 = 32.98, 

_ 800 Vi 
P " (32.98)^ 


This too is a case where cavitation is to be carefully guarded 
against, so we consider only Fig. 2 14. 

From this figure for p = 14.3 the best pitch ratio is 1.004 and 
8 = 60.3, the propeller efficiency being about 62 per cent. 

Then d = 6'. 036, p = 6'.o6. 

Consider now blade thickness, PI = 4167, and from Fig. 250 
Ci= 1015, alsod 3 = 220. 

Then from Fig. 251 

Ci-Pi _ 1015 X 4167 _ 
= Rd? ~~ '' 800 X 220 

This is a value of x beyond the limits of Fig. 251, but to use this 
method a designer should prepare an enlarged and extended copy 
of Fig. 251. In this case we wish to use a high stress, say 12,000 
Ibs. It will be found that using this stress in an enlarged copy of 
Fig. 251 we have for x = 24, c/zr 2 


In this case, too, we may put c = and we have h = .35. Then 
2 .0020 X 3 

T Z = - ^=.00857, T = .0026. 


Axial thickness = 6| ". 

In this case, too, there would be a decrease of diameter of about 
7 per cent and an increase of pitch of nearly 6 per cent from Fig. 
215. But with a tip speed of about 15,000 feet per minute there 
will almost certainly be cavitation, and it is not safe to reduce the 
diameter. It does seem advisable, however, to increase the pitch 
slightly to provide against excessive slip. So the dimensions indi- 
cated are: Diameter 6 feet inch, pitch 6 feet 5 inches, mean width 
ratio .35, axial blade thickness 6f inches. The propeller efficiency 
to be expected in the absence of cavitation is about 62 per cent, 
but this is a case where the actual efficiency depends largely upon 
the amount of cavitation. Some cavitation is almost unavoidable. 
The propeller in this case would be cast with solid hub. We thus 
lose the possibility of varying the pitch and hence adjusting the 
propeller to the engines after trial. In cases where there is uncer- 
tainty it is possible virtually to provide for this, however, by making 
the propeller originally a little large. If trials show it too large, 
blade tip can be cut off to suit, being careful not to throw the pro- 
peller out of balance. 

Third Case Design the propeller for a twin-screw gunboat to 
make 17 knots with 3700 I.H.P. at 156 revolutions per minute, the 
wake fraction being .08. 

Then VA= 17 X .92 = 15.64. We are dealing now with I.H.P. 
and must estimate the propeller power. Assume it .9 of the I.H.P. 

P = _ = I66s> 

_ 165 Vi66 5 , , . g (1665 X 15-64)* _. - . 
P ' (I5-64) 2 ' 5 (165)' 

This is a case where with proper blade section we need not seriously 
apprehend cavitation. Hence we should try all four pd diagrams. 
The results are tabulated below: 



pd diagrams, Fig. No 




Mean width ratio 

. 20 



Blade thickness fractions, (Stan- 




Maximum standard efficiency 

. 7OO 

. 706 

. 7O2 


Best pitch ratio 

I . 222 


I 2OO 

I 21 1 

Best 8 

Cj.. 6 

re . C 

c6 2 

rr 6 

Diameter d 



10 o; 

lo' 17 

Pitch p 

I2'. O7 

- W 5 

12 IJ. 

, i/ 
12 2O 






PI (power absorbed by each blade) 
C, from Fig. 250 





C P 
Value of x = - 



2 8l 


Assumed nominal stress 10,000 Ibs. 
Value of c&r 2 from Fig 251 

. OOO3 




Value of ch assuming c = $ 




07? 3 

Value of T 3 .. . - 



OOI ? 


Value of r 





Departure of T from standard 



. OO 1 1 

4- .ooso 

Per cent change of diameter from 
Fit? 21 c 


+ O.QI; 


O 74 

Per cent change of pitch ratio from 
Fie 2 i c . . 

1 .40 


O. 14 

+ 0.6C 

New pitch ratio 

1 . 204 

I . IQ7 


I . 2J.I 

New diameter 


IO . I S 

IO'. IQ 


New pitch 


12 . 1C 


12 . 7O 

This is a case where we have a wide range of choice of width with 
little change of efficiency. It is evident, too, from the pd diagrams 
that we may change diameter and pitch through a range of 10 per 
cent each without much effect upon efficiency. 

Where cavitation is not to be feared the best all-round mean width 
ratio is about .25, and using this we would finally adopt: Diameter 
10 feet 2 inches, pitch 12 feet 2 inches, M.W.R. .25, B.T.F. .0424, 
axial thickness of blade, 6 inches. 

Fourth Case. Design the propeller for a large single screw cargo 
vessel to make 12 knots with 4000 I.H.P. at 78 R.P.M., the wake 
fraction being .26. 

For this case VA = 12 X-74 = 8.88, and if we assume the pro- 
peller power to be .9 the I.H.P. we have P = 3600. 
Then _ . 


= ~~ = 



P= ~ = 

This being a case of moderate tip speed we will consider mean 
width ratios of .2 and .25 only. Results are tabulated below: 



pS diagram, Fig No 



Mean width ratio 



Blade thickness fraction, (Standard) 



Maximum efficiency for value of p 



Rest pitch ratio 



Best d 

6' 8 

67. o 



1 6' . 70 

17'. 7C 

Real slip per cent 

?o o 

?! e 

Apparent slip, per cent 

6 7 


if '. 

7? I 7 


p power absorbed by each blade 



C from Fig 250 


Value of x( C[P '\ 

2 2 

\cPR ) 
From Fig. 251 for cast steel, chr 2 = nominal stress 5000 
If c = ch = 

. I 111 

. 00048 




06 1 


The departure of T from standard values is too small to 
take account of. If we wish to use a 4-bladed screw we 
have from Fig. 2 1 6 Diameter co-efficient forp = 19.9 
Pitch coefficient for p = 19.9 




Diameter of 4-bladed screw 

18 ?6 

18 42 

Pitch of 4-bladed screw 

16. 21 


The pitch ratio being low. suppose we assume a pitch 
ratio of i.i. Then for a 3-bladed screw we have effi- 

. coo 




c8 i 


17 O2 

5- * 

17 OS 


IO 71 

10' od. 

For corresponding 4-bladed screw diameter 



For corresponding 4-bladed screw Pitch 

IO'. 14 

IQ'. 1 7 

The above example is a very interesting one and illustrates several 
facts apt to be lost sight of. 

In the first place, the large vessel of low speed as built has a value 
of p entirely too large for good propeller efficiency. The p value is 
materially larger than that of the destroyer case and the maximum 
possible propeller efficiency less. It is true that the large cargo 
vessel should approach the maximum, while the destroyer is apt to 
lose through cavitation. In spite of the low propeller efficiency the 
efficiency of propulsion of the cargo vessel may be good. Such 
vessels are apt to have a hull efficiency greater than unity, which 
brings up their efficiency of propulsion, but the fact remains that 
their efficiency of propulsion would be better still if they could be 
given more efficient propellers. In order to do this the p value 
must be less. Now p can be reduced by reducing the revolutions, 


but this will result in increased diameter, which is already by no 
means small. Also reduced revolutions are almost certain to be 
objectionable as regards the engine. Another practicable method 
of reducing p is to use twin screws, but this has obvious objections. 
The trouble is essentially the same as encountered with moderate 
speed turbine vessels, namely, that the desirable engine revolu- 
tions are too high for a propeller of high efficiency. There is a 
further trouble, namely, that the propeller of high efficiency may 
require an impossibly large diameter. Still, the best solution of the 
problem is the same as for the turbine, namely, a satisfactory speed 
reduction gear of high efficiency, so that both engine and propeller 
can be given the revolutions best suited to their needs. 

It will be observed that the propeller of best efficiency has to 
work at a very high real slip. This essential condition is masked 
in practice by the fact that the wake fraction is large, so that the 
apparent slip is very much below the real slip. In fact, for such 
vessels very good results may be obtained when the apparent slip 
is zero. 

The fact that the best we can do in such cases is to work a pro- 
peller of fine pitch, and hence low maximum efficiency, at a high 
slip, so that its efficiency is well below its maximum, is the main 
reason for the rapid reduction of efficiency with large values of p. 
For small values of p propellers can usually be worked much closer 
to their point of maximum efficiency. 

It will be observed that while for the .25 M.W.R. the best pitch 
ratio is .891, this can be made i.i with a reduction of possible effi- 
ciency from .600 to .585 only. But the diameter can be reduced 
thus from i9'-47 to 17'. 95 or over 18", the pitch rising from i7'-35 
to 19'. 74. If a four-bladed screw is used the diameter can be re- 
duced still more. 

32. Paddle Propulsion 

The vast majority of sea-going vessels are propelled by screws, 
and vessels using paddle wheels are practically all engaged in chan- 
nel, bay, lake or river service. 

i. General Features. It is obvious that a paddle wheel through 
its construction and method of operation approaches more nearly 
than the screw propeller the ideal frictionless propelling apparatus 


discussed in Section 22. If, for instance, we regard a paddle wheel 
as discharging directly astern a column of water of area equal to the 
area of a paddle float and with velocity equal to the difference 
between the peripheral velocity of the center of the float and the 
speed of advance of the ship, and make the further assumption that 
the action is frictionless and that the water is discharged without 
change of pressure we have an ideal propelling instrument to which 
Fig. 171 applies. 

This leads us to the 'conclusion that if A denote the area of a 
paddle float, V the speed of advance in knots and P the shaft 

horse-power absorbed by the paddle wheel = <f)(e) = (f>(s) =K r , 

where the coefficient K' is a function of the slip. For paddle wheels 
the slip is generally reckoned with reference to the peripheral speed 
of the paddle centers. If V p denote the peripheral speed of the 
paddle centers in knots and V the speed of advance of the vessel 
in knots, y _y 

2. Fixed Blades. -- The earliest paddle wheels had the blades 
on radical lines, as indicated diagrammatically in Fig. 253, and many 
paddle wheels are still of this type. 

Figures 254 and 255 trace out the successive positions of a 
single float with reference to still water for 30 per cent slip and 10 
per cent slip respectively. The direction and relative amounts of 
the velocities of the inner and outer edges of the floats are also 

The line marked W.L. indicates a water line such that the blade 
has its upper edge immersed in its deepest position about one half 
of its breadth. There is of course minimum obliquity of action 
when the blade is vertical, in its deepest position, and it is desira- 
ble that the blade should do as much work as possible when deeply 
immersed. That would require it to enter the water edgewise, or 
nearly so. It is evident from Figs. 254 and 255 that radial blades 
will not be moving edgewise with respect to still water at the time 
they reach the water surface. This result may, of course, be ac- 
complished by setting the blades at suitable fixed angles. But fixed 


blades so set are usually regarded as undesirable, perhaps without 
good reason. 

In the United States the development of wheels which will not 
suffer from excessive obliquity of blades at entering and leaving 
has been toward wheels of large diameter and wide narrow floats 
of small immersion. This line of development was facilitated by 
the type of engine usually fitted on paddle steamers. 

Furthermore, broadly speaking, paddle steamers in the United 
States have been for service in smooth waters, and hence could be 
designed for a small immersion of floats which would be inadvisable 
in rough water service. 

3. Feathering Blades. In Great Britain, influenced perhaps 
originally by the fact that many of the finest and fastest paddle 
steamers were for service across the English Channel and had to be 
prepared to encounter rough weather, paddle wheels are almost uni- 
versally fitted with feathering blades. 

As indicated diagrammatically in Fig. 256, a blade is pivoted about 
its center, the pivots being carried by the framing of the wheel 
proper, which revolves about A. Each blade has an arm perpen- 
dicular to it on its back, to which is attached a link, and the other 
end of the link is connected to a center K eccentric from A. The 
point K is very simply determined. The positions of H, G and F 
are obviously fixed by the positions desired for a blade entering the 
water, leaving the water and at maximum submergence. Then K 
is the center of the circle passing through H, G and F. 

It is very common in practice to fit feathering paddle blades as 
indicated in Fig. 256, where the planes of the entering and leaving 
blades intersect the circle of blade centers vertically above the shaft. 
Paddle wheels have been fitted where the blades remained vertical 
throughout the revolution, but this is not done now. 

It might seem very simple from Figs. 254 to 255 to determine 
the proper angles for blades entering and leaving the water, but the 
actual problem is one of extreme complexity. Figs. 254 and 255 
show velocities with reference to water at rest, and this is far from 
the conditions of practical operation. 

The water upon which a paddle wheel acts has been previously 
disturbed by the ship, the amount of disturbance varying with the 


speed. Moreover, each paddle enters water which has been dis- 
turbed by the preceding paddles. There is little question that in 
practically all cases of side paddle wheels the paddles enter water 
which has already a sternward motion. Stream line action and the 
action of preceding paddles will both give the water a sternward 
motion, and even if the wheel is located at a wave crest as is 
desirable the forward motion due to the wave motion will be less 
than the other two. 

For stern wheels stream line and wave action will give the water 
a forward motion, the action of preceding paddles a rearward motion, 
and it is not possible without extensive experiments to lay down any 
general conclusions. 

4. Comparison of Fixed and Feathering Blades. Paddle wheels 
with feathering blades are heavier, more complicated and more 
expensive than wheels of the same size with fixed blades. But in 
practice they can be made materially smaller in diameter for the 
same efficiency, and also can be given greater depth of immersion 
resulting in a larger virtual area of paddle for a given actual size. 
This is an important consideration for high-speed paddle vessels. 
The smaller the wheel the higher the engine revolutions, and it is 
usually desirable as regards weight and space to increase the revo- 
lutions of paddle boat engines when directly connected. In practice 
fast high-powered paddle boats are usually fitted with feathering 
blades, fixed blades being used when the revolutions are low and 
the diameter of wheel great, or for service in remote rivers where 
simplicity is essential. 

5. Paddle Wheel Location. While it is not proposed to con- 
sider structural details, some considerations affecting paddle wheel 
design will now be taken up. In practice, paddle wheel vessels are 
side wheelers or stern wheelers. In side wheelers the wheels are 
located somewhere near the center of length. It is advisable to 
locate them so that they work in a crest of the transverse waves 
caused by the ship, or at any rate not in a hollow. When working 
in a crest there is a virtual wave wake favoring efficiency, while in a 
hollow the wave wake is prejudicial to efficiency. The stream line 
wake in which side wheels work is prejudicial to efficiency, so that 
side paddle wheels usually have a virtual negative wake. Also the 


wash from the wheels increases the frictional resistance of the rear 
of the ship and produces a virtual thrust deduction. 

Side wheels cannot be placed very far forward or aft of the center 
of ships of ordinary form without danger of under or over immer- 
sion through changes of trim, incident to service. 

Stern wheel boats are of the wide flat type and the draft aft 
does not vary much in service. 

Stern wheels are so located that the wake due to stream line and 
wave action is in their favor, and they will cause but little thrust 
deduction as a rule, so that, broadly speaking, the stern wheel may 
be expected to be more efficient as an instrument of propulsion than 
side wheels. 

It is very desirable to fix the heights of all paddle wheels so that 
the desired immersion will be had when the vessel is under way. 
This can readily be done by model basin experiments in advance, 
and for the best results with feathering wheels the question of blade 
angles at entrance in and departure from the water should also be 
investigated experimentally. 

The immersion of paddles is varied somewhat with the service. 
For seagoing boats the immersion of the upper edge of the paddle 
in its lowest position is seldom less than \ its breadth and as great 
as .8 its breadth. For smooth water service the immersion is usually 
less, i to i the breadth. The desirable immersion depends some- 
what upon the type of float. A very long narrow float on a large 
wheel may have its upper edge immersed its whole breadth without 
loss of efficiency. 

6. Dimensions and Proportions of Paddle Wheels. One of the 
most important questions arising in the design of any type of paddle 
wheel is the determination of the dimensions of the blades, buckets 
or floats, as they are variously designated. 

These are sometimes curved, but seldom curved much, and may 
be taken as rectangular. The length or horizontal dimension of 
the float is always greater than its width or radial dimension. 
There is found in practice a difference in proportions between 
feathering and fixed floats. For feathering floats the length is 
usually about 3 times the width, though shorter floats have often 
been fitted. For fixed floats the length is seldom less than 4 times, 


and may be in extreme cases 7 or 8 times the width. This difference 
of practice naturally arises from the fact that floats are usually 
made as long as possible from practical considerations, as tending to 
efficiency, and then as wide as necessary to absorb the power. For 
side wheels, floats are, however, seldom longer than the beam 
even for vessels always in smooth water, and for seagoing vessels 
it is not regarded as good practice to make them longer than about 
f the beam. 

The float area is dependent primarily upon the power absorbed 
and the slip. We have seen that the theoretical formula involved 

P I 

is = K'. This may be rewritten A = K where A is area of 
AV 3 V 3 

two floats (one on each side) in square feet, 7 is indicated horse- 
power and proportional in a given case to P, V is speed of ship in 
knots and K is a coefficient depending primarily upon the slip and 
secondarily upon a large number of minor factors, such as wake, 
thrust deduction, float proportions, number and immersion, etc. 

Hence K may be expected to vary a great deal from ship to 
ship, but fortunately it is not necessary to know it with minute 

Analysis of a number of published trial results for paddle steamers, 
nearly all with feathering floats, appears to indicate that a reason- 
able expression for the average value of K will be, for slips used in 
practice ranging say from .10 to .30, 

K = 212.5 ~ 375 * 

From the nature of the case individual values of K may be 
expected to vary materially from the average. A long narrow blade 
deeply immersed may be expected to show a much smaller value 
of K than a short wide blade with its upper edge barely immersed. 

Then a suitable paddle area may be determined approximately 

by the formula .4 = (212. 5 3755) It must be remembered 

that in the above A is total area in square feet of two paddles when 
side wheels are fitted, and 5 is slip based upon the peripheral velocity 
of the centers of paddles. 

It is desirable to keep the slips of paddle wheels low. For feather- 


ing floats .15 is frequently aimed at, and for fixed floats .20. Know- 
ing the speed of the ship and the desired slip, the peripheral velocity 
of the mean diameter of the paddle wheel upon which slip is based 
is known, and this in conjunction with the desired engine revolutions 
fixes the mean diameter of the wheel. 

The desired float area being known, the float dimensions are 
determined, enabling all dimensions to the wheel to be fixed. If 
these are found suitable the desired blade angles at entry and depar- 
ture will govern the details of gear for feathering blades when such 
are fitted. 

As regards number of blades it is a very common practice with 
fixed blades to fit one for each foot of outside diameter of wheel. 
This number should not be exceeded for wheels of good size and may 
be reduced by 20 per cent or so without detriment. The spacing 
of feathering blades is greater than that of fixed blades, partly be- 
cause such blades are usually relatively deeper than fixed blades 
and partly because of the additional complications of feathering 
gear for blades close together. 

With feathering blades there are sometimes fitted one for each 
foot of radius but a greater number are usually regarded as desirable, 
say about 3 blades to each 2 feet of radius. 

33. Jet Propulsion 

i. General Considerations. Jet propulsion has never been used 
except experimentally. In jet propulsion water is taken into a ship, 
where it passes through some form of pump or impelling apparatus 
and then delivered astern through suitable pipes. Many schemes 
for jet propulsion have been brought forward in the past, usually 
including methods for diverting the jets sidewise as desired, in order 
to gain maneuvering power. 

While some schemes of jet propulsion have been actually tried, 
none has proved so efficient as the screw propeller or paddle wheel. 
Hence, jet propulsion is of academic interest only and will not be 
given detailed consideration. 

That any system of jet propulsion involving any form of impelling 
apparatus known at present must be inefficient will be evident from 
Fig. 171. It will be found from this that, even with frictionless 


impelling apparatus, if there is not to be a great loss through slip 
the pipes to get the water into and out of the ship must be so large 
that they will involve very serious increase in skin friction to say 
nothing of eddy losses. If pipes are made small there is unavoid- 
ably a great loss by slip, and still larger loss by friction in the pipes. 
Furthermore, any pump or impelling apparatus now known is not 
materially more efficient in communicating velocity to a given 
quantity of water than the screw propeller or the paddle wheel. 

Hence, jet propulsion, involving taking water in large amount into 
the ship and discharging it again, is with any known form of impell- 
ing apparatus necessarily less efficient than the screw and the paddle, 
which operate in the water outside the ship. 

Since the essential inefficiency of jet propulsion as a method of 
utilizing the power of ordinary engines has become evident, some 
inventors have attempted to devise apparatus specially adapted to 
jet propulsion in which power is developed more economically than 
in engines driving propellers and paddle wheels. Efforts along this 
line have not hitherto been successful. 



34. Measured Courses 

i. Features Desirable for Measured Miles. Trials for the 
determination of speed must be made over a course of known length, 
unless by trials already made over such a course the relation between 
revolutions of the propellers and speed through the water has been 
established so that a speed trial may be conducted in free route. 
The measured course may be long or short. The difficulties of locat- 
ing, measuring and marking a satisfactory long course are evidently 
much greater than for a short course, and nearly all accurately 
measured and marked courses are one nautical mile long. For a 
number of years, however, four-hour full-speed trials of United 
States naval vessels were held on long deep water courses extending 
to the northward of Cape Ann on the Massachusetts coast. The 
length used was carefully determined in each case so that the vessel 
would run about two hours in each direction and four or five vessels 
or more were anchored on the course for the double purpose of 
defining it and of making observations of the tidal current during 
trials. Of late years, however, four-hour full-speed trials have been 
made in free route by the standardized screw method. For stand- 
ardizing the screw or determining the relation between speed and 
revolutions, trials are usually held on a course one measured mile 
in length near Rockland, Me. This course is shown in Fig. 257. 

It is seen that the course is defined by four range buoys, one at 
each end of the measured mile and one a mile from each end. These 
buoys, however, are for steering purposes only. The ends of the 
course are fixed by ranges established on shore, each with a front 
and rear signal or beacon. When these signals are in line the 
observer is at one end of the course, which, as shown, is perpendicular 
to the range lines. 



The desirable features for a measured mile course in tidal waters 
are enumerated below. 

If they were all present in any particular case the course would 
be ideal. In practice it is necessary to be satisfied with a reasonable 
approximation to the ideal. 

1. The range marks on shore at each end of the course should 
be well separated say f the length of the course or more 
and should by the transit of the front signal past the back signal 
mark definitely and sharply the instant of crossing the range. This 
is best attained when both front and back signals show against the 

2. The situation should be such that the course is not far from 
shore and fairly well protected, insuring smooth water when the 
local wind conditions are favorable. 

3. There should be plenty of room at each end of the course for 

4. The course should be so situated that the ship making runs 
over it need never cross or obstruct a channel or fairway that is 
much used. 

5. The tidal current should be small and always parallel to the 

6. The depth of water should be sufficient, so that the resistance 
of the ship using the course is practically the same as in deep 

As regards most of the features enumerated above, the Rockland 
course, shown in Fig. 257, approximates fairly closely to the ideal. 
It has the disadvantage of being rather remote from most of the 
building yards whose vessels must use it. 

It would be better if the front and back signals marking the ranges 
were further separated and showed above the sky line. It may be 
noted in this connection that if the range marks do not show against 
the sky a course running north and south is not so good as one 
running east and west. If the ranges are to the west of the course 
the marks are difficult to pick up in the afternoon, and if they are 
to the east they are difficult to pick up in the forenoon. 


35. Conduct of Speed and Power Trials 

i. General Considerations. Vessels may be given many kinds 
of trials, as of speed and power, of fuel economy, maneuvering capa- 
city, etc. We need consider the first named only. 

Speed and power trials may be considered from the point of view 
of (a) the owner, (b) the designer, or (c) the builder. In some cases, 
as for vessels of war built in government establishments, the owner, 
designer and builder are one; frequently for vessels of war the owner 
and designer are one; and usually for merchant ships, and sometimes 
for vessels of war, the designer and builder are one. 

From whatever point of view we consider speed trials, however, 
they are primarily of importance for new and untried vessels. For 
such vessels the owner wishes to know what his ship will do in 
service and from the results of progressive speed and power trials 
he can generally closely estimate the results to be expected in ser- 
vice. The designer wishes to know what the ship actually does 
under known trial conditions in order that he may utilize the infor- 
mation in preparing subsequent designs. The builder is generally 
required to guarantee certain results to be demonstrated by trial 
before the ship leaves his hands and at times wishes to develop on 
trial certain results not exacted by his contract, but which may be 
of use to him in a business way. Apart from this he is apt to con- 
sider that trials conducted at his expense should be reduced to the 
lowest terms. 

As a result of various conflicting considerations the most that 
can usually be expected for speed and power trials of a new ship in 
the builder's hands is the determination of corresponding values of 
speed, revolutions, and power over a reasonable range from the 
maximum down, at one displacement and under favorable condi- 
tions of wind and weather. Such a trial is usually called a pro- 
gressive speed trial and appears to have been first developed in 
Great Britain by Mr. William Denny. Concerning this develop- 
ment Mr. William Froude said in a paper before the Institution of 
Naval Architects, April 7, 1876: 

" Mr. Denny has taken the bold but well-considered step of dis- 
carding the conventional type of measured mile trials which, as 


regards the speeds tried, have long been limited to full speed and 
half boiler power. Mr. Denny now tries each of his ships at four 
or even at five speeds; and the result is that he obtains fair data 
for a complete curve of indicated horse-power from the lowest to 
the highest speeds; whereas with trials on the ordinary system we 
obtain merely two spots in the curve, and these at comparatively 
high speeds, the intermediate or lower portion of the curve being 
left uninvestigated." 

2. Accuracy Possible in Progressive Trial Results. The deter- 
mination of accurate results on a progressive trial is by no means 
the simple matter it might seem at first. Approximate results are, 
of course, readily obtained, but for the results of progressive trials 
to be of real value for the designer they should be quite accurate. 
What we need are simultaneous values of speed of the vessel, power 
indicated by the machinery and revolutions per minute of the en- 
gines, determined for a sufficient number of speeds covering a good 
range to enable accurate curves of power and revolutions as ordi- 
nates to be drawn on speeds as abscissae throughout the range 
covered by the trials. 

If we had available a measured course in perfectly still, calm, deep 
water, and wished to determine the most reliable curves from a defi- 
nite number of runs, it would evidently be desirable to run back 
and forth, increasing and decreasing the speed or revolutions by 
equal amounts between successive runs. Observing on each run 
the time and revolutions on the course and taking indicator cards 
for the power determination, we could plot curves through points 
obtained by the observations. 

Progressive trials are not made on ideal courses, as above. Even 
if they were, it would seldom happen that the data obtained would 
be absolutely consistent and harmonious. It is probable that on a 
course in still water the time on the course would be determined 
with a good deal of accuracy. But even with a long straight run 
at each end before coming on the course an important point fre- 
quently neglected the speed on the course is seldom absolutely 
uniform. Unless steam is actually blowing off all the time the boiler 
pressure is always going up or down it may be very slowly with 
skilled firing, it may be with sufficient rapidity to cause quite an 


appreciable change in speed while on the course. Moreover, the 
rudder is constantly being used more or less, and even when put 
over to a small angle only it has a noticeable effect upon the speed. 
This is a matter of practical importance in the conduct of trials 
which does not always receive proper attention. 

Then the indicator even the best is not an instrument of 
precision. If several sets of cards are taken during a run the powers 
worked out from them will differ materially. Professor Peabody, 
an authority on indicators, considers that even " under favorable 
circumstances the unavoidable error of a steam engine indicator 
is likely to be from two to five per cent." 

If the indicated horse-power is determined on the measured course, 
not less than three sets of cards should be obtained and the average 
of all good cards used in determining the average power. At times 
some cards are obviously defective, and these should be thrown out. 

For single-screw ships the revolutions and speed vary together, 
and there are no serious complications from the inevitable slight 
variations in revolutions, except that sometimes there is doubt as 
to the proper revolutions to use with the indicator cards for the 
determination of power. But with twin-screw ships the revolutions 
and power of the two engines are not identical on any run. The 
only thing that can be done in such cases is to try to have the port 
and starboard revolutions during each run as nearly the same as 
possible and use the average of the two results. With two screws, 
unless the propellers differ more than they should, we may safely 
assume that at a given speed and the same revolutions, each engine 
will require the same power. In practice, owing to minor differences 
in propellers, and differences in engine friction, the assumption is 
not exact. But it is near enough, and is, in fact, the only one we 
can make. 

With three screws, however, the case is different. At full speed, 
with everything wide open, the central engine will differ from the side 
engines as regards both power and revolutions, even if identical in 
size with them. When it comes to the runs at reduced speed, we 
may for a given speed have enormous variations in the power dis- 
tribution. It would seem proper in such cases, where the engines 
are identical, to be careful to have the steam pressure in the H.P. 


valve chests and the linking up the same for all three engines on 
each run. Otherwise the curves of slip of the center and side 
screws will be very erratic. With four screws the case is even more 

For such vessels, where each engine is independent, it may be 
necessary to plot results upon speed plotting separate curves 
of revolutions for each engine. But even here equally good results 
can be obtained by plotting results upon the average revolutions 
of one pair of engines plotting, upon these revolutions, a curve 
of the average revolutions of the other engine or pair of engines. 
For turbine installations, where the turbines are in tandem, the 
steam passing from one turbine to another, this method is distinctly 

When we come to turbines we meet the difficulty of determining 
the actual power exerted by them. Several methods are used 
all based upon the fact that the twist of the shafting is proportional 
to the torque of the turbine. This twist is a small quantity in any 
case, and its accurate determination experimentally is difficult. It 
is probable, however, that as the use of turbines extends the accu- 
racy of their power determination will be improved. With an 
accurate torsion meter the determination of shaft horse-power will 
be much simpler and easier than the determination of indicated 
horse-power by means of indicators. 

3. Elimination of Tidal Current Effects. It is evident from 
what has been said that even on an imaginary still-water course a 
progressive trial would not be free from doubts and difficulties in 
connection with obtaining and plotting the results. 

Actual measured courses, however, must be laid off in a tideway 
where tidal currents varying in direction and magnitude are encoun- 
tered. No course is suitable for a progressive trial unless the tidal 
current is practically parallel to the course. Slight cross currents 
are nearly always present, however. When they are present the 
steering on the course should always be by compass and not by 
buoys or other fixed fore and aft ranges. By always steering a 
compass course parallel to the true range the effect of slight cross 
currents is eliminated. So we will consider from now on only the 
current parallel to the course. Suppose, first, that the current is 



constant and that we make two runs at the same true speed one 
with and one against the current. 

Suppose V is the true constant speed of the two runs, C the con- 
stant but unknown speed of current and V\, F 2 , the apparent speeds 
of the successive runs. Then Vi = V + C, V Z =V C whence V = 
% ( V\ + V z ) , or the true speed through the water is the average of the 
two apparent speeds with and against the current. Sometimes the 
true speed is taken as that corresponding to the average time of 
the runs with and against the current. This is incorrect. The true 
speed for two runs with and against a constant current being the 
average of the two apparent speeds, it is a common practice to 
make the runs of a progressive trial in pairs one run being made 
in each direction at the same speed. There are two objections to 
this. One is that the tidal current changes between runs. The 
other often more serious in practice arises from the fact that 
in practice the successive runs are made not at the same speed but 
at different speeds, and the average horse-power is not the proper 
horse-power for the average speed. Figure 260 illustrates this, in 
an exaggerated form. A and B are points on a curve of horse- 
power plotted on speed corresponding to two runs. C is the point 
on the curve corresponding to the average speed, while D, midway 
of the straight line joining A and B, is the average horse-power. 

The first source of error, the change of tidal current, can be largely, 
but not entirely, eliminated by making a series of runs over the 
course at one speed and obtaining the true speed from the apparent 
speeds by the method of successive means. This is illustrated below 
with four runs the apparent speeds being Vi, V z , V 3 , F 4 . 


First Means. 

Second Means. 

Final Means. 




Vl-r- Vz 

Vi+ zV z + V 3 

Vi+ 3 Vz + 3 V 3 + V 4 

Vz + V 3 

Vs + 2 V 3 + F 4 

V 3 +V 4 




The first means are simply the averages of the successive pairs of 
runs. The second means are the averages of the successive pairs 


of first means, and so on. There appears to be a difference of 
opinion as to whether, when there are more than four runs, the true 
speed should be taken as the final mean, or the average of the 
second means. As appears above, for four runs the two are the 

Now if n denote the number of a run of a series we can always 
express C, the strength of the current, in the form 

C = a + bn + cn*+ dn s + en 4 + . . ., 

using as many terms as there are runs in the series. Suppose, for 
instance, there are four runs. Then we have 

C = a + bn + cn z + dn 3 . 

Denote by C\, C 2 , C 3 , C 4 the actual current strength of the four 
successive runs. 

Then d= a + b + c + d, 

C 2 = a + 2 b + 4 c + 8 </, 
C 3 = a + 36 + gc + 27 d, 
C 4 = a + 4 b + 16 c + 64 d. 

These are four equations from which we could determine the four 
unknown quantities a, b, c, d. Hence, no matter what the current 
strength of the successive runs, we could always find values of the 
coefficients a, b, c and d such that we can represent the current by 

C = a + bn + cn z + dn 3 . 
On solving the equations above for a, b, c, d we have 

a = H 2 4 Ci- 36 C 2 + 24 C 3 - 6 C 4 ), 
b = $ ( - 26 Ci+ 57 C 2 42 C 3 + ii C 4 ), 
c = i(9 C"i- 2 4 C 2 + 21 C 3 - 6 C 4 ), 
<* = H-C" 1 4- 3 C 2 -3C 3 + C 4 ). 

Now consider further the final mean result. We have, if V 
denotes the true constant speed of the four runs, 

F 1= F + C 1= F + a+ b+ c+ d, 
V 2 = V - C 2 = V -a - 2b - 40- Sd, 
V 3 = V + C 3 = V + a + sb+ 9 c + 2 7 d, 
F 4 = V - C 4 = V - a - 4b - i6c - 64 d. 


Final mean = \ (V\+ 3 F 2 + 3 F 3 +F 4 ). Upon substituting for 
Vi, Vz, etc., in this expression, their values above in terms of V 
and the coefficients a, b, c and d, we finally have, after reduction, 

Final mean = V -ld=V -\(- Ci+ 3 C 2 - 3 C 3 + C 4 ). 
In case only three runs are made the current formula is 
C = a + bn + cn~, 

and the currents of the successive runs are Ci, C 2 and C 3 . For this 


Final mean =i (Vi+ 2F 2 + F 3 )= F + Jc = V + i(Ci- 2 C 2 +C 3 ). 

Then the final mean is not the true speed unless the rate of change 
of the tidal current and the timing of the runs is such that for four 

- (d-C 4 ) + 3(C 2 -C 3 ) = o, 

and for three runs 

Ci+ C 3 = 2 C 2 . 

This will happen exactly only by accident. Another way of express- 
ing the condition is that d, the coefficient of w 4 , should be = o, the 
actual error being 1 d. As a matter of fact, in most practical cases 
d would be very small and the final mean but little in error if the 
assumptions upon which the final mean method is based were cor- 

These underlying assumptions are two, namely, that the tidal 
current varies according to a fair curve and that all runs back and 
forth are made at the same speed. 

Every one who has often plotted results of speed trials in a tide- 
way will have encountered results which could be explained only on 
the theory that the tidal current varied by fits and starts rather than 
according to a fair curve. 

It is sometimes assumed that the tidal current varies from maxi- 
mum to minimum in a manner such that a curve of tidal strength 
plotted on a base of time would be a curve of sines. This is per- 
haps a reasonable approximation to the general outline of the curve, 
but observations of actual strengths of tidal currents appear to show 
that they vary erratically and would seldom plot as a fair curve 
closely approximating a mathematical curve of sines. 


A more serious error than that due to tidal current is liable to 
result from the fact that successive runs of a group are not made 
at the same speed. This is a matter of practical experience. It is very 
unusual, indeed, for four successive runs to be made over a measured 
course where the revolutions per minute, if accurately determined, 
do not vary appreciably. If the speed were constant, the revolu- 
tions should not change. Suppose now four successive runs were 
made aiming at a uniform speed of ten knots, while the actual 
speeds were 9.72, 10.24, 10.16, 9.88. The true average speed would 
be ten knots, but the final mean of the four speeds above would be 
10.1 knots. This is quite a large error. In the above I have not 
taken account of the tide. The error is not affected if the tide is 
such that the final mean would eliminate the tidal error if the runs 
were made at constant speed. For instance, suppose the tidal cur- 
rents were in knots .61, .74, .89, 1.06. For ten knots true speed 
the apparent speeds would be 10.61, 9.26, 10.89, 8.94. The final 
mean of these four speeds is 10 knots, as it should be. But if the 
true speeds of the successive runs were as given above, the apparent 
speeds after making allowance for currents, would be 10.33, 9-5> 
11.05, 8.82. The final mean of these is 10.1 knots, as before. 

Evidently, then, as the final mean method is equivalent to giving 
the two middle runs of a set of four a weight of three as compared 
with a weight of one for the first and last runs, when it is used for 
speed it should in theory be also used for revolutions and power. 
Thus, if a middle run of a series of four is made at a true speed 
above the average the excess speed in determining the average speed 
is given a weight of 3. This run will show excess power and revolu- 
tions, and if the average power and revolutions are properly to 
correspond with the average speed by the final mean method the 
power and revolutions should be given the same weight as the speed 
in determining the average. Practice in this respect appears to be 
somewhat variable. We often, but not always, find the final mean 
method used for revolutions. It appears to be seldom used for 

4. Methods of Conducting Progressive Trials. We seem war- 
ranted in concluding that when we attempt to get a spot on a speed 
and power curve by applying the final mean method to the data 


observed during a series of four runs, we by no means eliminate 
the probabilities of error. The question arises whether there are 
not better methods, or simpler methods equally good. We wish to 
determine curves as accurate as possible expressing the simul- 
taneous values of speed, revolutions and horse-power. Now in any 
particular case we can usually determine the revolutions with great 
accuracy. We can determine the indicated horse-power with reason- 
able approximation, and with good indicators the error is as likely to 
be in excess as in defect. For twin-screw vessels, when the two en- 
gines show different revolutions during a run, the best we can do is to 
take the total indicated horse-power as corresponding to the average 
revolutions of the two engines. For any run we can determine the 
speed over the ground with ample accuracy, but owing to tidal cur- 
rent we cannot determine accurately the speed through the water. 
Now in plotting our results shall we plot power and speed on revolu- 
tions, or power and revolutions on speed, or perhaps speed and revo- 
lutions on power? A little consideration will show that there are 
real advantages in using revolutions as the independent variable, 
so to speak, and from the trial data plotting on revolutions separate 
curves of power and speed. For the revolutions of a run can and 
should be determined exactly to all intents and purposes. 

Then by plotting our approximate data upon the correct revolu- 
tions we get rid of one element of uncertainty. We do not, for 
instance, plot a spot for power where the error is in excess over a 
spot for speed where the error is in defect. We will ultimately 
arrive at a more reliable relation between speed and power by deter- 
mining first the most reliable relation between each and the accu- 
rately determined revolutions. Starting, then, with the basic idea 
that we will in the first place plot speed and power as ordinates 
upon revolutions as abscissae, how should the progressive trial be 
conducted in order to determine most reliably the relation between 
power and revolutions ? 

We know from the Theory of Probabilities that if we wish to 
determine a single quantity as, for instance, the value of a fixed 
angle the best plan is to take as many observations as possible 
and use the average as the best obtainable approximation to the 
true value. Similarly, if we wish to determine a curve from experi- 


ment the best plan is to ascertain as many approximate spots as 
possible, plot them and draw the final curve as the average curve 
through the spots. Then to establish a curve of power on revolu- 
tions we should make numerous simultaneous determinations of 
power and revolutions, plot the results and draw an average curve 
through. To locate the curve of power as accurately as possible 
from a given number of runs, it would be better to have each run 
made at different revolutions. This would enable us to cover the 
curve closely with experimental spots. Here we encounter another 
weak point of the four-run final mean method. 

Sixteen runs are necessary to determine four spots on a power 
curve, and four spots are insufficient for the accurate determination 
of a curve of power covering a wide range of speed. On the other 
hand, sixteen spots distributed at approximately equal intervals 
over the whole length of the curve will locate it with great accuracy. 
Each spot may be in error, owing to limitations on accuracy of any 
determination of indicated horse-power, but if the errors are as 
likely to be positive as negative a fair average curve through sixteen 
spots will practically eliminate the indicator errors. If the indi- 
cators have a constant positive or negative error no number of 
experimental spots will eliminate it. I conclude, then, that as to 
the relation between power and revolutions about sixteen simulta- 
neous determinations of revolutions and indicated horse-power, 
made at approximately equal intervals of revolutions, will enable a 
satisfactory power revolution curve to be drawn. These observa- 
tions need not necessarily be taken on the measured course, when the 
speed revolutions observations are being made. It is usual, how- 
ever, to take the indicator cards while on the course. When the 
observing staff is adequate it is more convenient to make one job 
of it, and if the water on the measured course is somewhat shallow, 
so as to affect the results, it is desirable to determine everything 
under the same conditions. By doing this, too, we avoid the chance 
of the initial friction of the engines altering between two sets of 
runs, one to determine the power revolution relation, and the other 
the speed revolution relation. Finally, with an ample observing 
staff the time of a run over the measured course is generally of a 
length convenient for taking several sets of cards. There is, how- 


ever, something to be said in favor of making runs off the course 
for determining power revolutions spots. With a small observing 
staff indicator cards can be taken more at leisure and given revolu- 
tions can be maintained until a sufficient number of satisfactory 
cards are taken, even if indicator accidents crop up. Again, as 
soon as good cards for a given number of revolutions are obtained 
the revolutions can be changed at once up or down. This will 
not save much time at high speeds, but will at low speeds, so that the 
total time the staff must be kept at the indicators will be a good 
deal shorter. The preferable method really seems to depend in the 
end upon the observing staff available. With an ample staff of 
skilled observers, so that in addition to time and revolutions on the 
course three good cards can (barring accident) be obtained during 
each run from each end of each cylinder, it would seem advisable 
to make all observations on the measured course. With a small 
staff of observers, however, including many without good experience 
in such work, it would often be advisable to run separate trials, 
making the progressive power revolution trial before or after the 
speed revolution trial on the measured course. 

Fig. 258 shows trial spots and final curve of power on revolu- 
tions as drawn from the trial of an armored cruiser. 

Let us consider now the most suitable practical method of deter- 
mining the speed-revolution relation from trials on the course. In 
the first place, no method will give reliable results unless we have a 
sufficient number of runs. Each experimental spot is necessarily 
and unavoidably somewhat in error. Hence, in order to get a 
reliable curve we must have so many spots and have them so close 
together that the accidental and erratic errors are practically elimi- 
nated by drawing a mean fair curve. There are two methods which 
may be used with confidence. The first is probably the most accu- 
rate and reliable, provided the trial is conducted with special skill 
along the lines described below. It is also adapted to the determina- 
tion of the power revolution relation by the method just given. The 
second method is probably preferable -for the usual run of trials. 

Under the first method make a series of runs back and forth 
alternately with and against the tide and increasing or decreasing 
the revolutions by equal increments after each run. The curve of 


true speed then will fall midway between the two curves of apparent 
speed, one with and one against the tide. The advantages of this 
method are that if the curve of tidal variation is a fair curve and the 
trial skillfully run so that the interval between successive runs varies 
according to a fair curve, all spots of apparent speed will fall upon 
fair curves. Should, however, a spot be erratic, it will naturally 
fall off the curve and be given little weight in drawing the final 
curve of apparent speed. It is a very real advantage in such work 
to have a method of reducing the data such that bad spots show for 
themselves and are not incorporated in the final results. It is evi- 
dent, however, that to get reliable curves of apparent speed we 
should have a sufficient number of spots for each curve. Not less 
than sixteen runs in all should be made. Figure 259 shows curves 
of apparent speed with and against the tide and the mean curve 
from the trial of an armored cruiser. All experimental spots are 

There are some objections to the above method. One is, that at 
top speed, the most important part of the curve, we would have 
only one run, and the high speed part of the curve would not be 
defined so well as the lower portion. This difficulty should be over- 
come by making three runs at top speed two in one direction, and 
one in the other, and determining the final speed of the three by 
giving the middle run double the weight of the others. This is 
equivalent to taking the second mean of the three runs. The other 
objection to this method is that for thoroughly satisfactory results 
a trial once begun should be completely carried through without 
stopping. This sometimes introduces practical difficulties. A run 
may be lost through breakdown of the observing apparatus or inter- 
ference of some other vessel while on the course. This is not a 
very serious objection, because it is found in practice that even if 
the intervals between the runs are somewhat erratic the curves of 
apparent speed can be drawn tolerably well. Another objection of 
the same nature is that if a trial is interrupted after five or six runs 
the results of these runs are of little value, as they are not suffi- 
ciently numerous accurately to define the part of the curve to which 
they refer, and a whole new trial has to be made. 

The second recommended method of running a progressive trial 


is to make runs in groups of three, 18 in all for a fast vessel, 15 
for a vessel of moderate speed and 12 for a slow vessel. Each group 
should be made at a constant number of revolutions, as nearly as 
possible, the revolutions for the various groups covering the range 
desired. Then, taking for each group of three runs the second mean 
of speed and revolutions we have for a fast vessel six spots through 
which to plot a curve of speed and revolutions. This method in 
practice gives from each group of three runs a spot substantially as 
reliable as if four runs had been made. While it has the advantage, 
as compared with the four-run method, of giving more spots on the 
curve for a given total number of runs, it also has the advantage of 
beginning consecutive groups with runs in opposite directions. 
That is to say, if one group began with a run to the north the next 
group will begin with a run to the south. This is a desirable con- 
dition, as tending to eliminate some of the errors due to tidal current. 
This method has the advantage of requiring less skill and care in 
the conduct of trials, and each group of three runs stands by itself 
and is not wasted in case it is necessary to stop the trial. It is not 
quite so accurate as the method previously described, but the dif- 
ference in accuracy would not be appreciable in the majority of 
cases. A practical advantage is that it does not require readjust- 
ment of throttles and links after each run in order to change the 
revolutions. This adjustment, in order rapidly to change revolu- 
tions by a definite amount, is by no means the simple matter it 
might appear at first thought and requires quick and accurate work 
in the engine room. 

If there were no variations of tidal current between runs both 
methods above described would be theoretically exact. It is evi- 
dently desirable to time the progressive trial so that during it there 
shall be as little variation of tidal current between runs as possible. 
Now, when the tidal current is at a maximum, whether ebb or flow, 
the variation of current is at a minimum, while about the turn of 
the tide the rate of variation is about at a maximum. This state- 
ment would be exactly true if the curve of tidal current plotted on 
time was a curve of sines, as often assumed, and is substantially 
correct even as applied to actual tidal currents, varying by leaps 
and bounds rather than with definite progression. Then a pro- 


gressive trial should always be run during the strength of one tide. 
A trial can generally be run in four hours or less, and so should, if 
practicable, be begun about an hour and a half after the turn of the 
tide. Circumstances often render this inconvenient or impossible, 
and weather conditions frequently cause the turn of the tide to 
come before or after the time fixed by tide tables, but the best time 
for a trial should be used unless there are good reasons to the con- 

So far as accuracy of results is concerned it makes no difference 
whether we begin with the low speeds and work up or with the high 
speeds and work down. It seems advisable, however, as a rule to 
begin with the top speeds and work down. With clean fires and 
fresh men the top speed can be obtained and maintained with more 
ease than after several hours of running. Also, if the trial is spoilt 
by a breakdown it is more apt to come during the high speed runs, 
and if a breakdown must come it is better to have it come early 
than late. 

There may be mentioned here some minor points in connection 
with the conduct of trials which tend to produce accurate and sat- 
isfactory results. It is desirable after a run to shift revolutions 
promptly to the revolutions for the next run, if they are to be 
different. If there is a pressure gauge giving the pressure in the 
H.P. chest (beyond the throttle) it is easy by preliminary runs to 
establish a curve (or curves, if more than one valve gear setting is 
to be used) giving the relation between H.P. valve chest pressure 
and revolutions. Then it is necessary only to establish the proper 
pressure to insure that the revolutions are sufficiently near what 
is desired. Such a pressure gauge as above is apt to fluctuate vio- 
lently unless its cock is nearly closed. Systematic handling of the 
ship when off the course is desirable. Each time when coming on 
the measured course the ship should have made a long straight run 
with the minumum operation of helm. For most trials about a 
mile is a convenient and desirable length for the straight run, and 
it much facilitates trials if in addition to buoys at the ends of the 
measured course, moored closely on the ranges, there are planted 
buoys in the line of the course a mile from each end. Suppose we 
have the course thus buoyed as indicated in Fig. 257. Before begin- 


ning the trial proper while warming up steam over the course 
as indicated in Fig. 257 by ABCDEFCBGHA. 

When abreast the buoy D put the helm over to a moderate and 
definite angle, say 10 degrees. Steady the ship on the course EF 
which will cross the line of the course a little beyond the buoy C. 
While on this course note carefully the compass reading and deter- 
mine the reading of the steering compass which will give the opposite 
course FE. Then when coming of! the course at C after a run, put 
the helm over at once and steady the ship on the course FE. If 
the revolutions are to be changed for the next run the engine room 
force should immediately set to work on this. With skillful han- 
dling the new desired revolutions should be attained before the vessel 
is at E. If this is so, on reaching E abreast the buoy D put the 
helm over to 10 degrees. The vessel will, by the time she swings 
to the correct heading for the next run, be practically on the line 
of the course, requiring very little use of the helm to come dead on. 
If the revolutions are not adjusted by the time the vessel reaches 
E, she should at this point be steadied on the course EK, shown 
dotted in Fig. 257, and kept on this course until the revolutions are 
satisfactorily adjusted or the vessel has run so far that there will 
be ample time after turning finally to adjust the revolutions before 
the vessel reaches D. The methods are of course just the same at 
each end of the course. 

To conduct a trial in this way requires quick communication and 
complete understanding between the deck and the engine room, but 
results will be distinctly superior to those obtained by more hap- 
hazard methods. 

5. Trial Conditions. It is customary to make progressive trials 
with clean bottoms under good conditions of wind and sea. For 
men-of-war the trial is generally made at normal load displacement. 
For merchant vessels the displacement is ^sometimes the average 
displacement to be expected in service, but generally a less displace- 
ment and at times a very light displacement. 

The usual practice is at times criticised. As to men-of-war, for 
instance, it is alleged that they will never show in service such good 
results as upon trial. It is true that there is ever present the temp- 
tation to run trials at too light a displacement. This is largely due 


to the natural desire of those concerned to make the best showing 
possible. But the loss of speed in service due to increased displac- 
ment is apt to be exaggerated, particularly for large ships. More 
potent causes are rough water at sea, dirty bottoms, poor coal, or 
inability of the engineering personnel to get good power results. It 
is evidently desirable to have trials always run under uniform or 
standard conditions. The most easily attained standard trial con- 
ditions are obviously fair weather, smooth water and a clean bottom. 
From reliable results under such conditions the results which should 
be attained in service can be estimated with sufficient approxima- 
tion until they can be ascertained by experience. As a general 
thing, however, progressive trials cannot, and are not expected to, 
show exactly what a ship will do in service. This requires service 
experience. They furnish data to enable the performance of the 
ship under standard conditions to be determined and compared with 
other vessels, and in case the performance is poor careful progressive 
trials will not only determine that fact, but as a rule, upon analysis, 
indicate the line that should be followed to obtain improvement. 

36. Analysis of Trial Results 

i. Components of Indicated Horse Power. Figure 260 shows a 
curve of speed and power for the U. S. S. Yorktown, the powers as 
ordinates being plotted over the speeds as abscissae. 

The power is the indicated horse-power developed in the cylinders 
of the engines. We know that only a fraction of this power is finally 
utilized to propel the ship and it is important to gain some idea of 
the distribution of the remainder. 

The engine itself absorbs a certain amount of power through its 
own friction. This friction is usually classed under two heads, 
namely, " initial " or " dead " friction, due to tightness of pistons, 
valves, glands, bearings, etc., and " load " friction, or the friction 
due to the load upon the bearings and thrust block. 

The power required to work feed, air, circulating and bilge pumps, 
driven from the main engines, is usually classed with the initial 
friction. For reciprocating engines, the power P delivered to the 
propeller is the original indicated horse-power less the power as 
above absorbed by friction. For turbine engines the power is 


usually determined from the twist of the shaft, measurements being 
taken astern of the thrust block. All of this shaft horse-power is 
delivered to the screw except what is wasted in friction of line bear- 
ings, stern tubes, and outward bearings, if any. This is usually so 
small that the shaft horse-power is assumed to be the same as the 
propeller power P. 

Of the propeller power P a portion is wasted in friction and slip 
of the propeller. The remainder is used in developing thrust horse- 
power. Also there is added here a certain amount of power derived 
from the wake which also appears as thrust horse-power. Of the 
thrust horse-power a certain amount is used to overcome the aug- 
mentation of resistance of the ship due to the suction of the pro- 
peller, and the remainder is the effective horse-power, the net 
power required to drive the ship. 

The above components of the I.H.P. vary widely. The initial 
friction will absorb from as low as 3 or 4 per cent of the power in 
large well-adjusted engines with independent air and circulating 
pumps to 10 per cent or more in the case of machinery badly 
adjusted with air and circulating pumps driven off the main engines. 

The load friction is usually taken as about 7 per cent of the 
remainder obtained by deducting the inital friction power from the 
original I.H.P. With well-lubricated engines it is generally some- 
what less. Investigations of the shaft horse-power of reciprocating 
engines by means of torsion meters have shown as much as 92 per 
cent of the indicated horse-power delivered to the shaft, involving 
a loss of but 8 per cent for both initial and load friction. Engines 
seldom run any length of time with excessive load friction. It 
promptly causes hot bearings. 

The ultimate distribution of the propeller power the shaft 
horse-power for turbine jobs is a question of the efficiency of the 
propeller, the wake factor and the thrust deduction. 

It is evident from what has gone before that as a reasonable work- 
ing approximation we may assume that for a reciprocating engine 
of high-class workmanship about 90 per cent of the indicated horse- 
power is delivered to the propeller when independent air and circu- 
lating pumps are fitted, and about 85 per cent of the indicated power 
when all pumps are driven off the main engine. 


Accurate trial results can be analyzed to give an approximation 
to the resistance of the ship, and hence efficiency of propulsion, etc., 
but these quantities can be estimated directly with sufficient accu- 
racy and with much less labor by methods already given. It is 
very desirable, however, to determine accurately the initial friction 
of an engine, as then we know with close approximation the pro- 
peller power, P, and this power is an essential factor of the propeller 
design. Hence we will now consider in detail the initial friction of 
an engine and methods for determining it from progressive trial 

2. Initial Friction Determined by Curves Extended to Origin. 
Mr. William Froude, the pioneer investigator of this question, defines 
initial friction as " the friction due to the dead weight of the work- 
ing parts, piston packings, and the like, which constitute the initial 
or low speed friction of the engine." The initial friction, or internal 
resistance, is generally regarded as constant throughout the range of 
speed and power of the engine, thus differing from the load friction, 
which is generally regarded as absorbing a uniform fraction of the 
power developed. As a matter of fact, it seems altogether probable 
that the internal resistance varies slightly with power and revolu- 
tions, but the variation is probably so small as long as bearings run 
cool that we are justified in ignoring it. 

There is no doubt that the internal friction will alter materially 
from time to time, due to changes in tightness of various parts. 
The problem under consideration, however, is the determination of 
the initial friction at a given time. If the frictional resistance is 
constant the power absorbed by it will be proportional to the revo- 
lutions, so that if we denote by // the horse-power absorbed by 
initial friction and R denotes revolutions, we have //= R X (a coeffi- 
cient), where the coefficient at a given time for a given engine is 
constant. Suppose we denote the coefficient by C/, then //= C f R. 
Now analysis and consideration of the various absorbents or com- 
ponents of the indicated horse-power, such as the power utilized to 
propel the ship, the power wasted by the propeller, the power 
absorbed in ^load friction, etc., show that they all, except //, must 
vary as some power of the revolutions greater than unity. This 
being the case, it follows that if / denote the indicated horse-power 


at revolutions R, we may write / = C/R + < (/?), where we know 
that (f> (R) is some function of the revolutions which varies always 
as a power of R greater than unity. If, then, we plot a curve of 7 
on revolutions, as we approach the origin the curve of 7 will ap- 
proach the straight line I f = C f R, and at the origin will be tangent 
to this line. Hence C/ can be determined from the inclination at 
the origin of the curve of 7 plotted on R. 

Figure 261 shows for the U. S. S. Yorktown a curve of indicated 
horse-power plotted on revolutions, the curve being extended to the 
origin and the tangent at the origin being drawn in. It is desirable 
in plotting this curve to draw, as shown, a similar symmetrical curve 
in the third quadrant joining the real curve in the first quadrant to 
the imaginary curve in the third quadrant at the origin where there 
is a point of inflection. This facilitates drawing a curve which has 
the proper direction at the origin. Then drawing the tangent at 
the origin we determine the line for 7/= C f R, and taking at any 

point the simultaneous values of 7/ and R we have C/= 


Another method is to plot a curve of 7 divided by R in the first 
quadrant and a symmetrical curve in the second quadrant. Such 
a curve will not pass through the origin but cut the axis of R = zero 
at a point above the origin. Its ordinate here is evidently C/. The 

ordinates of the curve of bear a constant ratio to the ordinates of 


the curve of mean effective pressure. 

It is customary to reduce the initial friction or internal resistance 
of an engine to equivalent mean effective pressure in the low pressure 
cylinder or cylinders. This is the most convenient and probably 
the most reliable way of comparing engines of different types and 
sizes as regards internal resistance. 

Let n denote the number of L.P. cylinders, d the diameter of 
each in inches, 5 the stroke in inches, p m the mean effective pressure 
in pounds per square inch reduced to the L.P. cylinder area and R 
the revolutions per minute. Then 

7T 2 S 

, _ 4 m 12 



At the limit 7 = // = C f R. If p f denote mean effective pressure 
equivalent to internal resistance reduced to L.P. area, at the limit 

n nd?spf 2521000 

p m = p f or C f = - or p f = J - J - 

252100 nsd? 

It is seen from the above that when we have once determined a 
reliable value of C/ we can readily obtain the corresponding value 
of the mean effective pressure in the low pressure cylinder from the 
known data of the engine. If we could determine with accuracy 
the curve of indicated horse-power for a given engine to a very low 
number of revolutions the above method of determining internal 
resistance would leave little to be desired. However, we meet here 
with a number of practical difficulties. If we determine simulta- 
neous values of speed, power and revolutions, which is the usual 
practice in progressive trials, it is found that the low speed trials 
over a measured mile are very tedious. If we avoid this trouble by 
determining in free route at the lowest speeds the horse-power and 
revolutions only, we still encounter difficulties. No reciprocating 
engine will run at all below a certain speed, and as it approaches 
the limiting speed at which it will stick, its action becomes some- 
what erratic and uncertain. It is true that the less the friction the 
lower the revolutions at which the engine will stick, and that this 
is a rough measure of the initial friction; but even the smoothest 
running engines will seldom run steadily down to a speed sufficiently 
low to enable the internal resistance to be determined with accuracy 
by a curve extended to the origin. For determining very low speed 
powers of engines which use high pressure it is necessary to use 
special weak indicator springs, otherwise the indicator diagrams 
have such a very small area that the determination of the power is 
very uncertain. If, instead of determining a curve of power and 
revolutions for the ship under way we determine the same thing for 
the vessel tied up at the dock, we will get larger indicator cards and 
the engine will turn over at a slightly lower number of revolutions, 
but even then the results generally leave something to be desired. 

Torsion meter apparatus has been designed of late years to 
measure the power being transmitted by a shaft by determining 
the twist of the shaft. If we measure shaft horse-power by a tor- 


sion meter and simultaneously indicate the engine, we can determine 
the total frictional resistance of the engine, the power absorbed by 
friction being of course the difference between the indicated horse- 
power and the shaft horse-power. With accurate data this would 
probably be the most nearly exact method of determining the initial 
friction of the engine and would have the incidental advantage of 
enabling the load friction to be determined as well, but the accuracy 
of torsion meters at low speeds and powers is not sufficient to enable 
this method to be made use of except perhaps in very exceptional 
cases. It is evident that we need some method of obtaining the 
desired result from an ordinary curve of power and revolutions which 
does not go below a speed and power for which the data may be 
readily obtained and regarded as fairly reliable. It is natural to 
ask whether there is any inherent feature or property of curves of 
horse-power which would facilitate the determination of the internal 
friction. Mr. William Froude worked on these lines. He plotted 
a curve of indicated thrust upon the speed of the ship in knots, 
carrying the curve down as low as possible. Indicated thrust is a 
thrust which at the speed of the propeller will absorb the indicated 
horse-power. At zero speed and zero revolutions the curve of indi- 
cated thrust, whose ordinates are proportional to , will cut the 


axis of thrust at a distance above the origin proportional to the 
initial friction. To pass from the lowest point of his curve of indi- 
cated thrust, determined by observation, Mr. Froude made use of 
an essential property of these curves. He assumed that at these 
low speeds the resistance of the ship varied as the 1.87 power of the 
speed, and that all other losses except the initial friction loss were 
constant fractions of the power absorbed by resistance. It would 
follow that the curve of indicated thrust in the vicinity of the origin 
is a parabola of the 1.87 degreewhose ordinate at zero speed is 
proportional to the initial friction. 

Now, referring to Fig. 262, if the curve therein indicated is a 
parabola of the 1.87 degree it follows that the tangent at the point 
P will cut the horizontal tangent through the lowest point A at a. 


point M, so that AM divided by A N is equal to- Mr. Froude, 



then, having drawn his curve of indicated thrust to as low a speed 
as he could from the data, next drew the tangent at its extremity 

OTJ 87 

as KB in Fig. 263, and dividing OL at H so that 7 equals - 

OL 1.87 

he set up HB to intersect the tangent at K in the point B. A hori- 
zontal line, then, through B cuts the axis of thrust at the point 7 1 , 
and Or is the indicated thrust corresponding to the initial friction. 
This method makes use of a property of the curve, but as a matter 
of fact, it is hardly so reliable in practice as the method of extending 
the curve of indicated horse-power to the origin and setting off the 
tangent to it. While the low speed resistance of the ship would be 
reasonably close to the 1.87 power of the speed this is still an approxi- 
mation, but the principal objection to this method is that it requires 
a tangent to be drawn at the low speed extremity of the curve of 
indicated thrust. The difficulty of obtaining reliable values for this 
curve at the lowest speed have been pointed out and it follows, 
apart from the difficulty of drawing an accurate tangent at the 
extremity of any curve, that an error in the low speed spot would 
throw out the low speed tangent and introduce material errors. 

3. Initial Friction Deduced from Low Speed Portion of Power 
Curves. The question arises, then, whether we cannot make use of 
some inherent property of curves of horse-power which will enable 
us to determine the initial friction with reasonable accuracy without 
it being necessary to carry any curve to the origin. We know that 
the frictional resistance of a ship varies about as the 1.83 power of 
the speed, so that the horse-power absorbed by frictional resistance 
varies as the 2.83 power of the speed. The power absorbed by 
wave making varies as a higher power than the cube of the speed. 
The practical result is that at low speeds, when there is almost 
no wave resistance, the total effective horse-power will vary as a 
somewhat lower power of the speed than the cube, whereas at high 
speeds it will vary as a higher power of the speed than the cube. 
There is then some point at moderate speed where the effective 
horse-power is varying as the cube of the speed. 

Consider now the propeller. For a given slip the power absorbed 
by a propeller varies as the cube of the revolutions, or for constant 
slip as the cube of the speed. It follows, then, that starting from a 


very low speed, where the effective horse-power is varying at a lower 
power than the cube, the slip of the propeller falls off until we reach 
the speed at which the effective horse-power varies as the cube of 
the speed. At this point the slip of the propeller reaches a minimum 
beyond which it increases. The efficiency of the propeller at the 
point where the slip reaches a minimum will be constant, and the 
power delivered will vary as the cube of the speed or as the cube of 
the revolutions. Also all losses will vary as the power delivered to 
the propeller or as the cube of the revolutions, except the initial 
friction loss. Hence, at the point of minimum slip where the slip 
remains constant for a minute interval the following formula will 
express exactly the indicated horse-power: 

For some little distance on either side of the point of minimum 
slip the above formula will give a reasonably close approximation 
to the facts, especially for the speeds below the point of minimum 
slip. Now Cy and c in the above equation are both unknown, but 
from the curve of indicated horse-power plotted on revolutions we 
can determine any number of simultaneous values of 7 and R, and 
for each pair of such values we can draw a straight line on axes of c 
and Cf, constituting a focal diagram. If the equation above applies 
throughout to the curve of indicated horse-power and C/ and c were 
constant, it would follow that this diagram would have a perfect 
focus. Now we know that the above equation does not apply to the 
upper part of the curve of horse-power at which the indicated horse- 
power generally varies as a very much higher power than the cube. 
It seems reasonable from the nature of the case, however, that this 
equation should be fairly approximate over a tolerably wide range 
of the lower speeds, and that if we draw for this range a series of 
lines Cf and c, they should all pass reasonably close to a common 
point; in other words, should form a reliable focal diagram. Inves- 
tigation of practical cases shows that we do have such a focal 
diagram. The methods of calculation are very simple. The 
table below shows the calculations for the Yorktown, and Fig. 264 
shows the diagram for the Yorktown. 





















9 86 





71 .5 














if *-o>c,-i 



3- 2 75 






IfC/=o, c=-^- 3 



. 000836 






It is seen that taking the four lines corresponding to speeds of 
5, 6, 7 and 8 knots we get an excellent focal diagram. The Q-knot 
line does not give quite such a good intersection and the zo-knot 
line leaves the focus entirely. This is typical of such diagrams, and 
the lines themselves show very clearly which should be used and 
which should not be used in determining the focal point. C/ from 
Fig. 264 for the Yorktown is equal to 0.96, while its value from 
Fig. 261, taken from the curves extended to the origin, was made 
.95 a number of years ago, soon after the trial of the Yorktown 
in 1889. The practical agreement of the results of the two methods 
might seem in favor of the method of extension to the origin, which 
is the simpler. But special care* was taken on the Yorktown trial 
to obtain reliable power data at very low speed, and the results 
obtained by extending her power curve to the origin can be regarded 
with more confidence than those of subsequent trials of other vessels 
where it was found practically impossible to extend the power curves 
to the origin with certainty. It was this fact which impelled a 
search for a better method. Figures 265 to 269 show initial fric- 
tion diagrams for the United States ships Alabama, Kearsarge, 
Massachusetts and Maine and the revenue cutter Manning. The 
mean effective pressure in the L.P. cylinders equivalent to initial 
friction and the percentage of maximum power absorbed by initial 
friction are given below for these vessels and the Yorktown. 

Name of Ship. 







Mean effective pressure in L.P. equiv- 
alent to initial friction 


2. O7 


1 . 77 

?. 20 

2. C2 

Per cent of max. power absorbed by 
initial friction 




4- 33 


6. co 


The above vessels were all given careful trials and the results are 
as reliable as will usually be obtained. While the diagrams show 
lines for successive speeds, successive values of revolutions could 
have been used as well, and in fact the method can be readily applied 
to a curve of power and revolutions where the speed is not known. 
It is seen that in every case there is an excellent focus formed by 
the lines for the lower speeds, except in the case of the Maine, 
where the focus is not so well defined as would be desirable. The 
generally satisfactory determination of the focus in accordance with 
theoretical reasoning may be regarded as fairly strong evidence in 
favor of the method outlined above. To produce direct evidence 
for this method we can apply it to a case where the internal resis- 
tance is accurately known by some other method. The Yorktown 
was one such case. Fortunately, however, we can produce stronger 
cases. In the transactions of the Society of Naval Architects and 
Marine Engineers we find two cases of determinations of speed and 
power of double-ended ferry boats with a propeller at each end. 
Three curves are given for each case, one curve for both screws in 
use, one for only the stern screw in use, the bow screw being removed, 
and one for only the bow screw in use. One case was that of the 
Cincinnati, the data for which can be found in a paper by F. L. 
DuBosque, in the volume for 1896, and the other case was that of 
the Edgewaier, the data for which can be found in a paper by 
E. A. Stevens in the volume for 1902. Fig. 270 reproduces the 
curves of power plotted on revolutions for the Cincinnati and 
Fig. 271 the similar curves for the Edgewaier. It is seen that the 
three curves for each boat differ radically from each other, owing 
to differences of propeller efficiencies, etc., but it is evident that for 
each vessel the internal friction of the engine should not vary much 
for the three conditions, since the engines, shafting, etc., were the 
same and the only factors affecting frictional resistance were the 
presence or absence of one screw and the variations of initial friction 
between trials. Figures 272 and 273 show the frictional focal dia- 
grams for the Cincinnati and Edgewaier as deduced from Figs. 270 
and 271 and the curves of speed and revolutions. The original obser- 
vations for the Cincinnati do not extend to quite so low a speed as 
desirable for the initial friction determination, but it is seen that the 


several cases, in spite of the radical differences in the curves of power, 
give fairly satisfactory foci in adequate agreement. The average 
value of Cf for the Edgewater is .738, the highest value being 5.0 
per cent above and the lowest value 7.2 per cent below the average. 
Similarly for the Cincinnati the average value of Cf is .897, the 
highest value being 8.7 per cent above and the lowest value 13.0 
per cent below the average. 

I think, then, it may be safely concluded that the Focal Diagram 
method outlined above will give a definite determination of the 
initial friction which, with good data, may be expected to be within 
10 per cent of the truth. This approximation is ample for practical 
purposes, since at the higher speeds the whole initial friction power 
is but a small percentage of the total. It will be observed that the 
focal points are simply spotted by eye on the focal diagrams. The 
theoretical most correct focus of such a diagram can be determined 
by Least Square methods at the expense of not very much time and 
trouble. Since, however, the results obtained are approximate in any 
case, we gain no real additional accuracy by the extra calculations. 

4. Determination of Efficiency of Propulsion from Trial Results. 
-The efficiency of propulsion being the ratio between effective and 
indicated or shaft horse-power we need to know the effective horse- 
power in order to determine it for any speed. 

The effective horse-power may be that of the bare hull or include 
the appendages. In either case, given the curves of E.H.P. and of 
I.H.P., the determination of a curve of efficiency of propulsion is 
simple and obvious. 

Since initial friction absorbs a greater proportion of the power 
at low speeds we may expect to find for vessels with reciprocating 
engines the efficiency of propulsion falling off rapidly at low speeds. 
If propeller efficiency, wake factor, etc., were constant, the maxi- 
mum efficiency of propulsion would always be found at top speed, 
but propeller efficiency varies with slip, which is not constant as 
speed changes, and the wake fraction also varies with speed. Hence, 
we frequently find the maximum efficiency of propulsion below the 
maximum speed. But in most practical cases unless cavitation sets 
in the efficiency of propulsion does not change much either way for 
several knots below the maximum speed. 


If curves of E.H.P. are deduced from experiments with a model 
of the ship the resulting efficiencies of propulsion are of course more 
reliable than those obtained from estimated curves of E.H.P. If, 
however, model experiments are not available for a vessel for which 
we have reliable power data we should always estimate curves of 
E.H.P. from the Standard Series diagrams (Figs. 81 to 120) and 
deduce curves of what may be called nominal efficiencies of pro- 
pulsion. Such nominal efficiencies for vessels of a definite type are, 
when dealing with a new vessel of the same type, almost as useful 
as if they were derived from model tests. 

5. Analysis for Wake Fraction and Thrust Deduction. When 
considering the question of wake in Section 28 we saw how from the 
propeller power P and the revolutions and speed we could estimate 
the wake fractions by a curve of 5 from experiments with a model 
of the propeller or by the standard curves of 5 (Figs. 230 to 233). 

As the values of P and 5" used are at best experimental and ap- 
proximate, the most that can be hoped for wake fractions thus 
determined is that they will be reasonably good approximations. 

If there is cavitation the method fails, and there is reason to 
believe that propellers with blunt or rounding leading edges cavitate 
without it being discovered. The effect of slight cavitation, or in 
fact of any failure of the Law of Comparison, is to cause the wake 
deduced from Figs. 230 to 233 or by similar methods to be less than 
the real wake. This possibility should always be borne in mind 
when analyzing trial results for the determination of wake. Theo- 
retically we can determine thrust deduction factors from analysis 
of trial results in connection with accurate model results for ship 
and propeller tested separately. 

E.H.P. (i - iv) 

For i / = *- t 


where E.H.P. is effective horse-power of ship, e is propeller effi- 
ciency, P is propeller power and w is wake fraction. In practice, 
however, since every quantity on the right of the above equation 
is estimated or only approximated, the thrust deduction factors 
thus determined are seldom reliable. 



37. Powering Methods Based upon Surface 

i. Rankine's Augmented Surface Method. The methods that 
have been proposed and used to estimate the power required to 
drive a given ship at a given speed are many and various. One 
of the earliest English methods which broke away from the rule of 
thumb and attacked the problem in a logical and scientific way was 
Rankine's Augmented Surface method, brought out some fifty years 
ago. Rankine assumed that in a well-formed ship the resistance 
was wholly frictional, the water flowing past the ship with perfect 
stream motion and the frictional resistance varying as the square 
of the speed. 

But with perfect stream motion the average relative velocity of 
flow over the surface would be somewhat greater than the speed of 
the vessel with reference to undisturbed water, and Rankine devel- 
oped elaborate mathematical methods for determining an " Aug- 
mented " surface such that its frictional resistance at the speed of 
the vessel, neglecting stream motion, would be the same as the 
actual frictional resistance of the real surface of the ship when 
there was perfect stream motion. Rankine assumed .01 as a coef- 
ficient of friction, so by his method we would have Resistance = 
.01 V 2 X Augmented Surface. We know now that Rankine's fun- 
damental assumptions were wrong and would involve results vastly 
more erroneous in practice than the use of the actual surface instead 
of the slightly greater augmented surface. In his time, however, 
there were few fast ships, and the assumption that resistance was 
wholly frictional was not so much in error as it would be now. 
Furthermore, little was known of the actual coefficients and laws 
of frictional resistance, as William Froude's epoch-making experi- 
ments on the subject were subsequent to 1870. So Rankine's 
neglect of all resistances but friction was to some extent made up 



by his overestimate of the friction. The calculation of the Aug- 
mented Surface was, however, not easy, and for many years Ran- 
kine's method has been obsolete. 

2. Kirk's Method. A method of estimating power was brought 
out by Dr. A. C. Kirk of Glasgow nearly thirty years ago, which 
though resembling closely Rankine's method in basic underlying 
principles, is much simpler and easy of practical application. Dr. 
Kirk devised in the first place a method of approximation to the 
wetted surface S. He then assumed that the resistance would vary 
directly as the square of the speed and the indicated horse-power 

kSV 3 

as the cube of the speed, using the formula 7 = - - where 7 is 


indicated horse-power, V is speed in knots, 5 is wetted surface in 
square feet and k is a coefficient which must be fixed by experience. 

Kirk made k = 5 for merchant ships of ordinary proportions and 
efficiency, while for fine ships with smooth clean bottoms and high 
propulsive efficiencies it was as low as 4 and for short broad ships 
as high as 6. 

For the low speed cargo vessel for which Kirk devised and recom- 
mended his method it has many excellent features. 

For such vessels the residuary resistance is usually not a large 
proportion of the whole, and up to u or 12 knots the I.H.P. does 
vary approximately as the cube of the speed. 

Then the coefficient k was fixed, not by preconceived ideas or 
reasoning as to what it ought to be, but by experience of what it 
had been on other similar ships. Hence, Kirk's method is sound 
in principle. The main objection to it is that it is of little value 
for fast vessels, and even for the 10 to 12 knot cargo boat the coeffi- 
cient k is apt to vary erratically. 

3. Coal Endurance Estimated from Surface. The principle of 
Kirk's method may be utilized to advantage for estimating the low 
speed endurance of vessels of war. Such vessels, whatever their 
full speed, usually make passages at a moderate speed of 10 to 12 
knots in order to save coal or gain endurance. At such speeds the 
I.H.P. varies approximately as the cube of the speed V and as 
the wetted surface which is proportional to \/DL. Hence, I.H.P. 
varies as F 3 VDL 


Now at these low speeds the coal burnt for all purposes per I.H.P. 
varies inversely as some power of the speed and may be assumed to 

vary approximately as 

Hence, coal per hour varies as or as V \/DL. 

Hence, coal per mile varies as or as v DL. 

Hence, miles steamed per ton of coal vary as 


So if m denote the miles steamed per ton of coal and K a coal 


coefficient, we have m= - == If the approximate assumptions 

v L/ j^^ 

above were exact K would be constant for all ships and speeds. 
In practice K varies from ship to ship and with the speed of a 
given ship. It increases from a very low speed up to a maximum 
value nearly always for a speed below 10 knots which is the most 
economical speed for the ship. 

For speeds beyond the most economical speed KQ falls off steadily. 
Fig. 274 shows curves of K for some United States battleships, 
the average of the sister ships Kearsarge and Kentucky, the Wisconsin 
and the Oregon. 

These curves are averaged from consumption at various displace- 
ments with all kinds of coal, under all conditions of bottom and of 
weather and hence are from average service results. The Wisconsin 
data was not complete enough to make a reliable final average. On 
a given passage a vessel may well show values of K twenty "per 
cent above or below the average, with the varying conditions as 
respects quality of coal, state and management of the machinery, 
foulness of the bottom and the weather. 

On the voyage of the United States Atlantic fleet around the 
world the Kearsarge and the Kentucky showed an average K for 10 
knots of 6900 as against about 7130 in Fig. 274. The Wisconsin, 
however, which has a lo-knot K of only 6910 in Fig. 274, showed 
an average value of 7600 in the voyage around the world, the values 
on the different legs varying from 7300 to 790x3. For the whole 


fleet the average value of K was about 7200 at 10 knots. This 
figure may be regarded as fairly typical of large battleships with 
reciprocating engines, though it will be found that it will give such 
vessels endurances under average service conditions far below those 
usually credited to them in naval handbooks. 

A flotilla of United States destroyers on its way from the United 
States to the Philippines via the Suez Canal some years ago showed 
an average value of K at 10 knots of 5000. Merchant vessels 
designed for only 10 knots naturally show much larger values of 
KQ. Thus a large lo-knot naval collier on a voyage from Hampton 
Roads to Manila showed an average K of nearly 13,000. An 
i8,ooo-ton ten-knot freighter in the Atlantic trade showed about 
12,000 in three passages under moderate weather conditions, while 
on a passage made in exceptionally heavy weather throughout, its 
K dropped to less than 9000. 

4. Admiralty Coefficients. Perhaps the method most used in 
the past for powering ships has been the Admiralty Coefficient 
method. Here again the basic assumptions are that the resistance 
is all frictional and the I.H.P. varies as the cube of the speed. 
The wetted surface is not used directly, however. For similar ships 
the wetted surface varies as the square of the linear dimensions, or 
as D* where D is displacement in tons, or as M where M is area 
of midship section in square feet. Hence we write 


where I is indicated horse-power, V is speed in knots, C\ is 
the " displacement " coefficient and C 2 is the " midship section " 

Ci D* 
It is evident from the above that = , so that for a given 

62 M 

ship ~ is constant throughout the range of speed. But for dis- 
C 2 

similar ships the ratio between C\ and C 2 is different, so that two 
ships on trial may show the same values of the displacement coeffi- 
cient and very different values of the midship section coefficient, 
and vice versa. t 


In England, the displacement coefficient has been regarded as 
the most reliable, that is, as showing less change with variation of 
type of vessel. In France, on the contrary, the reciprocal of the 
midship section coefficient is largely used. It is evident, however, 
that any formula based upon the assumption that resistance varies 
as the square of the speed must be unreliable for high speeds unless 
there is available a large accumulation of data from trials of fairly 
similar high speed vessels. In such case, in spite of the faulty 
assumption, it may be possible to select a suitable coefficient. 

It is apparent, however, that the Admiralty coefficients ignore a 
number of factors which have great influence upon resistance. For 
instance, both coefficients ignore the length and the longitudinal 
coefficient, factors which are sometimes of enormous importance. 

So, in spite of the long use that has been made of the Admiralty 
Coefficient method, it must be regarded as reliable only when on the 
well-beaten track. Reliable trial results from a number of vessels 
of different types will give Admiralty coefficients which vary widely. 

When it is necessary to fix upon the coefficient to adopt when 
powering a new vessel, much experience and good judgment will be 

38. The Extended Law of Comparison 

i. Deduction of Extended Law of Comparison. The most 
accurate method known at present for the estimation of the resist- 
ance of a full-sized ship is to determine the resistance of a model of 
it and by using the Law of Comparison deduce the resistance of the 
full- sized ship. 

Evidently, then, we may regard a full-sized ship whose trial results 
we know as a model and power similar ships from its trial results. 
Thus, suppose we have a ship of displacement D whose resistance 
is R at speed V, whose effective horse-power is E, indicated horse- 
power 7 and efficiency of propulsion e. 

For a similar ship at corresponding speed let us denote the quan- 
tities enumerated above by D\, RI, V\, I\ and e\. 

We know by the Law of Comparison that 

1L = H. Z i/TL IT^\ = (R\- 

RI Di Vi \ Li V \ZV \Dj 


E RV ID " 

WhenCG E l -Wi = fe 

and if e = e\, which should be the case with sufficient approximation, 

EI e\I\ I\ 

This is the Extended Law of Comparison, so called. We may ex- 
press it by the statement that for similar models at corresponding 

speeds , is constant. 

2. Application of Extended Law of Comparison. There are 
various methods of plotting the trial data of a ship so that by using 
the Extended Law of Comparison it can be applied to new designs. 

7 V 

A simple method is to plot a curve of ^ over values of - = This 

D 6 v T 

J > 

eliminates the size factor. Thus, Fig. 275 shows a curve of 


for the U. S. S. Yorktown plotted on values of = 

The Yorktown is of 230 feet mean immersed length, of 1680 tons' 
trial displacement and made about 17 knots on trial. Suppose we 
wish, from Fig. 275, to determine the necessary I.H.P. for a vessel 
similar to the Yorktown, 289 feet long, of 3333 tons' displacement, 
and to make 17 knots. Then for the 289-foot vessel 

V_ 17 

VL V2&9 

From Fig. 275 when 

V I 

VL & 

Also (3333)*= 12,881, 

whence for the 289-foot vessel to make 17 knots 

7 = .415 X 12,881 = 5345. 
This is very simple, but for practical work it is convenient to 

plot our data a little differently. The curve of , in Fig. 275 is 

quite steep and varies a great deal as we pass from low to high 



So let us use instead a curve of 
/ ' / V \ 3 7 

Then / = N 


This is a convenient form. We may call N 

the Extended Law of Comparison coefficient. 

Figure 276 shows curves of N as deduced from trial results for the 
Yorktown and several other vessels. The curves are numbered, and 
the dimension and proportion of the corresponding vessels are given 
below : 



L i 














(-) 3 

\ !J 








2 1O 


602 1 


118 i 

? 6 oc 

i? 82 






Cutter . . . 

1 88 










5 680 


4"? .4 


4. ^J. 



Birmingham. . . 

Paddle Str. 
B S 



399 2 







181 8 


76 2 I 



21 7? 


Connecticut.. . . 

B. S 





1 70. 7 


2^ .OO 



North Carolina 
St. Louis 

Arm. Cruiser 
Prot. Cruiser 
B S 






I 1OQ 





26. 7O 



Tug .... 

02 . 5 


I O^T 



2O. Q^ 



Sheadle . . . 

Lake Fgt 


I 33O7 








Ocean Fgt. . 



. 2004 




IO. 21 

It will be observed that in any particular case N is proportional 
to , and hence is proportional to the reciprocals of the Admiralty 


coefficients, which are both proportional to 

The Extended Law of Comparison method of estimating power, 
though better than the Admiralty Coefficient method, is essentially 
but an improved form of the latter. 

The assumption that all resistances follow the Law of Comparison 
is in error as regards the Skin Resistance. This tends to make us 
overestimate when powering a large ship from the results of a small 


ship, and vice versa. The efficiency of propulsion is not constant, 
and the efficiency of the new ship may be different from that of the 
old ship. This source of error is common to all methods of esti- 
mating power from trial results. 

We have seen that resistance is materially affected by variation 
of the displacement length coefficient and of the longitudinal 
coefficient. The method of the Extended Law of Comparison takes 
no account directly of such variations and is subject to error accord- 
ingly. In fact, curves of N, as in Fig. 276, are of very little value 
without full information as to the ships to which they refer. Thus, 


suppose we wish to power a ship for which = is to be .8. For this 

speed length ratio we find in Fig. 276 values of N which vary radi- 
cally. Thus Nos. 6 and 7 would give N = .275. There are a num- 
ber of other values between .30 and .35. Nos. 4 and 5 would give 
.475, while No. 3 would give .72. These values are thoroughly dis- 
cordant. It is evidently desirable, when powering a new ship, to 
use curves of N from ships of the same type having approximately 
the same longitudinal and displacement length coefficients. 

3. Powering Sheet using Extended Law. If. a number of speed 
and power curves of various types of ships are available, their prac- 
tical use in powering is materially facilitated by reducing them to 
curves of N, as in Fig. 276, but plotting these curves as in Fig. 277. 
A large sheet should be used, section ruled vertically with lines 
representing equal intervals of longitudinal coefficient and hori- 

/ L \ 3 

zontally with lines representing equal intervals of D -J- ( 1 as 
. \ioo/ 


Curves of N are placed upon this sheet so that the termination 
corresponding to the maximum trial speed is located at the point 

/ r \i 

corresponding to the longitudinal coefficient and the D -f- ( ) for 


the ship. All curves terminate at their other extremity where =. = 

.5, and a vertical line is drawn down from this extremity to the point 
where N = o or has a given value. For greater clearness each curve 


is numbered, and the corresponding spot where N = o, - = = .5 is 

marked O with a subscript number the same as the curve number. 

When for the datum point = =.5 but N is not o, the value of N 



is indicated. The same scales of =and of N are used for all 


curves, and being drawn upon a separate piece of tracing cloth can 
be adjusted over or under the main sheet so as to apply to any curve. 
Thus, in Fig. 277 the dotted lines represent the scale in position for 
No. 6 curve of N. When reliable data is available for but a single 
spot not a curve it may be located on the powering sheet as 
the spots marked 10, n, 12 in Fig. 277. Each spot must have its o 
spot also located as shown. 

When powering a new design we will know the values we expect 

/ jr \s y 

to use for longitudinal coefficient, for D -*[-] and f or Lo- 

\ioo/ \/L 

eating on the sheet the point corresponding to the longitudinal coeffi- 

/ L \ 3 

cient and the value of D -r- ( ) , it is obvious that the best curves 


of N to use are those terminating nearest to the located spot. 
Having selected the curves of N to be used, adjust the scale to the 
chosen ones in succession and from each curve take the value of N 

corresponding to the = for the new design. Sometimes there may 


be reasons for giving more weight to some of those values of AT 
than to others. If not, the average of the values of N is the proper 
value to use for the new design, as a basis for an estimate of the 
neat power and the variation in the various values will assist in 
fixing the margin of power which should always be allowed over and 
above the neat estimated power. In practice several values of N 

should be taken from each available curve corresponding to definite 

values of - and an estimated curve of I.H.P. determined extending 

above and below the intended speed of the new design. 

The few curves of N in Fig. 277 are shown simply to indicate 
how a working sheet should be prepared. Such a sheet should have 


a large number of curves on it, the more the better, but no curves 
or spots should be used which are not derived from reliable results 
of careful trials. Published trials are not always reliable. 

The advantage of a powering sheet laid off as shown in Fig. 277 
is that when a designer is considering a question of powering it 
enables him to determine immediately whether his power data from 
previous ships is applicable to the case or whether he is working 
in a region not covered by reliable data in his possession. 

The error arising from the application of the results of a small 
ship to the powering of a large ship can be approximately corrected 
if estimates of the frictional effective horse-power at corresponding 
speeds of the two are made. By applying the Law of Comparison 
to the frictional effective horse-power of the small ship and deducting 
from the result the frictional effective horse-power of the large ship 
we determine the error in the effective horse-power incident to the 
use of the Extended Law of Comparison , and the error in the indi- 
cated horse-power will usually be about double that in the effective 
horse-power. By an obvious similar method we can correct when 
passing from a large to a small ship. 

The error due to variation of propulsive efficiency from ship to 
ship is not great when we use results of similar ships with somewhat 
similar types of propelling machinery. But caution should be exer- 
cised and liberal margins allowed if, for instance, we wish to power 
a turbine vessel and have available only data from vessels with 
reciprocating engines. 

The main difficulty with the Extended Law of Comparison method 
as a practical working proposition is the fact that few or no design- 
ers will have available reliable trial results which will cover the 
whole field of speed, longitudinal fineness and displacement length 

39. Standard Series Method 

i. Use of Standard Series Results. In addition to the com- 
paratively simple methods of powering ships described there have 
been many others proposed which are as a rule more complicated. 
Many involved formulae for resistance have been brought forward 
from time to time. 


Skin resistance is readily estimated by a formula using the coeffi- 
cients of Froude and Tideman, but no general formula giving resid- 
uary resistance accurately for any wide range of speed, proportions, 
and fullness of model has yet been brought forward. We have seen 
that the best approximate methods of powering hitherto used are 
all weak in leaving largely to the skill and judgment of the designer, 
to his guesswork, the effect of proportions and fullness of model, and 
that in order to make satisfactory guesses the designer must have 
an accumulation of data possessed by few. 

Now by the use of the data given in Figs. 78 to 120 it is possible 
to estimate with great accuracy the effective horse-power of a ship 
of any displacement, dimensions, and longitudinal coefficient upon 
the lines of the Standard Series. Furthermore, such a curve of 
effective horse-power will approximate fairly closely the E.H.P. of 
models upon different lines. For with displacement, length, mid- 
ship section area, and longitudinal coefficient fixed, any variations 
in shape that would be made in good practice will have a com- 
paratively minor effect upon resistance. Hence, with the aid of the 
Standard Series the problem of powering a ship is solved in two steps. 

First: From the Standard Series results get out a curve of E.H.P. 
for a ship of the same displacement, length, beam draught ratio and 
longitudinal coefficient. 

Second : From the E.H.P. estimate the I.H.P. by applying a suit- 
able coefficient of propulsion. 

When following this method there are two principal sources of 

First, there is the possibility that the lines used may differ so 
much from those of the Standard Series that the estimated E.H.P. 
may be materially in error. This source of error may be avoided 
by closely following the lines of the Standard Series unless lines 
positively known to be superior are available. 

Second, the coefficient of propulsion chosen may be in error. This 
is an unavoidable source of error, and it is on this point only that 
the designer, when using the Standard Series method, must use some 

2. Propulsive Coefficients to Use. When an accumulation of 
power data is not available, it is generally safe, when using lines 


closely resembling those of the Standard Series, to assume a nominal 
efficiency of propulsion in the vicinity of 50 per cent based upon 
indicated horse-power for reciprocating engines and somewhat less, 
say 46 per cent, for the usual run of turbine jobs, but using shaft 
horse-power in this case. These average efficiencies are based upon 
the E.H.P. of the bare hull and are sufficiently low to allow for the 
average run of appendages. 

The above is independent of accumulated data of experience and 
will enable fairly good results to be obtained without such data, but 
when such is available it should be made use of to the fullest extent. 

Thus, if we have a reliable speed and power curve of a vessel, we 
can estimate from the Standard Series the E.H.P. for a vessel on 
Standard Series lines having the same displacement, length, area of 
midship section and ratio of beam to draught. Then from the 
I.H.P. curve of the actual vessel we can determine the nominal 
efficiency of propulsion. The same nominal efficiency should be 
found for another vessel of the same general type as the vessel whose 
trial results are known, including type of engines and propellers. 
Or it may be that there is some change made in the new vessel which 
leads us to anticipate a certain reduction in nominal efficiency of 
propulsion. Knowing the old nominal efficiency and the probable 
reduction, the new nominal efficiency to be expected is determined. 

Any one who finds from reliable data of a given type of vessel a 
nominal efficiency of propulsion below 50 per cent, should be careful 
when powering a new vessel of the type to use the nominal efficiency 
based upon preceding results. Analysis of trial results by the aid 
of the Standard Series will disclose plenty of nominal efficiencies 
below 50 per cent. They may be due to lines inferior to those of 
the Standard Series, to inefficient propelling machinery, or to inac- 
curate power data. All trials are not handled so that the resulting 
speed and power data will be accurate. Still nominal efficiencies 
of propulsion of 50 per cent for indicated horse-power of recipro- 
cating engines and 46 per cent for shaft horse-power of turbines are 
often materially exceeded, and when it is found that they have not 
been reached endeavor should be made to locate the trouble. 

3. Advantages of Standard Series Method. The Standard 
Series method of estimating power has the great advantage that 


even if the resistance of a given ship is different from the correspond- 
ing Standard Series ship the variations of resistance with varying 
dimensions and shape of ships of the type will follow closely the 
variations deduced from the Standard Series. In other words, the 
Standard Series may be used as a reference scale to determine rela- 
tive resistances of ships of constant type of any dimensions and pro- 
portions. A tape measure need not be accurate to determine the 
ratio of two lengths, and even if from the Standard Series curves we 
cannot accurately estimate a priori the resistance of a ship of a given 
type, we can estimate with fair accuracy the ratio of the resistances 
of two ships of the type; and if we have accurate power data for one 
or more such ships we can use it to establish the proper nominal 
efficiency of propulsion from which, using the Standard Series, we 
can estimate with ample accuracy the power required for other ves- 
sels of the type. For this purpose it makes no difference whether 
the nominal efficiency is or is not the real efficiency of propulsion. 
If it is really typical of the type of vessel in hand it is adequate for 
powering purposes. 

The fact that the nominal efficiency of propulsion, which does 
not vary much without good reason, is the only quantity which must 
be estimated or guessed at from experience is much in favor of the 
Standard Series method. Furthermore, in the great majority of 
cases the efficiency of propulsion does not vary much in the vicinity 
of full speed. 

Hence, for practical purposes for use in future designs, we can 
characterize a complete trial by a single number, namely, the effi- 
ciency of propulsion whether actual or nominal. This is a great 
advantage where there is a mass of data to deal with. In the Stand- 
ard Series method of powering all other factors are taken care of by 
the method, automatically, as it were. 

While by the Standard Series method estimates of power are 
much simplified and should be made with more accuracy than by 
any of the other methods of approximation described, they are still 
estimated, and the designer should be careful always to allow a 
margin of power adequate to the necessities of the case. By any 
conceivable method of powering two sister ships would be given the 
same power for the same speed, yet sister ships do not always 


develop the same power on trial and do not always make the same 
speed for the same power. Changes from previous vessels made 
with a view to improvement sometimes turn out badly. Propellers 
frequently disappoint the designer, and the quick running propellers 
required by turbines are especially uncertain. 

The designer who is an optimist in choosing the efficiency of pro- 
pulsion to be expected may be very pessimistic after the trial. The 
time for pessimism is when the powering is being done, not when 
the trial is being run. 



Admiralty coefficient method, powering ships 294 

Air disengaged around moving ships 64 

Air friction, Zahm's experiments 82 

Air resistance 82 

Air resistance of planes 84 

Air resistance of ships 86 

Appendages, allowance for, in powering ships 126 

Appendages fitted on ships 1 23 

Area, coefficients of, for elliptical blades 134 

Area, developed, determination of 132 

Area of midship section, effect upon resistance 97 

Area of propeller blades, effect of 173 

Atlantic liner, design of propeller for 248 

Augmented surface method, Rankine's 291 

Babcock, measurements of settlement of ships in channels 1 20 

Back of blade 128 

Beam and draught, effect of ratio upon resistance 96 

Beaufoy's eddy resistance experiments, John's analysis of 69 

Bilge keels, resistance of 123 

Blade area, effect of 173 

Blade, back, variation of pitch over 159 

Blade sections, propeller, strength of 223 

Blade thickness, correction factors for 1 79 

Blade thickness, effect of 172 

Blade thickness ratio or fraction 130 

Blades, detachable, connections of 239 

Blades, propeller, forces on 216 

Blades, propeller, moments on 216 

Blades, propeller, number of 245 

Blades, propeller, stresses allowed 235 

Bow, change of level under way 108 

Bow, shape of, effect upon resistance 93 

Cargo vessel, design of propeller for 252 

Cavitation, accepted theory inadequate 183 

Cavitation, cause of 188 

Cavitation causes failure of Law of Comparison 151 

Cavitation, cure for 192 

Cavitation, effect of broad blades upon 190 


306 INDEX 


Cavitation, experiments with narrow and broad blades 191 

Cavitation, model experiments with 186 

Cavitation, nature of 182 

Cavitation, possible methods of experimental investigation 185 

Cavitation, possible theories of 184 

Cavitation, theory of 188 

Cavitation, visible phenomena in model experiments 188 

Centrifugal force, stresses due to 226 

Channels, shallow, settlement of ships under way. 120 

Coal endurance estimated from surface 292 

Coefficients of propeller performance, characteristic 161 

Comparison, Law of, deduction 26 

Compressive stresses on propeller blades 224 

Current, tidal, elimination of effect on trials 267 

Depth for no change of resistance 116 

Depth of various trial courses 118 

Design of propeller for Atlantic liner 248 

Design of propeller for destroyer 250- 

Design of propeller for gunboat 251 

Design of propeller for large cargo vessel 252 

Design, pS diagrams of 176- 

Design of propellers, reduction of model experiment results for 164 

Destroyer, design of propeller for 250 

Detachable blades, connections of 239. 

Developed area, determination of 132- 

Deviations of shafts, actual and virtual 211 

Deviations of shafts, virtual, due to motion of water 213 

Diagrams, pS, for design 176 

Diameter ratio defined 130 

Dimensional formulae 33 

Direction of rotation of propellers 245 

Disc area 132: 

Disc area ratio 132 

Displacement length ratio, influence on resistance 104 

Displacement length ratio defined 99 

Disturbance of water by a ship : 50 

Draught and beam, effect of ratio upon resistance 96 

Docking keels, resistance of 123. 

Eddy resistance, formulae for, inclined plates 71 

Eddy resistance, formulae for, normal plates 70 

Eddy resistance, formulae for practical use v 72 

Eddy resistance, limitations of rear suction formula 72 

Effect of foulness upon skin resistance 66 

Efficiency, effect of shaft inclination upon 214 

Efficiency^ hull 197 

Efficiency; maximum attainable in practice 177 

INDEX 307 


Efficiency, maximum of p8 diagrams 177 

Efficiency of a propeller, general considerations 138 

Efficiency of ideal propellers 153 

Efficiency of propulsion, determination from trial results 289 

Efficiency of propulsion, values for practical use 301 

Efficiency, propeller, deduction from experimental results 160 

Elliptical blades, coefficients of area for 134 

Endurance, coal, estimated from surface 292 

Expanded area 133 

Extended Law of Comparison, powering ships 295 

Face of blade 128 

Feathering paddle wheels 256 

Float area of paddle wheels 259 

Flow past vessel, lines of 52 

Focal diagrams 48 

Forces on propeller blades 216 

Four-bladed propellers, design from pd diagrams 180 

Four-bladed propellers, ratios connecting with three-bladed 180 

Friction, initial 279 

Friction, initial, determination of 281 

Friction in propeller action and head resistance 144 

Friction, load 279 

Froude, R. E., skin resistance constants 62 

Froude's Law 26 

Froude's propeller theory, formulas from 140 

Froude, W., skin resistance experiments 58 

Gaillard's experimental investigations of trochoidal waves 19 

Girth parameters 40 

Girths of sections 40 

Greenhill's propeller theory, formulae from 143 

Groups of waves 17 

Gunboat, design of propeller for 251 

Havelock's wave formulae 55 

Hovgaard's observations of wave patterns 55 

Hub, propeller, effect of size 169 

Hubs, propeller, fair waters to 125 

Hull efficiency 197 

Humps and hollows of resistance 78 

Immersion of propellers, effect on efficiency 210 

Inclination of propeller blades, effect of 168 

Inclinations of shafts, effect upon efficiency 214 

Inclination of shafts, effect upon vibration 214 

Inclination of shafts, virtual, due to motion of water 213 

Indicated horse-power, components of 279 

308 INDEX 


Indicated thrust 284 

Initial friction 279 

Initial friction, determination of 281 

Jet propulsion 260 

JoessePs eddy resistance results 68 

John's analysis of Beaufoy's eddy resistance results 69 

Keels, bilge, resistance of 123 

Keels, docking, resistance of 123 

Kelvin's wave patterns 53 

Kirk's method for powering ships 292 

Law of Comparison, application to centrifugal fans 31 

Law of Comparison, application to propellers 31 

Law of Comparison, application to ship's resistance 30 

Law of Comparison, application to steam engines 30 

Law of Comparison applied to propellers 150 

Law of Comparison, deduction 26 

Law of Comparison, formulae for simple resistances which follow 32 

Law of Comparison, not applicable to skin resistance 63 

Leading edges of propellers, fluid pressures at 194 

Length, effect upon resistance 98 

Length, illustration of influence upon resistance 105 

Level of vessel, change under way 52 

Level of water around vessel, change under way 52 

Lines of flow over vessel 52 

Load friction 279 

Location of propellers 241 

Longitudinal coefficient, effect upon resistance 97 

Longitudinal coefficient, influence on resistance in Standard Series : . . . 103 

Luke's experiments on wake fractions and thrust deductions 198 

Margin to be allowed in powering ships 303 

Material of propellers 247 

McEntee, limits of propeller efficiency 153 

Measured courses, desirable features of 262 

Measured miles, desirable features of 262 

Midship section area, effect upon resistance 97 

Midship section area, optimum for resistance 104 

Midship section coefficient, effect on resistance 95 

Midship section shape, effect on resistance 95 

Model basin methods 87 

Model propeller, analysis of experimental results 158 

Model propeller experiments, reduction for design work 164 

Model propeller, plotting experimental results 157 

Model propellers, experimental methods 155 

Model trial results applied to determine ship's resistance 88 

INDEX 309 


Moments on propeller blades 216 

Motion past a ship, differences from ideal 51 

Nominal pitch 158 

Nominal slip 158 

Number of blades of propellers 245 

Number of propeller blades, effect of 167 

Number of propellers 241 

Obliquity factor for wetted surface 38 

Obliquity of flow of water at propeller 215 

Obliquity scales for wetted surface calculations 38 

Paddle propulsion '..... 254 

Paddle wheel location 257 

Paddle wheels, dimensions and proportions 258 

Paddle wheels, feathering 256 

Paddle wheels, float area 259 

Paddle wheels, number of blades 260 

Paddle wheels with fixed blades 255 

Parallel middle body, curves for finding resistance of vessels with 107 

Parallel middle body, experiments on models with 106 

Parallel middle body, optimum percentages for resistance 107 

Parameters, girth 40 

Parent lines, derivation of models from 91 

Pitch angle 129 

Pitch, decreasing 129 

Pitch, increasing 129 

Pitch, nominal 158 

Pitch of back of blade 129 

Pitch of helicoidal surface 128 

Pitch ratio defined 1 29 

Pitch ratio, effect upon propeller action 171 

Pitch, variation over back of blade 159 

Pitch, variation of, for twisted blades 135 

Pitch, virtual 158 

Pitch, virtual, determination from experimental results 159 

Planes, resistance in air 84 

Plane, thin, flow past 67 

Powering ships, Admiralty coefficient method 294 

Powering ships, allowance for appendages 126 

Powering ships, extended Law of Comparison 295 

Powering ships, Kirk's method 292 

Powering ships, Rankine's method 291 

Powering ships, Standard Series method 300 

Practical application of model propeller results 175 

Pressure, fluid, at leading edges of propellers 194 

310 INDEX 


Progressive speed trials 264 

Progressive trials, accuracy of results attainable 265 

Progressive trials, conditions of 278 

Progressive trials, methods of conducting 271 

Projected area 132 

Propeller action, comparison between theories and experience 147 

Propeller action, comparison of theories 143 

Propeller action, formulae on various theories 146 

Propeller action, friction and head resistance 144 

Propeller action, Froude's theory of 138 

Propeller action, GreenhilPs theory of 138 

Propeller action, Rankine's theory of 138 

Propeller action, theories of 136 

Propeller, area of 132 

Propeller blade sections, strength of 223 

Propeller blades, forces on 216 

Propeller blades, moments on 216 

Propeller blades, stresses allowed 235 

Propeller blades, width of 247 

Propeller bossing or spectacle frames, resistance of 126 

Propeller, coefficients of performance, characteristic 161 

Propeller, delineation of 131 

Propeller design, reduction of model experiment results for 164 

Propeller efficiency, deduction from experimental results 160 

Propeller efficiency, ideal 153 

Propeller efficiency, McEntee's limits of 153 

Propeller immersion, effect on efficiency 210 

Propeller, obliquity of water flow at 215 

Propellers, location of 241 

Propellers, material of 247 

Propellers, model, analysis of experimental results 158 

Propellers, model experimental methods 155 

Propeller, model, plotting experimental results 157 

Propellers, number of 241 

Propeller suction 207 

Propulsion by jets 260 

Propulsive coefficients, values for practical use 301 

Propulsive efficiency, determination from trial results 289 

p5 diagrams for practical use 176 

Rake of propeller blades, effect of 167 

Rake ratio 130 

Ram bow, effect upon resistance 93 

Rankine's augmented surface method 291 

Rankine's propeller theory, formulae from 140 

Rayleigh's formula for eddy resistance 67 

Residuary resistance, analysis of curves 80 

Residuary resistance, curves of 79 

INDEX 311 


Residuary resistance from Standard Series 101 

Residuary resistance, method of plotting for analysis 90 

Resistance, air 82 

Resistance, air, defined 58 

Resistance, air, of ships 86 

Resistance coefficients and constants, variables used in plotting 98 

Resistance, depth for no change 116 

Resistance, disengaged air, effect upon 64 

Resistance, eddy, defined 57 

Resistance, eddy, formulas for 67 

Resistance, eddy, of inclined plane 67 

Resistance, factors affecting 92 

Resistance, increased in rough water 121 

Resistance in shallow water, percentage variations 118 

Resistance, kinds of 57 

Resistance of ship, deduction from model results 88 

Resistance, residuary, analysis of curves So 

Resistance, residuary, curves of 79 

Resistance, residuary, from Standard Series 101 

Resistance, residuary, method of plotting for analysis 90 

Resistance, shallow-water effects 112 

Resistance, skin and wave, relative importance 58 

Resistance, skin, defined 57 

Resistance, skin, determination for ships 99 

Resistance, skin, effect of foulness 66 

Resistance, skin, Law of Comparison not applicable 63 

Resistance, skin, of ships, deduced from plane results 61 

Resistance, skin, R. E. Froude's constants 62 

Resistance, skin, Tideman's constants 63 

Resistance, skin, variation of coefficients 60 

Resistance, skin, W. Froude's experiments 58 

Resistance, wave, defined 57 

Rota's experiments on depth and resistance 116 

Rotation of propellers, direction of 245 

Rough water, reduction of speed in 121 

Screw, true 128 

Sections, girths of 40 

Settlement in shallow water HQ 

Settlement of ships in shallow channels 120 

Shaft bossing, effect on wake 205 

Shaft brackets, effect on wake 205 

Shaft deviations, actual and virtual 211 

Shaft deviations, virtual, due to motion of water 213 

Shaft inclination, virtual, due to motion of water 213 

Shallow water, changes of trim and settlement 119 

Shallow water, effect upon resistance 112 

Shallow-water resistance, percentage variations 118 

312 INDEX 


Shape of bow and stern, effect upon resistance 93 

Shape of midship section, effect upon resistance 95 

Sink and source motion 4 

Sink and source motion in uniform stream 5 

Skin resistance determination for ships 99 

Slip angle 130 

Slip-angle values 148 

Slip, effect of, upon propeller action 174 

Slip, nominal 158 

Slip of paddle wheels 255 

Slip percentage. . . 131 

Slip ratio 131 

Slip, variation because of shaft inclination 211 

Slip, virtual I5& 

Spectacle frames or propeller bossing, resistance of 126 

Speed and power trials, general considerations 264 

Speed of advance 130 

Speed of propeller 130 

Speed of slip 130 

Speed ratio 131 

Speed, reduction in rough water 121 

Speed trials, progressive 264 

Squat in shallow water 119 

Squat under way 108 

Standard Series method of powering ships 300 

Standard Series method of powering, advantages of 302 

Standard Series method of powering, margin to allow 303 

Standard Series of model propel'ers 1 70 

Standard Series, residuary resistance from 101 

Stanton's eddy resistance results 70 

Steady motion formula i 

Steady motion formula, failure of 3 

Steady motion past ships 2 

Stern, change of level under way A 108- 

Stern, shape of, effect upon resistance 93 

Stream forms 6 

Stream lines i 

Stream lines around sphere 10 

Stream lines past elliptical cylind rs 7 

Stream motion past a ship, ideal 51 

Strength of propeller blade sections 225 

Stresses allowed in practice on propeller blades 235 

Stresses, compressive, on propeller blades 224 

Stresses due to centrifugal force of propellers 226 

Stresses, tensile, on propeller blades 226 

Struts, resistance of 1 24 

Suction of propellers 207 

Superposition of trochoidal waves 17 

INDEX 313 


Tensile stresses on propeller blades 226 

Thickness of propeller blades, effect of 172 

Thrust deduction 197 

Thrust deduction, approximate determination of 200 

Thrust deduction coefficient 197 

Thrust deduction, determination from trial results 290 

Thrust deduction factors 197 

Thrust deduction, variation of 198 

Thrust, indicated 284 

Tidal current, elimination of effect on trials 267 

Tideman's skin resistance constants 63 

Trials, progressive, conditions of 278 

Trials, progressive, methods of conducting 271 

Trial results, analysis of 279 

Trim, change of, in shallow water 119 

Trim, change of, under way 108 

Trim, effect upon resistance 94 

Trim of vessel, change of, under way 52 

Trochoidal theory of waves, applicability of 18 

Trochoidal theory of waves, Gaillard's investigations 19 

Trochoidal wave theory 1 1 

True screw 128 

Twisted blades 135 

Two-bladed propellers, ratios connecting with three-bladed 181 

Vibration, effect of shaft inclination upon 2 14 

Virtual pitch 158 

Wake, components of 195 

Wake, effect of shaft brackets on 205 

Wake factor 197 

Wake fraction, approximate determination of 200 

Wake fraction, determination from trial results 290 

Wake fraction, estimates from trial results 201 

Wake fractions 196 

Wake fractions, variation of : 198 

Wake, frictional 195 

Wake, how it affects propulsion 196 

Wake, percentage, Froude's expression 197 

Wake, stream line 195 

Wake, wave 195 

Wave formulae, Havelock's 55 

Wave formulae, Kelvin's 53 

Wave groups 17 

Wave patterns, Hovgaard's observations 55 

Wave patterns, Kelvin's 53 

Wave patterns of ships 55 

314 INDEX 


Wave resistance 73 

Wave resistance, general formula for 76 

Wave resistance, humps and hollows in 78 

Waves dimensions of actual 23 

Waves, energy of trochoidal 15 

Waves, formulae for trochoidal 13 

Waves o c translation 23 

Waves, relation to wind causing them 24 

Waves, shallow water, trochoidal 21 

Waves, so'itary 21 

Waves, superposition of trochoidal 17 

Waves, trochoidal 1 1 

Wave system, resultant 74 

Wave systems, bow and stern 73 

Wetted surface calculations, correction factors for 40 

Wetted surface calculations, form for 39 

Wetted surface coefficients, average 46 

Wetted surface coefficients, variation of 44 

Wetted surface, factors affecting 47 

Wetted surface, formulae for 43 

Wetted surface, obliquity factor for 38 

Wetted surface of appendages 37 

Width of propeller blades 247 

Winds, relation to waves produced 24 

Yorktown model results compared with Standard Series 103 

Zahm's experiments on air friction .,,,,,,,,.,, 82