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AFCRL-63-596 


STUDY DIRECTED TOWARD OPTIMIZATION OF OPERATING 
PARAMETERS OF THE EMAC PROBE FOR THE REMOTE 
MEASUREMENT OF ATMOSPHERIC PARAMETERS 


By 

Clayton H. Allen* 
and 

Stephen D. Weiner* 


*BOLT BERANEK AND NEWMAN INC. 
50 Moulton Street 
Cambridge 38, Massachusetts 


FINAL REPORT 

Contract No. AF 19(628)2774 
Project 6672, Task 667205 


September 19c3 


Submitted to: 

Air Force Cambridge Research Laboratories 
Office of Aerospace Research 
United States Air Force 
Bedford, Massachusetts 





Report No. 1056 


Bolt Beranek and Newman Inc 


TABLE OF CONTENTS 

Page 

1. INTRODUCTION... 1 

1.1 Statement of the Problem . .. 5 

1.2 Atmospheric Parameters of Interest ....... 5 

1.3 Means for Remote Measurement . 3 

2. RADAR REFLECTION COEFFICIENTS . 10 

2.1 Reflection From a Sharp Dielectric 

Discontinuity.„. 10 

2.2 Reflection From Gradual Dielectric 

Variations. 10 

2.3 Reflection From a Sinusoidal Wa.a Train. .... 17 

2.4 Normal Incidence Reflection From a 

Train of Plane Shock Waves. . .. 19 

2.5 Normal Incidence Reflection From 

One Plane Shock Wave. 21 

3. VARIATION IN REFLECTION DUE TO INHOMOGENEITIES. ... 23 

3.1 Wavefront Deformation Due to 

Atmospheric Inhomogeneities . 23 

3.2 Reflection From Deformed Wavefronts. 37 

4. SOUND PROPAGATION IN THE ATMOSPHERE. 50 

4.1 Spherical Divergence ..... . 51 

4.2 Directivity. 52 

4.3 Atmospheric Absorption . . 56 

4.4 Nonlinear Sound Propagation. 61 


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5. SOURCE CHARACTERISTICS FOR MAXIMUM RANGE. 80 

5.1 Acoustic Source. .. 80 

5.2 Electromagnetic Source . ..... 82 

6 . ACCURACY OF MEASUREMENT OF ATMOSPHERIC PARAMETERS . . 84 

6.1 Wind Speed In Direction of Search. 84 

6.2 Wind Direction. 86 

6.3 Turbulence. 88 

6.4 Possibility of Differentiation Between 

Inhomogeneities of Various Kinds. ...... 91 

6.5 Temperature Discontinuities. 91 

6.6 Humidity Changes . 92 

6.7 Maximum Range of EMAC Probe. .. 92 

7. PRELIMINARY EXPERIMENTAL SYSTEM . 95 

8 . CONCLUSIONS .. 100 


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LIST OP FIGURES 

Following 

Page 

Fig. 2,1 Index of Refraction vs. Distance 

Arbitrary Variation .... . 22 

Fig. 2.2 Index of Refraction vs. Distance 

Linear Shock.... 

Fig. 2.3 Relative Reflection Coefficient vs. A/?v 
for Shocks With Two, One, and e 

Zero Sharp Edges.. 

Fig. 2.4 Radar Pulse Shape ..... . 

Fig. 2.5 Relative Power Reflection Coefficient vs. 

X g /X a for Different n . 

Fig. 2.6 Index of Refraction vs. Distance Repeated 

Sawtooth. ...... . 

Fig. 2.7 Radar Power Reflection Coefficients vs. 

Shock Pressure Level... 

Fig. 2.7a Correction for Radar Reflection from 
Pressure Discontinuity as Plotted 
in Fig. 2.7 . 

Fig. 3.1 Wavefront Shapes in Steady Wind. 49 

Fig. 3.2 Path of Sound Ray . 

Fig. 3.3 Acoustical V/avefront Shape and Sound Ray 

Path in Constant Wind Gradient (0.2 ft/sec)ft 

Fig. 3-4 Acoustic WavefrGnt Shape and Sound Ray 

Path in Turning Wind. 

Fig. 3.5 Sound Ray Path in Layered Medium. 

Fig. 3.5 Acoustic Wavefront Shape For Constant Tempera¬ 
ture Gradient l°C/400 ft . 


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LIST OP FIGURES (CONT) 


Pig. 3.7 
Pig. 3.8 
Fig. 3.9 

Pig. 3.10 

Pig. 3.11 
Pig. 3.12 

Fig. 3.13 
Pig. 4.1 

Fig. 4.2 
Fig. 4.3 

Fig. 4.4 

Fig. 4.5 
Fig. 4.6 

Fig. 4.7 
Fig. 4.8 


Following 

Page 

Wavefront Distortion by Turbulent Eddy. . . 49 

Radar Ray Path in Steady Wind . 

Radar Reflection From Deformed Acoustic 
Wavefronts . 

Diameter of Illuminated Portion of 

Wavefront vs. Range. .... . 

Reflection From Rough Wavefront . . 

Normalized Received Intensity, Y vs. 

Normalized Range, X . 

Reflection From Two Wavefronts. 

Experimental Value of Sound Pressure 
Level on Axis of Plane Piston Source 
5 A Diameter. 79 

Attenuation for Plane Sound Wave. ..... 

Divergence and Attenuation for a Spherical 
Sound Wave. 

Absolute Humidity for Maximum Molecular 
Absorption vs. Frequency. 

Plot of a/a Jnax vs. h/h m (After Harris)^. . 

Extended Plot of Maximum Molecular 
Absorption Coefficient a Versus 
Frequency at Various Temperatures .... 

Chart for Converting Units of Humidity. . . 

Sound Pressure Level In Air Averaged 
Over the Face of a Plane Circular 
Radiator, 5 Wavelength in Diameter, 
in a Baffle (Dia. 4.8", Frequency 
14.6 kc) System Gain Adjusted to 
Give Equal Trace Height . 















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LIST OF FIGURES (CONT) 

Followin 

Page 

Fig. 4.9 Reciprocal Pressure in a Sawtooth 
Acoustic Wave vs. Distant' in 

V.'avelengtns. 79 

Fig. 4.10 Plot of SPL for the Limiting Pressure p^ 
for a Plane Wave and the Calculated 
Pressure for Two Plane Waves Having 
Pressures p 1 and p 2 at the Source. . . . 

Fig. 4.11 Plot of 10 log ( --- 

Fig. 4.12 Plane and Spherical Wave Finite 

Amplitude Limits . .... 

Fig. 4.13 Experimental Values of Sound Pressure 
Level in the Far Field of a Plane 
Piston Source pA in Diameter . 

Fig. 4.14 Sound Pressure Level Expected With 

Midwest Research Sound Source Con¬ 
sidering Molecular Absorption 
Neglecting Finite Amplitude Limits . . . 

Fig. 4.15 Sound Pressure Levels Expected With 
Midwest Research Sound Source Con¬ 
sidering Both Finite Amplitude 
Limits and Molecular Absorption. 

Fig. 4.16 Sound Pressure Level for 1140 cps 

Signal Radiated From 10 ft Diameter 
Source 'with Average SPL of l4p db 
Near Source. 

Fig. 4,17 Sound Pressure Level for 114 cps 
Signal Radiated From 10' Dia. 

Source ’With Average SPL of 175 db 
Near Source. 

Fig. 4.18 Oscilloscope Trace of Sonic Boom 

Signature Boom No. 7 (Table 4.1) . . . . 










Pig. 6 

Fig. 6 
Fig. 6 

Fig. 6 

Fig. 6 

Fig. 6 
Fig. 6 
Fig. 6 
Fig. 7 


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Beranek and Newman Inc. 


LIST OF FIGURES (CONT) 


Following 

Page 


Figure For Finding Components of 
Acoustic Velocity Along Radar Ray. 


2 Function 


, V 2 
cos9 - ~ sin 
a 


vs. v, 


3 Measurement of Horizontal Wind 

Components With Single Probe .... 

4 Relative Error in Horizontal Wind 

Components vs. Angle Be* nTr, en 
Probing Directions ......... 

5 Possible Curve For Phase Difference 

Between Transmitted and Received 
Radar Signal as Function of Time . . 

6 Possible Freouency Spectrum of Doppler 

Shift. 


7 Spread in Doppler Shift vs. Diameter 

of Echoing Region. 

8 Sound and Radar Reflection From 

Temperature Discontinuity. 

1 Schematic Views of the Proposed System 

For Radar Detection of a Sonic Boom. . . . 


94 


99 


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Acknowledgements 

Grateful acknowledgement is made of the help provided by Dr. David 
Atlas, Dr. Roger Lhermitte and Mr. Kenneth Glover, members of the 
Weather Radar Research Branch of the Air Force Cambridge Research 
iAiL/C-iatoi,., xii Gisv.wbo.Uig the immediate problems of radar usage 
for weather studies and in providing innumerable references which 
served as valuable background for the present study Particular 
mention is also made of contributions of Samuel J. Mason, Professor 
of Electrical Engineering at Massachusetts Institute of Technology 
and Dr. John Ruze of Lincoln Laboratory in their discussions relat¬ 
ing to radar antenna performance and of Karl Pearsons of Bolt 
Beranek and Newman Inc., in reviewing and analyzing his measure¬ 
ments of sonic booms in order to show details of the wave signatures. 
Gratitude is also expressed for the many contributions from 
colleagues in discussions of particular aspects of sound generation, 
long range sound propagation and the effects of impulsive sound on 
personnel. 


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ABSTRACT 


The concept of utilizing sound waves as reflectors for pulsed 
Doppler radar as a means for measuring wind velocity, turbulence, 
and air temperature has been examined theoretically. Any 
extension of the initial and successful, small scale experiuK its 
performed by Midwest Research Institute to a practical system 
for atmospheric probing is snown to require a change in the 
operating concept of the acoustic system. This change involves 
the abandoning of the concept of coherent reflection reinforce¬ 
ment from a multiple wave train and the substitution of 
reflection from a single acoustic shock front with the introduc¬ 
tion of coherent integration of the pulsed Doppler radar signal. 

A preliminary experimental approach to a practical system is 
proposed. 


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1. INTRODUCTION 


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The use of radar to detect meteorological disturbances is now 
-iuite commonplace. Weather fronts can be observed, wind dis¬ 
turbances behind a front carrying humid air off the ocean upward 
and mixing with relatively dryer air are clearly visible under 
some conditions-/— the problem of "an .ais" and clear air turbu¬ 
lence detection by radar are being stuuied with some success 
Back scattering from turbulence and precipitation are being used 
to study storms and evaluation of vertical winds in storm centers. 


In storm centers radar is reflected from the interfaces where 
gros 3 difference in refractive index exists between air and water 
or ice particles. In the study of weather fronts and turbulence, 
use is made of the much smaller index variations caused by changes 
in density as between warm and cold air, moving and stationary air 
in turbulence or between dry and humid air These variations in 
index though small amount uo several N-units, i.e., several parts 
per million in the index of refraction of air which itself is of 
the order of 1.000,320 . 

The present study considers the use of a sound wave as a reflecting 
surface. Such a surface has the great advantage of being available 
ujon command and having a relatively large area oriented exactly 
or nearly in such a way as to focus the reflected radar beam back 
toward the receiver. Unfortunately, a sound wave which can be 
tolerated by personnel and buildings in the vicinity of the sound 
source can provide a change in Index of refraction wh_eh is small 
compared with the changes associated with normal variations 
occurring naturally In the atmosphere. Near the source sound 
levels higher than about l60 db would not be tolerable; these 
would create index variations of about 10 N-units. 









Bolt Beranek and Newman Inc. 


As sound waves propagate away from the source, they decrease in 
amplitude because of divergence of the sound, normal absorption 
of sound as heat in the atmosphere, and because of excess absorp¬ 
tion caused by finite amplitude losses when large sound waves 
are used. All of these factors limit the amount of sound which 
can ce propagated any great distance from the source. In order 
to observe sound waves which travel 2 miles or more from the 
source it appears necessary that the radar system be able to 
detect changes in index of refraction which are small compared 
with an N-unit. 

The fact that a sound wave gives a large, substantially coherent 
surface from which to reflect radar aids in making possible the 
detection of its small change in index of refraction. 

Pioneering theoretical and experimental work on the £HAC probe 
carried on by Midwest Research Institute has shown the feasibil¬ 
ity of this tool for measuring wind velocity. Experiments have 
checked well with theory. 

In summary, the efforts of MRI have been directed toward over¬ 
coming the limitations of the small change in index of refraction 
associated with a sound wave by using a train of many waves and 
obtaining coherent reinforcement of the reflections from the 
individual waves by matching the radar and acoustic waves accord¬ 
ing to the equation 

X = 2X (1.1) 

e a 

where is the electromagnetic wavelength is the acoustic 
wavelength. 


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With an exact wavelength match, a reflection with zero phase 
change occurs at each index rise and a reflection with 180 ° phase 
change occurs at each index fall along the wave train. Thus, a 
train of 100 waves gives the effect of 200 mirrors. If coherence 

is maintained throughout the entire train of r. waves the net 

2 ? 
reflected power will be n times the power from 1 wave or 4n 

times the power from 1/2 of a sinusoidal sound wave. 

* 

The advantage of coherent reinforcement resulting from a train of 
waves Is indeed inviting, but it has serious limitations in a real 
atmosphere with wind, turbulence, and other inhomogeneities. It 
Is necessary that the wavelengths of radar and sound match accord¬ 
ing to Eq. 1.1 to within 1/4 A over the full length of the train 

2 a 

in order to obtain the n advantage. Such a match made at any 
location generally will not remain a match as the sound train 
passes into a region of different temperature or a region where 
wind changes the sound ground speed. 

Thus, in order to obtain a match at a new location the radar 
frequency must be altered so that the radar wavelength tracks 
the sound wavelength at the location of the reflection. 

This, at best, involves complicated tracking circuits and requires 
expenditure of some tracking time for the optimization of the radar 
frequency. Inherent in the fact that the air is generally turbu¬ 
lent and otherwise inhomogeneous Is the concomitant fact that the 
acoustic wavelength along a wave train will not be constant and 
in general there will be no one radar frequency which can satisfy 
the requirement for wave matching over more than a very limited 
wave train length. The seriousness of this limitation increases 
with the inhomogtneity cf the air being studied. 


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3olt Beranek and Newman Ine. 


In order to localize the region of the atmosphere being studied 
the wave train should not be more than several feet long. In 
order that such a wave train contain many wavelengths the waves 
must be short. Midwest Research Institute used an acoustic sig¬ 
nal of 22 kc for which the wavelength was approximately 1/2 inch. 
For such a beam the length of 100 waves is only 50 inches, a very 
reasonable length permitting fine detail in atmospheric probing. 
However, sound at this high frequency and small wavelength Is 
rapidly attenuated in air. In their experiments, measurements 
could not be carried beyond about 93 feet. 

Calculations in Section 4.43 show that the experimental system 
used by MRI was very nearly optimized for the acoustic frequency 
used, and that no increase in range may be expected by increas¬ 
ing the size or power of the acoustic source. Some gain might 
be secured from an increase in radar power but at 100 ft the 
acoustic wave had a sound level of the order of 100 db and 
decreased so rapidly that within a fetv feet It would be at the 
noise level expected in a turbulent atmosphere. 

The present study extends the concept explored by MRI, and con¬ 
siders the use of individual shock fronts as the reflector for 
the radar signal, since such shcck fronts can be made to propa¬ 
gate and maintain useful intensity for ranges of several 
thousand feet. This study indicates the direction which should 
be taken in developing the EHAC Probe into a practical tool. 


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1.1 Statement of ti 


A 
A Vi 


u n ~ — 

X X \J U J-CJll 


The present study of the parameters governing the operation of 
the £HAC Probe has been motivated by the need for the measure¬ 
ment of the atmospheric conditions at distances remote from the 
measurement position on the ground. Specifically, this study 
is directed towards measurement of wind and temperature in the 
atmosphere as an aid in leather observation and as an aid in 
aircraft and missile guidance problems where such detailed 
information within a range of a mile or two from the source is 
needed on a substantially instantaneous and continuous basis. 


1.2 Atmospheric Parameters of Interest 

In addition to the measurement of wind it is desirable to measure 
or at^ least to obtain qualitative description of the wind shear, 
turqu.lence, humidity, and temperature variation throughout this 
field of search. The present study considers the possibility 
of observing these parameters with the 2MAC Probe. 


1.3 Means for Remote Measurement 

Conventional techniques for remote measurement of wind velocities, 
such as the visual observation of free balloons or clouds, and 
radar interception of chaff or naturally occurring inhomogenei¬ 
ties in the atmosphere have serious limitations arising either from 
the delayed response or the relatively small and highly unpredict¬ 
able region which may be covered by such measuring techniques. It 
is desired to be able to measure the wind velocity at any height 
and in any direction from a fixed observation point in a substan¬ 
tially continuous manner in order that the total wind field in 
the vicinity of the measuring point can be determined completely. 



Bolt* Bersnek and Nsvnnsn Inc. 

Several means for Indirect probing of the atmosphere are under 
consideration by various agencies. A good discussion of these 
probing methods has been published.-^/ They include the use of 
infra-red radiometry, optical lasers, and radar of several types 
including coherent pulse Doppler radar. All of these depend 
upon observing particles, inhomogeneities or density irregulari¬ 
ties in the atmosphere. Interpretation of the reflected signal 
in many cases can give significant information concerning the 
nature of irregularities causing the reflection and about their 
motion in the atmosphere. However, since these irregularities 
are random in nature and since there is no control over their 
position, the detailed measuring of atmospheric parameters 
throughout the region surrounding the measuring point is generally 
incomplete and therefore the mapping of such parameters necessarily 
involves large extrapolation of the observable data. This process 
gives insufficient accuracy for many purposes. 

Wind velocity and turbulence can be measured by obtaining radar 
reflections from density irregularities in the atmosphere or 
solid particles such as rain, snow, chaff and fog suspended in 
the air. Such measurements approximate the motion of the air 
since the particles observed follow the motions of the air 
fairly accurately If their size is less than 1 mm in diameter. 

In some instances the observed objects are dropping through the 
air at speeds which are large compared with the velocity of the 
air Itself. Chaff, which is light and can fall slowly, has 
limited application since it must be carried to a position 
above the point of observation and allowed to drift at the mercy 
of the elements hopefully into the region of Interest. 




Bolt Beranek and 


KT r» 

jiu man 


Inc • 


1.4 EMAC Probe Technique 

The electromagnetic acoustic (EMAC) probing method provides a 
reflecting surface which moves through the atmosphere with the 
speed of sound altered only by variations in wind 3peed and 
air temperature. The reflected signal carries information by 
which the speed of the sound wave can be determined in the 
direction of the radar beam. By combining the information gained 
from reflected signals in various directions it appears possible 
and practical to calculate, not only the speed of the wave in 
the direction of the radar but to deduce the actual wind speed 
and direction, as well as to determine the air temperature and 
estimate the amount of turbulence existing in various regions 
within the range of the system. 

The use of radar to observe or interrogate vibrating media, 
surfaces, or objects is not nevj. Basic patents-^/reading on the 
art of detecting and measuring the velocity or vibration of air, 
liquids, or solid objects were issued on disclosures made during 
World War II. 

The application of this art to the specific problem of measuring 
wind velocity by reflecting radar pulses from intense 3ound 
waves as described in reports by Midwest Research Institut e^* —^ 
demonstrates the feasibility of the process. Comparison of the 
theory and the experimental results indicates that as yet the 
theoretical limitations on the useful range of the velocity 
measuring technique have not been approached. This report pro¬ 
vides a theoretical discussion of the parameters influencing the 
optimization of the range and sensitivity of the EMAC Probe. In 
particular, theory and experience available regarding acoustic 
wave propagation in the atmosphere and the more subtle finite 


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amplitude phenomena associated with the propagation of sound 
waves of large amplitudes indicates that the range of the 
EMAC Probe can be greatly extended by proper choice of the 
acoustic wave parameters. 

The electromagnetic acoustic probe as described by Midwest 
Research Institute has proved successful out to distances 
approaching 100 ft but the extension of its useful range requires 
significant changes in its operating parameters. The choice of 
high frequency sound waves is its greatest limitation. A suffi¬ 
ciently large reduction of the frequency, however, results in 
such long acoustic wavelengths that it becomes impractical to 
use a radar wave which is comparable In length with the acoustic 
wave and still maintain the degree of beam definition which is 
necessary for fine scanning and analysis of wind and turbulence 
structure. Thus, the radar wavelength must be kept relatively 
short compared to the acoustic wavelength. 

This change appears at first to imply that the reflection 
coefficient for the acoustic wave will drop severely, but it 
is possible to utilize a sawtoothed sound wave which has a 
steep leading edge. This will have the reflectivity of a pressure 
discontinuity less than 1 ft' in thickness. Such a discontinuity 
will act as a good reflector for a radar wave of the order of 
2 ft in length, i.e., a 400 megacycle frequency. 

The use of a long acoustic wave necessitates abandoning the 
concept of coherence between acoustic wave fronts. This loss, 
however. Is not as serious as may appear from theoretical con¬ 
siderations of ideal wave propagation conditions. Such coherence 
would be effective only in homogeneous air masses which are of 
small interest and highly improbable in a real atmosphere 





Bolt Beranek and Newman me. 


outside of the laboratory. It is more realistic to substitute 
coherent integration between radar pulses since the radar speed 
is substantially unaffected by the atmospheric variations of 
wind turbulence, etc., and within the time interval of $0 radar 
pulses (repetition rate of 3000 pps) the sound wave will travel 
only 15 ft and the wind and turbulence velocities will remain 
substantially constant over all points within the radar beam 
cross section. By this means, full advantage can be taken of 
a fifty pulse coherent integration. Such a change in operating 
technique will more than make up for the loss in potential gain 
from multiple wave reflection. This mode of operation will also 
eliminate the need for frequency variation in the radar which 
was required to match lengths between radar and acoustic waves. 
The use of a fixed radar frequency will eliminate one search 
dimension and permit the more rapid accumulation of data and 
the more thorough search of the dimensions which are of direct 
interest. 

The results of the present analysis show that an EKAC Probe 
system employing proper sound pulses can provide a substan¬ 
tially continuous sweep scan of the hemispherical atmospheric 
region around the observation point and provide a relatively 
complete map of the wind velocity and turbulence in this region 
out to a distance of about 2 miles. The ultimate range will 
depend primarily upon the weather conditions, the amplitude of 
the sound waves permitted at the source as determined by 
community and personnel considerations, and upon the sensitivity 
of the radar receiving system. 


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2. RADAR REFLECTION COEFFICIENTS 

2.1 Reflection From a Sharp Dielectric Discontinuity 

The simplest case of reflection of electromagnetic waves is by a 
plane discontinuity in the index of refraction n. A change in n 
is "sharp" if it takes place over a distance short compared with 
one quarter of the electromagnetic wavelength. For reflection 
at normal incidence, the Fresnel formula for reflected power gives 


n l “ n o 

n i + n 0 


( 2 . 1 ) 


where P^ , P r are incident and reflected radar powers respectively 
, Uq indices of refraction on opposite sides of the discontin¬ 
uity. 

In the atmosphere, n is nearly unity and the variation in n 
obtainable with usable pressure discontinuities is so small com¬ 
pared with unity that Eq. (2,1) can be replaced by the simpler 
form 


= (^f 

\ £ J 


( 2 . 2 ) 


where 5n = - 


2.2 Reflection From Gradual Dielectric Variations 

For a gradual change in n, reflection of the radar will occur 
at all points in the region of variation. Since, as the radar 
wave progresses, its phase changes from point to point, the 


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reflected contributions of each point must be added in proper 
phase to obtain their sum. This requires determination of the 
reflected electric field in order to determine the reflected 
power, rather than calculation of the power directly. The 
reflected wave is found by replacing the continuously varying 
n{x) [Fig. 2.1a] by a series of steps [Fig. 2.1b], 


The Fresnel formula for the electric field amplitude reflected 
from each step dn(x) is 



E i (x) dn(x) 

2 


(2.3) 


By allowing the incident electric field amplitude E^x) to vary 
with x, we can take into account effects of different pulse shapes 
as discussed in Siegert and Goldstein.-^/ For an electromagnetic 
wave have a wavelength a the electric field contribution 
reflected from point x has a phase of 2(2mx/?v ) relative to the 
contribution reflected from x = 0. Thus, the amplitude and 
phase of dE i3 given by 

A 

E.(x) dn(x) 

dE = ———5 - exp [-4*±xA ] (2.4) 

A c u 


As dx —> 0. the sum of the reflected contributions from all 
elements, dx, between x Q and x^, can be expressed as the integral 


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n. 


r n i E i (x) 


exp 


- ^ dn(x) 

*e J 



E^x) exp 



dx 


(2.5) 


if multiple reflections are neglected. The reflected power 
is the square of the magnitude of the reflected E field. 


There are many combinations of pulse shape and index of refraction 
variations which are of interest. We will consider a few of these 
special cases below. 


Case A. Radar pulse length Infinite (2., = const). 
In this case Eq. (2.5) becomes 



"i / an 

2 J Hx 

x 0 


exp 



4irlx 

^e J 


dx 


( 2 . 6 ) 


A sub-case of Case A is that of linear variation of n from 
Uq to n^ over a distance £ as shown in Pig. 2.2. 


In this case 





p - _ 

a i n 


8Ti 3- 


exp 



( 2 . 7 ) 


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or 


P,&n 2 


The value of 4P r /[P i (&n) 2 } Is plotted vs. A/A e In Fig. 2.3. 

This gives the reflection coefficient relative to that for an 
infinitely sharp shock. For A « this case reduces to that 
of the 3harp discontinuity discussed in Sec. 2.1. 

The variation in n shown in Fig. 2.2 and used in calculations 
above has two "sharp" edges at x = x, and x = x + A . The 
interference between these edges produces maxima and minima In 
the reflection coefficient as shown in Fig. 2.3 We can also 
calculate the reflection from shocks with one or zero sharp edges. 
A shock with one sharp edge at x = 0 and the other rounded such 
that 


sin (£*A/Jv e ) 


T C. 


( 2 . 8 ) 


n = n Q x < 0 

(2.9) 

n = n Q + &n [1 - exp (~x/A)j x > 0 

for which the shock width is defined by A. For this variation 
in n, the power reflection coefficient is 


P r / &n\ 2 _ I 

' 2 / 1 + 16tt 2 A 2 /* 2 

' e 


( 2 . 10 ) 


which is also plotted in Fig. 2.3. This curve does not show the 
maxima and minima of Eq. (2.8) but falls off as (?vg/A) 2 as does 
the average value for wide shocks described by Eq. 2.8. 



Bolt Beranek and Newman Inc. 


to/ 

Friend^has calculated the reflection from index of refraction 
changes with no sharp edges. He considers a change, of "width" 
A, of the form 


n 


n o + 


6n ■ ex P (»,$/*),-■ 
1 + exp (4 x /A; 


( 2 . 11 ) 


for which the reflection coefficient is 


P r = /6n\ 2 9 V A e 

\ 2/ sinh (WX e ) 


( 2 . 12 ) 


which is also plotted in Fig. 2.3. The reflection from this 
shock falls off much more rapidly with shock thickness than the 
reflection from the shocks with either one or two "sharp" edges. 

This fret may he of practical significance in regard to reflection 
from acoustic shock waves. The leading edge of the shock front, 
at which the density begins to rise, appears experimentally to be 
much sharper than the crest of the wave at large distances from 
the source. This matter is discussed in more detail in Sec. 4.43. 

Case B . Finite radar pulse length 

To show the importance of the radar pulse shape, we will consider 
a linear variation in n extending over all space. Thus = constant. 
In this case Eq. (2.5) becomes 

00 

± (X) exp [- Mi ] dx (2.13) 

—00 ® 


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Bolt Beranek and Newman Inc. 


> 


int. 


Since the radar pulse length is finite, is zero at the lower 
and upper limits and we can integrate (2.13) by parts obtaining 


P 1 dn *e f ^1 

2 r " S c£ m: J 33T exp 

•00 

Siegert and Goldstein consider the case of a trapezoidal pulse 
as shown in Pig. 2.4. Por this pulse 

dE. 

= 0 except fc-r 0 < x < a and b < x < (b + a) 


4irlx 


'e J 


dx 


(2.14) 


so Eq. (2.14) becomes 


„ 1 dn E o ^e ! f _ f 47rixl f _T 4rixl 

E r = a^-rra|J o e - p ['~J J b exp r~J 


Integrating gives 


E r 3 11£ -r ( 5i )2 [ X - exp (- )][*• " exp (* - 


4»lb V 

*7/J 


which results in a reflected power P r ® |E r | 2 of 


(2.15) 


(2.16) 




(2.17) 


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Writing this with the notation of Case A gives 


P 1 (5n) <; [ sin (27ra/A e )‘ 

2 

[A 1 
e 

2 

sin 2lrb 

* r 5 L (2ra/A e ) j 


.2i&j 


L 


In actual practice, a pulsed radar will probably be used to conserve 
power, to produce minimum interference with the reflected signal, 
and permit the measurement of the range of the acoustic wavefront. 
Then Eq. (2,l8) shows that if A>b>a, the maximum reflection will 
occur for a value of a as small as possible and a value of b equal 
to an odd multiple of A /4. Siegert and Goldstein also show that 

c 

for a smooth radar pulse (not a trapezoid), the reflection is 
greatly decreased if a»A . This is analogous to the difference 

v? 

shown in Pig. 2.3 between sharp and smooth variations in n. 

The use of pulsed radar thoiJgi. useful, for reasons sited above, 
will involve some loss in returned signal. It can be seen from 
a comparison cf Sqs. 2.8 and 2.18 that a radar pulse of any shape 
gives less return signal intensity than a continuous radar wave 
from a gradual change of index in which A»A e . it can be similarly 
shown that changing the radar pulse shape cannot enhance the reflec¬ 
tion from a "sharp" change in n. For the type of variation of n 
shown in Pig. 2.2, the reflected E field described by Eq. (2.5) 
becomes 


E r = l I if E i < x > ex P [-^r*] dx < 2 - 19) 

o e 

In the case of A«A e , the exponential in Eq. (2.19) is almost 
constant and the maximum reflection results for any shape pulse 
with its largest amplitude between x = 0 and x = A. Thus, chang¬ 
ing the pulse shape will not change the reflection from a sharp 
variation in n. 


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Bolt Beranek and Newman Inc, 


3) 


srve 


h 


i/ • 


il 


•iy 


ec- 


) 


/ 

In the following sections we will calculate the reflection from 
different forms of variation in n. The pulse shape will not 
affect these results significantly and the radar signal will be 
taken as an infinite wave train for simplicity of calculation. 

2.3 Reflection Prom a Sinusoidal Wave Train 

The normal incidence reflection from a train of plane 3ine waves 
has been calculated by means of a transmission line analogy.-^/ 
A more direct derivation makes use of the methods of Sec. 2.2. 
With a constant radar pulse amplitude, Sq. (2.5) becomes 


S r 


1 

*5 


f 1 dn 

r 4xix i 

/ Hx exp 

x_ 

r J 


dx 


( 2 . 20 ) 


For a train of N sine waves of wavelength A , we have 

cl 


f n = n + bn.sin 2v x/A for 0 < x < N V 


I n “ n c 


otherwise 


( 2 . 21 ) 


Putting 


Sn cos — for 0 < x < N X 


( 2 . 22 ) 


dn 
L 3x 


= o 


otherwise 


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4 

XU w 


iSQ • 


/ n 

V 


AA \ 

cyj) 


z Ives 


f cos |E exp (.Mij dx (2.23) 

o 

The power reflection coefficient i3 the absolute square of this 
ratio. This reflection coefficient has two different forms 
depending on whether X = 2 cr not. 

4 6 a 

Case 1 V = 2 

■ c cx 

In this case the power reflection coefficient is 

{£ = iSsjLl? n 2 (2.24) 

The power reflected increases as the square of the number of 
wavelengths in the train. However, It is important to realize 
that this expression is valla only if the radar and acoustic 
waves remain in phase throughout the entire length of the train. 

Fluctuations in phase and amplitude of the acoustic wave occur be¬ 
cause of propagation tnrough inhomoger*eitie3 in the atmosphere. 

The radar wave is affected much less by inhomogeneities than Is 
the acoustic wave and consequently the acoustic and radar waves 
may get out of phase seriously even within the length of the 
acoustic wave train. 



Bolt Eeranek and Newman Inr 

i 

Case 2 / 2X„ 

—— c cl 

In this case the power reflection coefficient is 



The value^^/ of for values of "K q about 2A & are plotted in 

Fig. 2.5 for two values of N. 


It is apparent that the reflection coefficient drops to zero when 

the phase difference between acoustic and radar waves increases 

to ir over the wave train. Equation (2.25) represents a diffraction 

2 

pattern whose height increases as N and whose width decreases as 

p 

l/N . The height of the secondary maxima in Fig. 2.5 is at least 

13 db below that of the principal maximum. The results show the 

effect of a deviation from X = 2A but do not show the effect of 

e 3. 

phase fluctuations in the acoustic wave train. 

2.4 Normal Incidence Reflection from a Train of Plane Shock Waves 
In this case we take n(x) as shown in Fig. 2.6. 

The power reflection from each steep wavefront will be 



(2.26) 


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as calculated in Sec. 2.1. The power reflection from each 
sloping portion of the wave will be 


P r 6n/ sln 

Pl = t V"»W7 


(2.27) 


as calculated in Sec. 2.2 and plotted in Fig. 2.3 (A takes the 
place of V in Fig. 2.3). It is apparent that the power reflected 

a 

from the more gradual slope can be neglected compared with that 
reflected from the steep leading edge if X i3 comparable with or 

CL 

larger than * e /2. Therefore in considering a train of shock 
waves, we need only consider the leading edges of the shocks. 

From Eq. (2.20) for a train of N shocks, we find 

|-' = f I expf-4rtmM (2.28) 

m=o 

For N » 1, there will be strong reflection If ZK /\ = s where 

s is any integer. This is to be contrasted with the condition for 
reflection from a train of sine waves where strong reflection can 
be obtained only when 2\/\ - 1. 

a 0 

For the shock waves 



(2.29) 


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The maximum useful wave train j.engtn ia s-cin limited to that 
for which the waves remain in phase. For the train of shock 
waves, we must have N>> & * NsA g /2 to within about If 

product Ns is much greater than one, this requirement is very 
stringent and will probably prevent coherent reinforcement in 
a wave train. In this case the reflection, on the average, 
will be approximately equal to that from a single shock front 
and there will be no enhancement of the power reflection due 
to multiple shock waves. 


2.5 Normal Incidence F.eflection From One Plane Shock Wave 

p 

We have already seen that = (5n) /4 for reflection from 

one plane shock. It is now necessary to calculate 5n as a 
function of the shock strength. For shocks that are not too 
strong, density and pressure are related adiabatically 


$ f 2 - ^ (2.30) 

where y = the ratio of specific heats. 

The index of refraction n is related to density by 


d(n-l) dp _ 1 dp 
n-1 p . y p 


For weak shocks in air. 


(2.31) 


bn ~ (n - l) h 

o y p Q 


( 2 . 32 ) 


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Since p /P.^ depends on (5n) , the reflection coefficient will 
depend on the square of the shock strength. Since the shock in- 

p 

tensity is proportional to (op) also, the curve of reflection 
coefficient vs. shock intensity will be linear. For typical 
atmospheric conditions of p Q = 1000 millibars, T = 15° C., and 


65 %, the index of refraction n Q = 1.00032 


R.H. 

reflection coefficient vs 


The curve of 
shock intensity is plotted in Fig. 2 .1. 


When n Q differs from 1.000320 the reflection losses as given in 
Fig. 2.7 will change. Such change in n Q results from changes 
in absolute humidity and from changes in pressure primarily due 
to altitude. The gross magnitude of such changes in N units and 
the corresponding effect upon the radar reflection in db for 
typical air masses as a function of altitude is shown in Fig. 2.7a. 


-22- 



n 



n 


t 



FIG.2.1 INDEX OF REFRACTION VS. 01 
ARBITRARY VARIATION 
















BOLT 


BLRi,. 


vum nu n 


n 



. . 2.2 


INDEX OF REFRACTION VS. DISTANCE 
LINEAR SHOCK 













BOLT BERANEK a NEWMAN INC 


E 



FIG. 2.4 RADAR PULSE SHAPE 




BOLT BERANEK a NEWMAN INC 














































Report No. IO 56 


Bolt Beranek and Newman Inc. 


3. VakIAtIGN IN REFLECTION DUE TO INHOMOGSNEITI.&S 


3.1 Wavefront Deformation Due to Atmospheric Inhomogeneities 


The distortion of the acoustic wavefront caused by large scale 
atmospheric inhomogeneities such as steady wind, wind shear, and 
temperature gradients can be calculated using geometric acoustics. AV 
The location of a point on the wavefront is given by 



(a + V ) dt 


where a 


sound velocity 


V = wind velocity 


(3.1) 


3.11 Wind direction and magnitude 

If the temperature and wind velocity are constant in the region 
considered, then a” and V can be taken outside the integral giving 


r = (a + V) t (3.2) 

This is shown in Fig. 3.1 3 for V/a = .1 

The wavefront is a sphere whose center is located a distance Vt 
downwind from the source. 

3.12 Wind Shear 

Consider a wind in the x-direction whose magnitude depends on 2 . 

A sound ray will propagate as shown in Fig. 3-2. 


1 


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The sound speed a” is adirected along the ray and the wind 
speed V is always along the x-axis. Defining 3 a s the angle the 
ray makes with the x-axis, and considering only motion in the 
x-z plane, the equations for the path of the ray are 


" a sin 9 

= a cos d + V (z) 


(3.3) 


As the wave progresses Q changes. 

Equation (3.3) can be solved by successive approxirations taking 

z = z^ + z, . . . 

o 1 

x — x + x, . . . (3.'0 

O I 

3 = 9 4 -^, ... 

o 1 

The zero order equations are 


dz 

o 

3t~ 


a sin Q q 


to o 

at 


a cos 3 

o 


(3.5) 


with the solution 



Bolt Beranek and Newman Inc. 


z = at sin 9 
o o 


x = at cos 9 
o o 


(3.6) 


corresponding to a ray in the direction . In the first order 
equations, we can set cos 9-^=1 and sin 9 ^ = 0^ since V/a « 1 
so that 


dz. 

dt- = a a i 003 


0 

O 


dx- 

3T 1 = - a®! sin 9 0 + V (z Q ) 


(3.7) 


Equations (3.7) can be solved most easily by first neglecting 9^ 
and then correcting for it using 


tan 9 ~ tan (9 + S 1 ) 

O j. 


dz dz, 

o , _1 

dz __ dt' dt 

dx dx dx, 

o , 1 

dt" dt 


(3.8) 


For the winds V{z) used in this section, the terms involving 
in £q. (3*7) will be smaller than the term V(z) and will in- 

-A. 

volve only a small correction to x^ and z.,. This can be made 
more explicit by considering various types of wind. 


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Bolt Beranek and Newman Inc. 


Constant wind Shear 

In this case V = Gz and the ray equations become 



o 


dx, 

-gr=- = Gz q = Gat sin 


0 

o 


with solution 


(3.9) 


z = z = at sin 0 
o 


x = + X-, = at 

o l 


cos 9^ + —|— sin 9 
o 2 c 


(3.10) 


The ray paths are found by eliminating t from (3.10) giving 

Gz 2 


x = z cot 9 + ——— — t — — g - 

o z a sin & 


(3.11) 


The wavefront configuration as a function of time is found by 
eliminating 9 q from (3.10) giving 

X - \/a 2 t S - z 2 + (3.12) 

Equations (3.12) is plotted for a wind shear of G = (.2’/ se c)/ft 
in Fig. 3.3. It is apparent that the acoustic wavefronts are no 
longer spherical but are slightly blunt nosed downwind and Increase 
in deformation with range. The terms involving ^ in Eq. (3.7) 
can now be calculated. Equation (3.8) gives 


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Bolt Beranek and Newman Inc. 


a sin 9^ tan 

tan 9 “ a cos 0 -r Gat sin 9” * 1 + Gt tan 9 

oo o 

(3.13) 

tan 8 + tan 8. 

tan 8 = tan (8 + 8^= — tan y- ^~ 5 T 

o 1 


which gives 

e i = - Gt sin 2 9 q (3-14) 

Thus the correction terms in Eq. (3.7) are always less than the 
term V(z,). Since the distortion as shown in Fig. 3.3 is not 
very large, there will not be much error in using Eq. (3.9) instead 
of Eq. (3.7). 

Turning Wind 

A turning wind assumed to have a velocity increasing steadily 
with height will have a sinusoidal variation in velocity with 

the xz plane as 


relating to wind strength and b is a 
tightness of turning. The wind profile 
one vertical plane. The ray equations 


at sin 9 

9 0 sin -g-2 (3.14) 

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height when projected on 


V x = Hz sin £ 


in which H is a constant 
constant relating to the 
is shown in Fig. 3-^ for 
becomes 


dz 1 


= 0 


dx^ 

cT6~ 


= H at sin 













The solution is 


Bolt Beranek om 


*•*'■* me. 


z = at sin S 


y ^ ^i. a 3^ sin ^ 

X - at COo o ht) ~t COS -__ 2 

L D 


+ -~_ 

a sin 


at si 


in 3 
_o 


(3.15) 


The ray paths are 


x. = z cot 9 + 


H b z 


o a sin s i n -g- - cos 


(3.16) 


and the wavefront configuration is given as a function of t b y 


x = \ja 3 t Z - z S + Hb t 


Sin -g- - COS 


(3.17) 


rnese results are plotted in Pig. 3a f3P b . mo , 

al J,’° 2 • = 23 ' /SeC - The '^vefront has sorce dents but 
almost spherical. 


remains 


3.13 Temperature gradients 

reiarior) 0 SP6ea ’ dePen<iS “ the temperature . T, through the 


= \jyRT 


(3.18) 


where y = C p /C v , R 


gas constant 


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r f 


--•a + -ia tnon 



AT 


(3.19) 


Aa is given as a function of AT in Tat 1 3.1 for T = 27°C. 

z 


at ! 

1°C. 

2 °C. 

5°C. 

10°C. 

20°C. 

50°C. 

Aa J 

1.8'/sec 

3.7'/sec 

9.2'/sec 

18.4/sec 

37*/sec 

9 2* /sec 


Table 3.1 


To find the ray naths and wavefronts we make use of the analogy 
with geometric optics.—^ Since the speed of propagation at any 
point is independent of the direction of propagation, we can 
define an effective "index of refraction," n = a^a ~ 1 - Aa/a. 

In the atmosphere, temperature varies primarily with height so 
v;e can take n = n (z). As ir. geometric optics, we can use Snell's 
law of refraction to obtain the ray p =th. Consider a ray In the 
layered atmosphere shown ir. Pig. 3.5. 

At each interface. Snell's law states that 

n 1 cos 9^ = ru cos = n^ cos 9^ = cos Sj, etc. (3.20) 

These relations hold regardless of the number of layers and we 
have the general equation 

n(r) cos 9{z) = constant = cos 9 q 


(3.21) 




I 


Bolt Beranek and I.’ewman Inc. 

The ray paths and wavefront shapes can be found by integrating 
Equation (3.21) and using the fact that a wavefront is a surface 
of constant acoustical path length from the source. 

Constant Temperature Gradient 

For this temperature distribution, v/e have 


AT = - Gz and 


n (z) 




for temperature decreasing with height. 

Since n (z) > I , the ray will have no turning points for which 
3 = 0° as is seen from Eq, (3.2l). The ray paths and wavefront 
shapes can be found exactly through integration of Eq. (3*21) using 


tan 9 



(3.23) 


but since Gz/2T « i, it will be simoler to solve e:uations 

o 

similar to (3-3) approximately. The ray equations are 


dz 

at 


a 


~ sin 0 
n 




ax 


—2- cos 9 
n 


cos 3 
o o 



(3.24) 


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Taking n (z) = n (z ) = n (at sin 9 ) in Eq. (3.24) gives 


dz 

oT 


a sin 0 
o o 


2 

G a^ t 


(sin^ 9 - 4} 

' O 


ax 

at 


~ a 


cos 0 


G Al t o, 
T_ 


sin 9 cos 9 
o o 


(3.25) 


Tne solution is 


2 2 
G ci f-~ 

2 ■ a 0 t 3ln S o - -ST- 1 (sin2 9 o - |) 


x = a t cos Q 
o o 


2 2 
CX v 

o 

2T 


sin 9 cos 9 
o o 


(3.26) 


C- a t 

which is valid for —»- « sin 9 . Eliminating t gives the 

'o ° 

ray paths 


a 

G cos 9 

x = z cot 9 - -> 5 — 

° 4 T sin- 5 9 

r\ f 


(3.27) 


Eliminating 9^ gives the wavefront configuration as a function of 
time 


x =t 1 -%! v a o t2 - 


,2 Gz 2 1 2.2, 

z - y- i * ^ V ) 
o 


(3.28) 


The wavefronts are plotted In Fig. 3»o ‘‘or G = l o C/40O* 



Bolt Bersnek arid Newman Inc. 


3,14 Wavefront Roughening by Turbulence 

The distorted wavefronts considered in Sec. 3.11-3-13 and plotted 
In Pig. 3.1* 3.3* 3.4, and 3.6 are all fairly "smooth.” The dis¬ 
tance over which significant changes occur in the rays or wave- 
fronts is much larger than the acoustic or radar wavelength. It 
is this property which allowed us to use the geometric acoustics 
approximation. For propagation in a turbulent atmosphere, however, 
the temperature and wind speed will vary almost randomly over 
shorter distances {although still larger than a wavelength). Since 
turbulence Is a random phenomenon, its effect can be predicted only in 
a statistical manner, Chernov-^/is an excellent reference on wave 
propagation in turbulence. We will refer to his work frequently 
In the following sections. Propagation through turbulence will 
cause the amplitude and phase of a wave to deviate from their 
values for propagation in a homogeneous medium. Since a wavefront 
is a surface of constant phase, knowledge of the phase fluctuations 
will determine the distortion of the wavefront. If the fluctua¬ 
tions are small, the amplitude of the wave will be approximately 
the same as in a homogeneous medium. Thus, for our purposes, phase 
fluctuations are much more important than amplitude fluctuations. 

The phase of the wave, , is defined by writing the wave amplitude 
(a plane wave in this case) In the form 

p (r, t) = A (r) exp f-i(a;t-kx) + iv"(r)3 (3.29) 


where = frequency 

k = 2rr/k a = wavenumber 
A (*“) as amplitude 

For a homogeneous medium, f = 0 and for the case of isotropic 
turbulence <$> = G where < > denotes average value. 


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The mean local speed of propagation is <a> + <V> while the 
instantaneous speed of propagation is a + V. We can define 
a "turbulence strength" P as 


a + V. 


= 1 + P 


(3.30) 


y in 


where is the component of V in the direction of propagation. 
From this expression P is a random function of position with 
average value zero and is given approximately by 


n = £* + 1 ® 

a o 2 T o 


(3.31) 


where AV, At are the differences \ r r - <V> r > T - <T>. The phase 
fluctuations in the wave are determined by the space correlation 
function, N(r-,, r^), of the turbulence defined as 


N [r 19 r 2 ) = <P (r.^ P (r 2 )> 


(3.32) 


For homogeneous turbulence N (ru , r.~) = N (r^ - r^) and for 
isotropic turbulence N (r-, - r^) = N(r) where r = jr^ - r^j. 

The functional form of N{r) is not well known, but there usually 
exists a correlation distance, s, such that N(r) is very small 
for r > s. This correlation distance can be associated with the 
scale of the turbulence. For many of his calculations, Chernov 
assumes a Gaussian correlation function for P so that all possi- 
ble information is given by the values of CP- ;> and s.- Chernov 
considers the phase fluctuations for a plane wave but most of his 
results are equally applicable to tne case of a spherical wave. 


-33- 


Bolt Beranek and Newman Inc. 


There are two dimensionless parameters which are important for 
determining phase fluctuations. The first is ks or 2tts/A , the 

a 

ratio of the turbulence scale to the acoustic wavelength. While 
the inner scale of turbulence may extend down to centimeters, 
most of the turbulent intensity is found In larger scale inhomc- 
geneities. Golitsyn, Gurvich, and Tatarskii-^/found that most 
turbulence has a scale of between 100’ and 10,000*. Since the 
smallest turbulence produces the greatest wavefront distortion, 
we will choose s » 100'. For acoustic wavelengths considered 
(114 cpsj A = 10*) the ratio 2 tts/A » 1. The second dimension- 

2. ct 

less ratio is called the wave parameter d and is given by 


d = llR/ks 2 



( 3 . 33 ) 


where R Is the distance of propagation through the turbulent 

medium. Physically, d is the ratio of the size of the first Fresnel 

zone to the scale of the turbulence. For A = 10* , s = 100* , we 

a 

find 

_ R 

d = TStSF 

so that 3mall R corresponds to d « 1 while large R corresponds to 
d » 1. With a Gaussian correlation function for d, Chernov finds 
for the mean S 4 uare phase fluctuation^^ 



2ir 2 s R 



(1 * -y tan“' L d.) 
a 1 


( 3 . 34 ) 


For the case d « 1. 


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Bolt Beranek and Newman Inc. 


<* 2 > 


4 7 r 2 /tT<hS s R 



a 


(3.35) 

\ 


while for d » 1 



2t r 2 /7 <3i 2 > s R 



In these cases the rms phase fluctuation varies as 


(3.36) 


v rms rms 


/sR 


(3.37) 


The R dependence of i? is the same regardless of the form of the 

rms 

correlation function. In addition to the mean square phase fluctua¬ 
tion <tf 2 >, we are interested in the correlation distance of the 
phase fluctuations along the wavefront. Chernov^/found that the 
correlation distance for phase fluctuations Is approximately the 
same as the correlation distance for turbulent fluctuations. In 
fact, with a Gaussian correlation function for turbulence, the 
correlation function for phase fluctuations is also Gaussian with 
exactly the same correlation distance. Thus, the effects of turbu¬ 
lence on the acoustic wavefront are given in Eq. (3.34) - (3.37) 
together with the fact that for phase fluctuations, the correlation 
distance along the wavefront is s. To get an order of magnitude 

estimate of <V r2 >, we can substitute A 10’, s « 100’, M- ~ .01 

a rms 

giving 


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1 


****$«!* 


Bolt Beranek and Newman Inc. 


'i S3 

• rms 


where range is in ft 


For „ ** -001 
rmw 




rmw 


5t? 


TOT = 53J \f®" 


so my be large or small depending on P „ 

ms rms 


Essentially the same results may be obtained from a highly simpli¬ 
fied model of propagation in turbulence. Consider the propagation 
of sound through a turbulent eddy cf size s and turbulent wind 
strength AV. Some parts of the wavefront are speeded up by AV 
while other parts, within a distance s, are slowed down by AV. 

Thi3 difference in velocities obtains for a time s/a producing a 
distortion in the wavefront which may be considered as a phase 
fluctuation. The 3ize of the phase fluctuation is 


A* 


rms 


2ttAx 

•x - 



2i r s 


AV 


(3.37) 


where Ax is shown in Figure 3.7. 

In traveling a distance R, the wavefront passes through R/s such 
eddies. Since the direction of each wavefront distortion is ran¬ 
dom, the problem of finding the total phase fluctuation Is a 
random walk problem. For a sequence of N fluctuations of A-^ each 

in random directions, the rms total deflection is A-^ \fjf. The 

p 

final result for Ctf > Is thus 


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*■* 






Bolt Beranek and Newman Inc. 


<V^> 


4tt 2 s R 




4tt 2 <r*- 2 > s R 


V 


(3-33) 


which has the same functional dependence as Eqs. (3.35) and (3.36) 
and differs only by a numerical factor between 0.9 and 1.8. 


3.2 Reflection Prom Deformed Wavefronts 
3.21 Single Wavefront 

The radar beam will be incident on a certain region of the acoustic 
wavefront. This region can be characterized by its location, area, 
orientation, curvature, and roughness. This section will consider 
the conditions affecting the reception of a reflected signal but 
will not consider the interpretation of the information carried by 
the signal. To study the effects of wind, temperature, turbulence, 
and humidity, it is necessary to first look at the reflection from 
an acoustic wave propagated in a completely homogeneous, isotropic 
atmosphere (no wind, constant temperature, no turbulence). The 
wavefront will be a smooth sphere of radius, ta, centered at the 
acoustic source which is assumed to be collocated with the radar 
antenna. According to either geometric optics or wave theory, the 
entire radar signal reflected from the wavefront will return to 
the antenna. This follows from ray theory because all rays strike 
the wavefront at normal incidence and retrace their paths when 
reflected. According to wave theory, the acoustic wavefront is 
also a surface of constant phase for the radar signal. The solu¬ 
tion of the wave equation then gives a transmitted diverging 
spherical wave and a reflected converging spherical wave. Since 
the transmitting antenna has a finite area, the reflected signal 
will not focus to a point but will cover an area at least eq *al 
to that of the antenna. 


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Bolt Beranek and Newman Inc. 


Any deviation from homogeneity or isotropy in the atmosphere will 
change the wavefront characteristics from those of a smooth sphere 
centered at the antenna. A steady wind will keep the wavefront 
smooth and spherical but will cause the center to move. Wind shear 
will cause the curvature of the wavefront to change as will tempera¬ 
ture gradients. Turbulence will cause the wavefront to become rough 
(because of phase fluctuations). In thio section we will consider 
radar reflection from the types of distorted wavefronts discussed 
in Sec. 3.1. 


Steady Wind 

The wavefront in a steady wind V is a sphere of radius ta whGse 
center is a distance Vt downwind from the source. The wavefront 
still acts like a spherical mirror but the antenna is no longer 
at the center. This is seen in Pig. 3.8. 


0 Is the angle between the wind and direction of search and 


0 

w 


V sin 0 
a 


(3.39) 


using the approximation V « a which Is certainly valid. The 
reflected beam comes to a focus at a distance 2V't from the antenna 
(t = time at which the radar reflects from the wavefront). All 
rays within the radar beam width will be focused at the image 
therefore increasing the beamwidth will not increase the image 
size significantly for such a spherical reflecting surface. The 
reflection of a finite-width radar beam from a curved wavefront 
produces essentially the same result as reflection of a single 
radar ray (no beam width) from a plane wave-front whose nomal 
makes an angle 9 with the incident ray. 


- 38 - 



Bolt Beranek and Newman Inc. 


If the diameter of the antenna is D then the diameter of the 
image will be at least D. Examination of Pig. 3*8 shows that 
the reflected beam passes within a distance 2Vt sin 0 of the 
antenna. Thus if D > 2Vt sin 0 , part of the reflected beam 
will fall on the antenna. Expressing t in terms of the range 
R, the condition becomes 


D v 2 V sin 0 
IT > -a- 


(3.40) 


If this condition is not satisfied, then no reflected signal 
(to this approximation) will be received. This condition Is 
too strict, however, since the reflected beam usually has a 
finite width at its closest approach to the antenna, making 
the area of the beam at that point much greater than that of 
the antenna. Also, as we will see In the discussion of reflec¬ 
tion from "rough" wavefronts, the reflected signal will cover 
an even larger area if there is turbulence in the atmosphere. 

The same results for the reflected si^.al obtained above using 
geometric optics, can be obtained using the wave theory. The 
wavefront may be regarded as an aperture which is illuminated 
by the incident radar beam. The pattern of the signal reflected 
from the wavefront is known for the case 9 = 0 where the wave- 

W 

front is a surface of constant phase. When 9^ ^ 0, the relative 
phase of the radar wave varies linearly with distance along the 
acoustic wavefront which Is acting as a mirror (the linear varia¬ 
tion holds approximately if the illuminated portion of the wave- 
front is not too large). Silver^/ has shown that if the 
relative phase distribution on an aperature differs by a linear 


- 39 - 


» 



Bolt Beranek and Newman Inc. 


function of distance from that on an aperture with a known radiation 
pattern, then the pattern of the new aperture is identical to that 
of the old aperture but rotated through a constant angle. In this 
case the reflected beam makes an angle 2© w with the incident beam 
which agrees ’with the relation obtained using geometric optics. 

Wind Shear, Temperature G ^dlents 

These two conditions affect the wavefront by displacing it, changing 
its orientation, and changing its curvature. The results of changes 
in location and orientation have been considered above and in this 
part we will consider only effects of curvature. 

At any point, she wavefront has two principal radii of curvature. 

For reasonable values of wind shear and temperature gradient, both 
these radii of curvature are approximately equal to the range R. 

The beam sent out from the antenna will be focused at a point be¬ 
tween the effective wavefront center and the wavefront if range is 
greater than the radius of curvature. If range i3 less than the 
radius of curvature, the beam will be focused further from the wave- 
front than the effective wavefront center. These cases are shown 
in Fig. 3.9. In both cases shown in Fig. 3-9* the reflected beam 
at the antenna is much larger than the size of the antenna (the 
size of the image). Figure 3.9 is drawn for rays in one of the 
principal planes of the wavefront. If the two principal radii of 
curvature differ, the rays in the two principal planes will focus 
at different points and there will be no well-defined image of the 
antenna. The area of the reflected beam at any point can be found 
from knowledge of the radar beam width, the range of the wavefront, 
and its two principal radii of curvature. 


- 40 - 






Bolt Beranek and Newman Ire. 


The fact that the reflected beam covers a larger area for distorted 
wavefronts can cause a decrease or an increase in the received sig¬ 
nal under different conditions. If the beam falls on the antenna 
satisfying Eq. (3.40) then the received signal will decrease for a 
distorted wavefront since the power in the reflected beam is spread 
over a larger area thus giving a smaller intensity in the beam at 
the antenna. On the other hand, if the beam does not fall on the 
receiving antenna the beam spreading caused by a distorted wavefront 
would increase the intensity striking the antenna. 


These results also follow qualitatively from the wave theory. Again 
considering the wavefront as an illuminated aperture, a change in 
curvature will change the relative phase symmetrically about the 
center line. This is seen In the fact that the drawings in 
Fig. 3.9 are symmetric about their center lines. The relative 
phase will be a quadratic function of distance along the aperture 
(wavefront). The reflected pattern depends on the shape and Illu¬ 
mination of the aperture but some general results can be found. 
Silver^/calculates the radiation patterns of several apertures 
for zero phase differences and for quadratic phase differences. 

He finds that the reflected pattern is wider for the quadratic 
phase difference aperture regardless of whether the phase differ¬ 
ences were positive or negative. This agrees with the resuls of 
Fig. 3.9 that the reflected beam is wider regardless of whether 
the radii of curvature are larger or smaller than the range. 


The result that deviations from a spherical wavefront produce a 
broadening of the reflected beam will be encountered again in the 
sub-section on turbulence. This will not be unexpected since 
turbulence is composed of small scale wind and temperature gradients 
and should produce roughly the same effect. 


- 41 - 





Bolt Beranek and Newman Inc. 

3.23 Reflection From Roughened Wavefronts 

The principal difference between what we call "turbulence" and 
what we call "wind and temperature gradients" is one of scale. 
Turbulence has a smaller scale than other inhomogeneities. There 
are many characteristic lengths of Importance for the reflection 
of radar from acoustic wavefronts. The smallest of these is the 
radar wavelength V which is of the order of 1 or 2 feet. The 
acoustic wavelength X, is about 10 feet. The diameter of the 
Illuminated portion of the wavefront is BR where B = radar beam- 
width angle, R = range, and BR may be as large as 1,000 feet. 

The largest scale is the range itself which extends to about 10,000 
feet. The scales of wind and temperature gradients considered 
(larger than 1,000 ft) are larger than the Illuminated portion 
of the wavefront while the scale of turbulence considered 
(s ~ 100 ft) is larger than A but may be smaller than BR. A 

o 

graph of BR vs. R for various beamwidths is shown In Fig. 3.10. 

If BR < s then the phase fluctuation will vary smoothly over the 

illuminated portion of the wavefront and the radar reflection 

will be similar to that considered in the section on wind shear 

and temperature gradients. However, If 3R > s, the illuminated 

portion of the wavefront will appear rough producing a more diffuse 

reflection. It Is this case (BR > s) which will be considered in 

21 / 

this section. Chernov—' considers the type of image produced when 
the illumination of a lens has random phase fluctuations of magni¬ 
tude, ^ rjns and characteristic scale length s. 

l(y) = I 0 exp [-(y/y 0 ; 2 ] (3.41) 

where l(y) = received intensity at a distance y away from the 
focus In the focal plane 
I = received intensity at the focus 


- 42 - 






Bolt Beranek and Newman Inc. 


and 



rms 

ir 


(3.42) 


I depends on y Q since approximately the same total power is 
reflected, regardless of the area over which it is spread. Using 
the fact that 




I(y) d3 = 2ir I Q 



exp [-(y/y a ) 2 ] y dy (3.43) 


is the total power reflected from the wavefront, we find 



(3.44) 


Equation (3.41), (3.42), and (3.44) thus serve to detennlne the 
received intensity at a distance y from the focal point of the 
reflected beam. Putting in y = 2Vt 3in <f> = 2KV sin 0/a as is the 
case for reflection In a steady wind, and ^ from (3.36), we 
obtain 


or 


I Pr3 

exp 

V 2 s 3 in 2 0 1 

(3.45) 

2ir \fir <J^> R 3 

. 2 \JV a^ <P?> R J 

1 - P r exp [- 

C 2 1 
TTJ 


(3.46) 


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•c 

Bolt Beranek ana Kewman Inc. 



f 

* 


With 



~ _ V 2 s sin 2 0 

2 


(3.-47) 


The same qualitative results may be obtained from the simplified 

model introduced in Sec. 3.14. The rms phase fluctuation is 

Si „ with a correlation distance s. Thus any two points on the 
mb 

wavefront within a distance s from each other may be advanced or 
retarded with respect to each other in space by a distance: 


2Ax 


rms -x 
55r" 


(3.48) 


The rms angle that the turbulent wavefront makes with the average 
(smooth) wavefront is 




(3.49) 


We cam assume that the reflected beam will have a half-width 

of 20,p. Since the direction of reflection from a smooth wavefront 

makes an angle 2S with the incident beam, we have the situation 

w 

shown in Fig- 3.H. 


The angular distribution in the reflected beam is assumed to be 
Gaussian 


% 


!'(*) 



exp 



(3.50) 


L 


- 44 - 






Bolt Beranek and Newman Inc. 


As in Eq. (3.43), 1^ 



4ir 



(3.51) 


O 

The additional factor* 1/R accounts 
in the reflected beam. Putting Eq. 
for <; 2 > and 29 w for 9 , Eq. (3.50) 


for the spherical divergence 
(3.49) for e T , Eq. (3.36) 
becomes 


I' = 


K 3 


4ir ]/w <\^> R 3 


exp 



V 2 s sln 2 0 
Vttr^ <dS R , 


(3.52) 


This expression has the same functional form as Eq. (3.45). The 
numerical constant is Identical and C-, differs by a factor of 1/2. 
For future calculations we will use Eq. (3.45) - (3-47) 3ince they 
are probably more accurate. 


The expression for I can be considered as the product of an 
amplitude term and an angular distribution term. The amplitude 

. o 

term has a factor 1/R from spherical divergence, and a factor 

p 

3/R <M- > from spreading of the beam. The angular distribution 
term has a maximum value of unity, and becomes very small if 


p 2 

V~ s sln g 9 
2 \/ir" a 2 <M-~> R 


(3.53) 


Equation (3*53) is the condition for misalignment of the reflected 
beam and the receiving antenna and poor reception. When 0 = 0°, 
l80° condition Eq. (3.53) will not occur and the received signal 
will be detectable for all values of the parameters. Thi3 occurs 
when the direction of search and the wind direction are either 







Bolt Beranek and Newman Inc. 


the same or opposite* From Eq. (3*53) we see that Tor fixed 

o 

V. s, <P, and <M-“> there is some "blind range" R Q below which 

Sq. (3.53) may occur and signals may not be received at close 

range. If this "blind range" is less than the maximum range* 

R , of the orobe. then a detectable signal will be received 
max * 

for R Q < R < . From Eq. (3.46) we see that 0 o has the 

dimensions of a length and is directly proportional, to the R^ 
mentioned above. 0 o 13 made smaller by decreasing V, decreas- 
ing s, or increasing <M- >* 

From the above analysis, it appears that the most serious 
misalignment problems occur for small R. However, we must re¬ 
member that the radar antenna has a finite diameter, D. From 
Eqs. (3.42) and (3.36), the reflected beam has an approximate radius 

y 0 = f T 1 ^ ^ R (3.54) 

To receive a signal it is not necessary that the reflected beam 
fall on the center of the antenna but only that the beam fall 
on some part of the antenna. This condition may be stated as 

§ + |fF ir 1/4 R ]j R/a > 2 Vt sin i 

or 

~ + 2^ //2 7 r 1 ^ M- l/~£~ > 4 — sin <t> (3.55) 

R rms J s - a ' 

For sufficiently small R, this condition i3 always satisfied. (It 
is also satisfied for sufficiently large R as was seen above.) 


- 46 - 







Bolt Beranek and Newman Inc. 


To obtain an estimate of the reflected intensity as a function 
of range we can make £q. (3.^6) dimensionless. For this purpose, 
let the normalized range variable be 



and the normalized Intensity variable be 



Then substituting into Eq. (3.^6) 

Y = - 3 - exp (- |) (3.56) 

X 

This is plotted in Fig. 3*12. 

In order to estimate expected values for intensity and range from 
Eq. {3.56) or Fig. 3*12, the following method can be applied. 

p 

First choose (or measure) values for V, <M- >, 0, and s thus giving 
C,, C^. The value of provides the conversion from X to H 
giving the graph a horizontal scale. The value of I calculated 
from 




Y 


(3.57) 











Bolt Beranek and Newman Inc. 


is the Intensity received at the radar antenna in terms of the 
power reflected from the acoustic wavefront. The value of P^/P^ 
depends on the acoustic wave intensity and shape as discussed 
in Sec. 2. The dependence of acoustic wave intensity and shape 
on range, frequency, acoustic power radiated, and source geometry 
is discussed In Sec. 4. 

The calculation of received power at any range takes place as 
follows: 

p 

1. Use assumed (or measured) values of V, <h >, 0, s, and B 
to calculate C-,, and to determine the range of R for 
which R < aD/(2V sin 0), i.e., for which the reflected 
beam falls on the antenna. 

2. For R < aD/2V sin 0, the received Intensity will be high 
provided the sound wave is strong enough (r < R x ) 

3. For R > aB/(2V sin 0) continue as follows: 

4. Find X from R by X = R/C 2 

5. Find Y from X using Fig. 3.12 

6. Find the acoustic wave intensity and shape at range 
R using the results of Sec. 4 

7. Find P J /P i for reflection from the acoustic wave 
using the results of Sec. 2 

8. Calculate I from Eq. (3.57) and knowledge of 

9. Power received -lx Area of radar antenna. 


( 


-48- 









Bolt Beranek and Newman Inc. 


3.24 Off-Normal Reflection From a Train of Plane Shock Waves 

As with a single shock wave, if a train of waves remain perfectly 
plane, there will be little or no off-normal reflected signal re¬ 
turning to the antenna when the angle between the radar beam and 
the normal to the acoustic wave exceeds 1/4 of the radar beam- 
width. There will be a reflected signal if the wavefronts are 
sufficiently rough to have a part of their area normal to the 
radar beam. For the case of two wavefronts shown in Fig. 3.13 
normal reflection occurs at points A and B* although not at A* 
and B. For the signals reflected at A and B* to interfere con¬ 
structively, we must have 

Rb» ' r a = 1 X e m = integer 

Since Rg, - » X a » , it should not be difficult to adjust 

so that (1) is satisfied. However, even though the radar fre¬ 
quency is adjusted to give optimum reflection from such areas 
back to the source, the fact that these areas exist at random 
locations over the region of the shock wave train illuminated by 
radar, they will be as a group incoherent in that direction. On 
the other hand, in the direction of specular reflection, all such 
irregularities will have coherence. It Is apparent that a single 
shock wave front may be deformed sufficiently by irregularities 
in the atmosphere to direct a significant fraction of power back 
toward tne source by scattering. A train of waves however, is 
relatively insensitive to such irregularities and therefore re¬ 
flection from such a wave train tends to be highly specular with 
very little energy directed back to the source when the waves are 
not normal to the radar beam. Thus a train of waves tend to 
support specular reflection in an inhomogeneous medium but cannot 
be made to improve reflection at an arbitrary angle by choice of 
the radar wavelength. 




80IT SERANEK a NEWMAN INC 

















BOLT BERANEK a NEWMAN INC 





FIG. 3.4 ACOUSTIC WAVEFRONT SHAPE AND SOUND RAY PATH IN TURNING 
WIND 






BOLT BERANEK 6 NEWMAN IN 



FIG. 3.5 SOUND RAY PATH IN LAYERED MEDIUM 







BOLT BERANEK & NEWMAN INC 



FIG. 3.6 ACOUSTIC WAVEFRONT SHAPE FOR CONSTANT TEMPERATURE 
GRADIENT l°C/400FT 




BOLT 3ERANEK a NEWMAN INC 






FIG. 3.8 RADAR RAY PATH IN STEADY WIND 










BOLT BERANEK a NEWMAN INC 



o 

o 

o 


133d NI as 


u 100 1000 10,000 

R IN FEET 

FIG. 3.10 DIAMETER OF ILLUMINATED PORTION OF WAVEFRONT VS. RANGE 


















4AN INC 


L 



ITY. Y VS. NORMALIZED RANGE, 







BOLT BERANEK a NEWMAN INC 





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4. SOUND PROPAGATION IN THE ATMOSPHERE 

Sound in air is a longitudinal wave motion of the medium which 
propagates from its driving source at a speed determined by the 
physical characteristics of the medium. The directions of sound 
propagation away from the source are determined by the geometry 
of the source and its confinements. As the wave propagates 
through the air irregularities such as wind, wind shear, turbu¬ 
lence, temperature gradients etc., modify the local velocity of 
the sound causing significant alterations in the directions 
originally taken by the sound wave as it left the source. 

As sound radiates it carries energy away from the source. The 
rate at which energy radiates from a source source Is expressed 
in terms of power level PWL defined as 

PWL = 10 log M/V do re 10" 13 watt (4.1) 


where: W is the sound power radiated from the source and W „ 

_ 1 ^ - ej - 

is a reference power unit conventionally taken as 10 watt. 


The amount of power radiated per unit area normal to the direction 
of the wave propagation is the sound intensity expressed in db 

IL = 10 log I/I f db re 10" lo watt/cm 2 (4.2) 


For many purposes the pressure variations in a sound wave are of 
more direct concern than the intensity. In a free progressive 
wave the sound intensity and the rms sound pressure p in the wave 
are related by 


I 

x pa 

where p is the 


air density and a is the speed of sound. 


;«.3) 


Report No. 1056 


Bolt Beranek and Newman Inc. 


The sound pressure level SFL is defined as 

SPL = 20 log — db re 0.0002 n bar (4.4) 

*ref 

The reference pressure is chosen to make the sound pressure 
level and the intensity level numerically equal for sinusoidal 
sound waves under conditions near room temperature and pressure. 

Other pressure levels 3uch as the peak pressure level and the 
peak to peak pressure level will be used subsequently in the 
following discussion. They will all employ the same reference 
pressure 0.0002 bar and so they will not be numerically equal 
to the intensity level of the sound wave. 

As sound in air propagates away from a source it may undergo 
little change in amplitude and wave form or it may suffer a 
large decrease in amplitude and a radical change in its wave 
form depending upon the geometry of the source, the atmospheric 
attenuation characteristics, the sound frequency, and the ampli¬ 
tude of the sound wave. 

4.1 Spherical Divergence 

A sound source which is physically small compared with the 
wavelength of the sound acts as a point source and radiates 
uniformly i:i all directions. The sound intensity I at any 
distance r from such a source is therefore related to the total 
sound' power, W, radiated by the equation 



Report No. 1056 


Bolt Beranek and Newman Inc. 


v«av A««VVi(UX vjr 


M _ 

JLO 


IL = PWL - 10 log 4 t tv 2 (4.6) 

This same relation holds for any phy. .cal spherical source which 
radiates uniformly in all directions. 

4.2 Directivity 

A sound source which is comparable with or larger than a wavelength 
does not radiate uniformly and is therefore said to be directive. 
The directivity factor for such a source is defined in any 
direction Q as the ratio of the power radiated in that direction, 

W q, to the average power, W Q , radiated in all directions. 


% w e/ w aV g 


(4.7) 


Near any real source it is generally not possible to specify a 
directivity factor because the directly of energy flow is not 
known. However, at large distances the energy flow is radial and 
the sound intensity along any radiu3 decreases inversely as the 
square of the distance from the source. In this so-called far-field 
the directivity factor in any direction can be determined from the 
geometry of the source. 

For the present purposes it is of importance to know the directivity 
at a large distance along the axis of a plane piston radiator such 
as a parabolic radar antenna or acoustic horn. The directivity for 
such a radiator of diameter D is 

«=(x ) 2 (*- 8) 

This is the relation which is called antenna gain in radar 
applications 


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Bolt Beranek and Newman Inc. 


xne total beam width to the half power point for such a source at 

/ 


23/ 

1 ^-4 of 4 o .>4 r —_/ 

aqa UJ.W vauvvu xu ^xvw»u ujr 


0 - 70X 

0 = IT 


in degrees 


(4.9) 


In the near-field of a plane radiating surface the 3ound may radiate 
nearly as a plane wave but edge effects cause 3mall ripples in amp¬ 
litude along the wave front and corresponding small undulations in 
phase. From a practical standpoint, the near field of a plane 
radiator acts like a plane wave field in most respects over an 
area corresponding approximately to the area of the radiator. In 
this near-field the average sound intensity remains substantially 
constant along the axis. The division between the near-field and 
the far-field is not sharp and indeed it does not have a unique 
definition. 

For the present purpose the end of the near-field will be defined 
as the radius R n for which the far-field equation gives a sound 
intensity equal to the average intensity over the face of the 
piston radiator. 

The far-field intensity I f at the end of the near-field R n of a 
circular piston of diameter D is given by 



Bolt Beranek and Newman Inc. 


the near-field intensity l n is given by 

4 l 



Equating and I and using Eq. (4.t ; for Q gives 


tr - wJ) 

% ~v r 


(4.10) 


Within the near-field the sound intensity exhibits a number of 
maxima and minima determined by the source geometry and wavelength. 
On the axis of a plane circular source the maxima all have a sub¬ 
stantially constant value. This is illustrated in Fig. 4.] for an 

24 25 / 

experimental source— i —5 wavelengths ln diameter. The average 

sound intensity in the near-field is approximately 6 ab below the 

intensity peaks as Indicated by the dashed horizontal line. The 

calculated far-field sound intensity for this source is shown as 

the dashed line having a slope of -6 >.b per distance doubled. 

The intersection of these dashed curves determines the distance 

R to the end of the near-field. At the end of the near-field 
n 

the measured sound intensity falls substantially 6 db per distance- 
doubled. Farther from the source atmospheric attenuation (dis¬ 
cussed in Section 4.3) causes a more rapid decrease in the 
intensity of the experimentally measured sound. 

Directivity gain obviously increases with increase In the diameter 
of the source relative to the radiated wavelength. Increase in 
directivity has advantage from two major aspects: 

(1) It decreases the main beam angle thus enabling a more 
detailed searching pattern and (2) it permits the radiation 
of increased intensities in the desired directions with a 
given total radiated power. 


-54- 



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For a radar signal, the amount of energy which can be transmitted 
by the main beam is limited only by the power capabilities of the 
source and by the degree to which side lobe radiation Is sup¬ 
pressed. Thus the intensity of the main beam and therefore the 
total power incident upon a target which is small compared with 
the beam cross section can be increased directly as the cross 
section of the pencil beam is decreased, i.e., in proportion to 
the directivity. 

For a sound wave, the advantages expected from an increase in 
directivity are modified by other factors not encountered with 
radar; these greatly affect and limit the extent to which a gain 
in performance is secured by Increase in directivity. When a 
stationary and homogeneous medium exists around the source and 
when the sound waves do not carry much energy, the relations 
governing directivity are much the same a3 for radar waves. How¬ 
ever, when the medium has a velocity as is the case of the real 
atmosphere with wind, the sound beam Is swept down stream with 
the velocity of the wind. Although the wind may be slow, several 
feet per second compared with the speed of sound over 1000 ft/sec, 
the drift may be sufficient to throw a narrow beam seriously out 
of alignment with the radar bearr and result in the need for intro¬ 
ducing searching and tracking complications Into the radar control 
system in order to follow the sound waves. 

A much more stringent limitation upon the use of directivity 
arises from the nonlinear nature of air as a transmitting medium 
for sound. The air, in enect, will overload and will not 



u 


Bolt SeraneK and Newman Inc. 

propagate sounds above a limiting intensity regardless of the 
source power or the influence of directivity. This limitation 
is more fully discussed in Section 4.4. 

4.3 Atmospheric Absorption 

A sound wave traveling through air undergoes a decrease in 
intensity in addition to spherical divergence discussed above. 
This additional decrease in intensity results from an absorption 
of energy from the sound wave by heating the air or from disslpa 
tion of sound energy by scattering. 


Absorption causes a decrease in intensity of the form 


x = 


I 

o 


-mx 

P 


(4.11) 


where I and . are intensities at x ana x = 0 ft respectively 

m is tne attenuation coefficient in ft J ‘ 


The atcc r:_ou constant a in cb p^*r ft is given by 

g — e-. 3-»; i oo/ft (4.12) 

A normalized plot of attenuation in do for a plane wave is 
presented m Figure 4.2. A similarly normalized piot for- a 
spherical wave is givex. in Fig. 4.3. 

4.31 Classical Absorption 

At audio frequencies minor losses occur as a result of classical 
absorption including, 1) viscous losses, 2) heat conduction from 
the warn regions of the pressure peaks to the coder regions of 

- 56 - 


■k 


' 



i 

» 



Bolt Beranek and Newman Inc. 


the pressure minima, 3) heat radiation between regions of different- 
temperature and, 4) diffusion of molecules from the faster moving 
regions of the sound wave into slower moving regions. 

All of these losses are insignificant in magnitude compared to 
molecular absorption at frequencies below 10 kc. 2 §/ 


4.32 Molecular Absorption 

As sound vibrations pass through air containing small amounts of 
water vapor the molecules of water are set into vibration and ab¬ 
sorb energy from the wave. The amount of absorption depends upon 
the sound frequency, the absolute humidity and the temperature 
in a complex way.-^/ 


1) At any chosen frequency f, a maximum absorption 
occurs at a value of absolute humidity h^ 
which is independent of temperature 



where f is in kc. 

■a 

h m is in gm/nr 

This relation is plotted in Pig. 4.4. 


(4.13) 


2) For any chosen frequency and humidity the ratio 

w of the molecular absorption to the maximum 

molecular absorption c^ x is given theoretically 

in terms of the ratio of the absolute humidity 

h to h by the relation 
m 


w = “mo/Vix 


( h /h max > 2 + 


< W h >‘ 


(4.14) 




-57- 




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the experimental values of absorption represented 
by the curve in Pig. 4.5 are higher than those 
predicted theoretically at high and at low values 


of humidity but are in excellent agreement in the 


region around h . 

m 


3) The value of - increases linearly with 

ITlciA 

frequency as shown by the curves in B’ig. 4.6 
for <2 Jnax V3. f, with temperature as a parameter. 


4) The value of a ^ is obtained by multiplying 
the a max obtained from Fig. 4.6 by the w 
obtained from Fig. 4.5. 


Absolute humidity h can be determined conveniently from measured 
relative humidity by use of Fig. 4.7. 

4.33 Scattering 

At low audible frequencies where molecular and classical absorption 
both become very small, there is more attenuation of sound observed 
experimentally in long range signaling than can be accounted for by 
these processes. Some of this may result from a scattering of 
sound by inhomogeneities in the atmosphere. Experimentally the 
attenuation seems not to fall below approximately 0.001 db per 
foot. 


Such scattering has two effects of Importance in relation to the 
EFAC Probe. First, the scattering causes a withdrawal of energy 
from the progressive sound beam and a resultant increase in atten¬ 
uation by redirection of the sound energy. Second, it tends to 
promote a broadening of the steep front of a shock wave by causing 


(' 


-58- 




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slight variations in arrival time of wave contributions which 
have passed through slightly different paths of the inhomogeneous 
medium. Neither of these relations has received much theoretical 
or experimental study. The following discussion exposes the 
problem, presents plausible values related to some experimental 
observations, but indicates the need for experiemntai verifica¬ 
tion of results. 

Inhomogeneities in the atmosphere cause variations In the speed 
of sound and thereby cause variations in the direction of propa¬ 
gation of the wavefront of any sound disturbances passing through 
these inhomogeneities. The effect of such variations In the 
wavefront is to cause a redirection of sound energy in a random 
manner from various points along any wavefront. It is possible 
to calculate the subsequent position of the wavefront and the 
3hape of the shock wave by adding the contributions from all 
points on the wavefront during its entire path of travel from the 
source to the point in question substantially following the 
method of Section 3.14. Such an addition can be carried out 
only on a statistical basis because the lnhomcgeneities within 
the air are in themselves predictable only on a statistical 
basis. The net result is a reduction in the sound intensity at 
a distance by the direction of sound out of the direct path. 

The inhomogeneities also tend to cause a broadening of the steep 
front of a shock wave but this broadening process Is opposed by 
the finite amplitude distortion process discussed in Sec. 4.4l. 
Whereas the broadening effect of turbulence is independent of the 
sound wave amplitude, the distortion effects tending to steepen 
the wave are directly proportional to the wave amplitude. There¬ 
fore, it is expected that turbulence will have little effect in 
broadening the wavefront i r the wave has sufficient amplitude. 
However, when the amplitude drops below a level at which the 





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Bolt Beranek and Newman Inc. 


steepening and broadening effects are equal the wavefront will 
broaden rapidly. This amplitude is apparently dependent- upon 
the magnitude of the turbulent velocities, the structure of the 
turbulence, the geometrical configuration of the wavefront, the 
Initial form of the sound pulse, etc A theoretical determina¬ 
tion of shock wave structure as a func.'on of all these variables 
would be very difficult and apparently has never been done. 
However, for use with the SMAC Probe, all that is required of a 
shock wave Is that it be relatively thin compared with a radar 
wavelength, and that its level remain sufficiently high. A 
rough method for calculating level and shock front thickness is 
presented in Sec, 4.4A which gives the shock thickness produced 
by attenuation alone. Since these results are In good agreement 
with experimental measures of shock structure, it is fairly safe 
to assume that turbulence broadening is not the most Important 
cause of shock thickening. 

4.34 Precipitation and Fog 

Suspended particles in the atmosphere produce acoustic losses 
by two mechanisms. First, there will be viscous dissipation 
and heat conduction near the suspended particles, and second, 
there are relaxation losses because the time lag between evapo¬ 
ration and condensation on the part icier, as the local pressure 
and temperature changes when the wave passes. 

Experimental studies of sound attenuation in atmospheric precipi- 

2f> 28 / 

tation and fog——^show that these losses can be neglected. 
Attenuation in fog changes slowly with frequency and is below 
.3 db/1000 ft for frequencies less than 2 kc even in heavy fog. 
Absorption by water droplets exceeds molecular absorption at 
low frequencies (below about 300 eps) when both are small but 
at higher frequencies, absorption by droplets can be neglected 
compared with molecular absorption. 

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zgg&BK £ 12 . 










*4*1 



Report No. 1056 


Bolt Bersn(?k and Newman Inc. 


4.4 Nonlinear Sound Propagation 

Sound waves of any shape or harmonic content tend to deform 
toward the sawtoothed shape which is the stable wave form for 
high amplitude sound. The leading edge of the stable wave is 
a shock front whose thickness depends upon the amplitude of 
the wave and the attenuation characteristics of the medium but 
not upon the frequency of the fundamental component of the wave. 

4.41 Wave Distortion 

Finite amplitude distortion of this 3ort is important on two 
accounts. First, energy is transferred from the fundamental 
component into the higher harmonics; since these are more 
rapidly attenuated than the fundamental, an excess attenuation 
of the wave results which drains energy from the sound beam in 
direct proportion to the magnitude of the pressure discontinuity 
and the number of discontinuities per unit distance along the 
sound beam. Second, the distortion of the sound wave creates 
a sharp pressure discontinuity at its leading edge. This dis¬ 
continuity provides the optimum condi.ion for the reflection 
of radar waves from a pressure variation of a given pressure 
amplitude. This last fact is of utmost importance in the per¬ 
formance of the sound wave as a reflector in the electromagnetic 
acoustic probe. 

There are two causes for the change in shape. The first relates 
to the fact that sound consists of longitudinal vibrations and 
as such the alternating particle velocity of the medium is 
parallel to the direction of wave propagation. In such a wave 
the maximum positive particle velocity corresponds In time and 
space to the maximum excess pressure. The maximum negative 


Bolt Beranex ana liewman Inc. 


particle velocity corresponds to tho minimum pressure ol‘ tne 
wave. Therefore, the pressure- peaks and troughs of an acoustic 
wave travel respectively wifcn the velocity of sound plus and 
minus tne particle velocity. The second cause of finite ampli¬ 
tude distortion is that an acoustic wive is adiabatic, i.e., 
the local temperature of tne air increases as the pressure in¬ 
creases. Since the speed of sound increases as the sousre root 
of absolute temperature, the local wave velocity is greater 
than average at pressure maxima and less at the minima. In a 
normal gas the results of these two factors are additive causing 
pressure maxima to overtake pressure minima and create a steep 
pressure front at the leading edge of an acoustic wave. 

As the wave front steepens, the energy of the wave Is converted 
from the fundamental ana low harmonics into higher harmonic 
components. The steepness of the wave front is limited by the 
balance between tne rate of transfer of energy into the nigher 
harmonics and tne loss of energy from the nigher narmonics by 
means of attenuation wnich converts a- ustic energy into neating 
of the air through which the wave passes. The mechanism of the 
absorption is unimportant. The magnitude of the absorption as 
a function of frequency will determine the ultimate sharpness 
of the shock front which is necessary to create the balance of 
energy flow into the harmonics and from the harmonics into heat. 
The lower the rate of absorption from each harmonic, the closer 
the wavefront will approacn a theoretical discontinuity ana the 
closer the amplitude of each harmonic will approach the theo¬ 
retical aosolute limit of 1/n compared with the amplitude of 
the fundamental. 


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Inc. 


bo:* ov.raneK anu Newman 


ii s\ >>\* rj « f*rrn ^ A £Tt? 

. « w wj SA l i UOO»-.|I.U ^a^jV 


This action is snown dramatically in Fi 
of oscilloscope wave traces depicting the pressure as experienced 
jy a micropnor.e located in an intense 14 kc sound wave at several 
distances from a plane piston circular source for four sound out¬ 
put levels. 


Tiie traces in Fig. u.g nave oeen adjusted all to the same height 
by increasing tiie gain in tne oscilloscope so ti*at tne wave shapes 
could be compared directly. Trie widening of tne trace at 200 cm 
for the lowest souna level is caused by circuit noise which be¬ 
comes evident at tr.e nigh ^ain setting since tn_ display system 
uses a oroadoand circuit with no filtering. 

it can be seen that at tne lowest sound level (140 db rms averaged 
over tne face of the source) the wave progresses with little ob¬ 
servable distortion tnroughout the range of observation, 200 cm 
(approximately 90 ’wavelengths). At the highest level (13t do rms) 
although the wave is equally pure at the source, it distorts 
rapidly and becomes sawtoothed in a l ,w -wavelengths. 

At tne ibt ub level the wave oecomes noticeably sawtootnsd at a 
distance of approximately 6 wavelengths wnereas at InO db the 
same amount of distortion requires approximately 20 wavelengths. 


It is fcur.c t.iv.or-'tically uni experimentally mai, f r geometricail 
similar sound fields, tne am unt of distortion ootalr.'d : 'or any 
gicen sound intensity , is a function only of the distar.:, from the 
source measured in wavelengths of the fundamental free: . -y of the 


Bolt Beranek and Newman Inc. 


These considerations indicate that in order to obtain a maximum 
range with the EMAC system it is necessary to utilize a low 
frequency signal so as to reduce ordinary atmospheric attenuation 
and finite amplitude attenuation to an acceptable value for the 
chosen range. It is then necessary :o increase the source power 
to the point where the acoustic wave will reach and maintain a 
sawtooth wave form in order to take advantage of the high reflec¬ 
tivity of the sharp pressure discontinuity at the leading edge 
of a finite amplitude wave. Mathematical relations governing 
the frequency and source power are discussed in the following 
section. 


4.42 Finite Amplitude Limits 

A high amplitude plane wave of stable form (i.e., sawtooth shape) 
will attenuate^/in amplitude according to the relation 



(4.15) 


where: u is the particle velocity amplitude 

x is the distance 

y is the ratio oi specific heats 

a is the velocity of sound 

A Is the sound wavelength 



where: P Q is atmospheric pree_u^ _ 

p is excess sound pressi ':~plitude 


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Bolt Beranek and Newman Inc. 


For sound fields which are not plane the change in sound amplitude 
involves .the divergence of the wave. A general treatment of non¬ 
plane fields has been considered by Rudnick in relation to the 
transmission of sound in horns of varying cros3-section. If we 
consider a horn in which the area, S, of an equiphase surface of 
the wave depends upon the distance of propagation of the wave, 
then the area S at any distance x i3 given by: 


Sg 2 (x) = S Q (4.17) 

where S, = S Q at x = x Q and thus g (x q ) = i. 

f 

Combining the relation for divergence and the attenuation from 
Equation (4.15) 



or from Equation (4.1b) 


dp _ dg (-y-KL)pq ^ 

P g Y*P q 


(4.18) 


(4.19) 


Continuing now only with the equation for pressure and letting 
p = vg where v is a new variable Eq (4.19) reduces to 



(4.20) 


which can be integrated to rive, 

•* . 


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Bolt Beranek and Newman Inc. 


d = 


P 0 g(x) 


1 + ^P 0 


g(x)dx 


/ j, 

\' t 


C.JL 


\ 

/ 


where p is the excess pressure amplitude In a sound wave at distance 
x from the source and p Q is the excess pressure amplitude at x = x Q . 

From this equation it can be seen that there is a limit to the 
value of p at any distance x which cannot be exceeded regardless 
of the amplitude of p Q at the source and this value we shall call 
the limiting pressure p^ 


7* P Q g(x) 

p i = 

(y+i) 


/ g(x)dx 


(4.22) 


There are two cases of interest here for which the evaluation of 
p^ Is instructive. The first case is that of a plane wave. 
Although a truly plane wave cannot be generated and used in open 
space, its performance is descriptive of the process of finite 
amplitude limitation of the pressure in a sound wave as it pro¬ 
gresses away from the source. The second case is that of a 
spherical wave. Here the limiting relation will be seen to 
involve an additional term modifying the limit for a plane wave 
in such a way that the two limits can be handled separately to 
advantage in real applications. 


-DC- 


Bolt Beranek and Newman Inc. 


I7rtr» a r\1o«n 

* v* u auaa v> nave gj\ A / 


X 



(4.23) 


which indicates that the limiting pressure may be unlimited at 
the source where x = x Q but at any other distance this pressure 
must decrease as j / x i * e '> inversely as the distance 

measured in wavelengths. 


The limiting pressure is proportional to the atmospheric pressure 

6 p 

and for a normal atmosphere of 10 M-bar (1 M-bar = 1 dyne/cm ) the 
limiting pressure is 



(4.24) 

(4.25) 


where n is the number of wavelengths from the source. This 
relation is indicated as tie heavy solid line in Fig.(4.9) where 
the reciprocal of is plotted against n. 

When the exc<_- pressure in the wave is not infinite at the source 
but has some initial value p 2 , the pressure at a distance n wave¬ 
length from the source is given by 


P 



(4.26) 


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Bolt Beranek and Newman Inc. 


which is seen to be represented in Pig. 4.9 by lines parallel 
to that for an infinite shock at x » x Q but Intersecting the 
n • 0 axis at values of ~~ corresponding to the sound pressure 
radiated at the source. * 

Since this analysis assumes that the waves considered have reached, 
or are generated with the stable sawtooth form, they remain saw¬ 
toothed as they propagate. 

Before proceeding to the spherical wave case it is helpful -to 
replot the results shown in Fig. 4.9 in a more conventional form 
as shown in Fig. 4.10 where the sound pressure level is ex¬ 
pressed in decibels against the log of the distance from the source 
expressed in wavelengths. In this representation the limiting 
pressure for an infinite shock at n • 0 is a straight line having 
a negative slope of 6 db/dlstance doubled and passing through 189.3 
db at a distance of one wavelength from the source corresponding 
to Eq. (4.25). The curve representing the variation of pressure 
level for a wave having a preassigned amplitude at the source will 
be a curved line starting horizontally at the left with a value 
approaching the assigned value at the source and approaching the 
limiting pressure asymptotically toward the right. 

It is interesting to note that the slope of the limiting pressure 
curve as plotted in Fig. 4.10 for a plane wave is the same as 
that for the sound pressure in a spherically diverging wave of 
low amplitude. In the latter case the pressure amplitude falls off 
as 1/r because of divergence but that process involves no loss of 
energy. We may therefore expect a steeper slope when finite ampli¬ 
tude losses are considered in a spherical field. 


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Bolt Beranek and Newman Inc. 


To determine the effect of finite amplitude limitation upon a 
spherical wave, return to Eq. (4.22); substitute r for x and 
1 /r for g(x) representing she spherical divergence and then 
integrate. 

This gives 



l P o 

y + T 


3 


r 

X 


log^ ~ 
e r_ 


(4.27) 


where r is the distance from the center of divergence, r Q is the 
distance from the center to the surface of the sound source. This 
equation, obtained by Laird^^ also using a somewhat different analy¬ 
sis is similar to that for a plane wave, but has the extra factor 
log r/r in the denominator. This i actor becomes unity when 

9 Q 

r/r Q = e (i.e. r/r Q = 2.7). A plot of this factor in decibels 
is given in Fig. (4.11). A plot of Eas. (4.24) and (4.27) in 
Fig. (4.12) compares the limiting pressures for a plane and a 
spherical wave. The straight line is the limiting pressure for a 
plane wave starting at r = 0 and the curved line is the limiting 
pressure for a spherical wave having its center at r * 0 but start¬ 
ing from a spher*cal source whose radius is one wavelength. It is 
apparent that (as in Fig. 4.10) the amplitude of the plane wave is 
unlimited at ^ = 0. The spherical wave is unlimited at the surface 
of the spherical source, r ~ r . The spherical wave and the plane 
wave have the same value of limiting pressure when the spherical 
wave has progressed to a radius 2.7 times the radius of the source. 

The two curves of Fig. (4.12) are useful in combination because a 
simple translation of the spherical wave limit along the plane 
wave limit can be made to account for an arbitrary change in size 


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Bolt Beranek and Newman Inc. 


of the spherical source. For example, if the source radius is 
2 A instead of A as assumed in Fig. 4.12 the spherical wave 
limit may be translated to the right diagonally along the curve 
for the plane wave limit until the source position corresponds 
to 2 A instead of X. It will be seen that the two lines will 
then cross at 5.4 wavelengths instead of 2.7 

4.43 Applications to Experiment 

As was noted in Section 4.2, real sources can seldom be considered 
as strictly plane wave generators or as spherical wave generators. 
A plane piston moving in a rigid baffle approximates a plane 
source near its surface and a spherical source at large distances. 
The dividing distance R between the near-field and far-field for 
acoustic purposes was established in Section 4.2 as 



(4.10) 


where D is the diameter of the piston source. 

In applying the finite amplitude limits to real sources we may, 
with good approximation, apply the plane wave limit to the 
near-field and the spherical wave limit to the far-field by 
matching the two limits at the distance R n . 

Experimental data taken with the same 14.5 kc piston source 
described earlier in relation to Figs. 4.1 and 4.8 are compared 
in Fig. 4.13 with the theoretical limits calculated for that 
source. The upper four experimental curves in Fig. 4.13 corres¬ 
pond to the four sound intensity levels shown in Fig. 4.8. There 


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Bolt Beranek and Newman Inc. 


is a o db difference because Pig, 4.8 refers to average intensity 
level in the ncrr-fleld which is 6 db below the peak intensity 
level. It is Seun once that the sound pressure level measured 
’ *-he far-field behaves as Ic should for a spherically diverg¬ 
ing field, i.e., a 6 db decrease per distance doubled for the 
lower sound levels recorded. However, when the sound level is 
raised at the source it is seen that the sound level in the far- 
field increases only so as to approach but not exceed the limiting 
pressure levels. At 100 wavelengths the sound at the highest 
level of operation is more than 8 db below the value expected 
if the finite amplitude limit were disregarded. Substantially 
no increase in level at this distance could be obtained by In¬ 
creasing the source power. 


Even if the real source could be replaced by a theoretical plane 
source the plane wave limit would still limit the increase in 
sound level at large distances. The actual sound pressure would 
be a few db greater than for the spherical field but the total 
power loss would be very much greater. 


It should be noted that the data shown in Pig. 4.13 apply to the 
fundamental component of the 14.5 kc signal. All harmonics were 
filtered out. The finite amplitude limits have therefore been 
drawn to indicate the rms level of the fundamental in a sawtooth 
wave having the peak amplitude indicated by the limits in 
Fig. 4.12. The relation between the two is 


rms 


VT 

fundamental = —~ 


(4.28) 


SPL = 20 log p £ - 7 db 


(’4.29) 


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xidt Leranek ar.c I,awn an Inc. 

The finite amplitude limits may be applied in the same way to tne 
experiments of Midwest Researcn institute. Tnere, tne frequency 
of 22 kc reflected from an lb" paracolic mirror gives an effective 
near-field distance of 34 ft. 

First, however, we shall consider only the effects of molecular 
absorption as indicated in Fig. 4.14. If the total radiated 
power of 30 watts is assumed to be uniformly distributed over the 
beam area e:pial to the area cf the reflector, the average SPL in 
the near-field will be 144 db as indicated by the horizontal line. 
The far-field SPL (neglecting absorption) will be represented by 
a line having a slope of b ab phasing through the point 144 db 
at 34 ft. Molecular absorption is accounted for by use of the 
curve 4.3 and setting I db absorption at a distance of 5 ft since 
the attenuation of tne 22 kc signal is approximately 0.2 db per 
foot. The attenuated level in the far-field is represented by 
the light dashed curve. The attenuation expected in the near-field 
can be obtained similarly by use of curve 4.2. 31ending these two 
attenuated curves gives tne SPL expected from tne source shown as 
the heavy solid curve. The one measured value of 140 db at 10 ft 
falls 2 db below the curve so constructed. 

We now consider the effects of the finite amplitude limits on the 
MRI experiment. Figure 4.15 shows the near-field SPL as before, 
a horizontal line at 144 db. The wave is here assumed to be sinu¬ 
soidal as generated. The plane wave limit for the near-field is 
presented as the rms limit which would be measured by a sound 
level meter for a sawtooth wave. It is related to p, 

P 0 = —==. p^ for a sawtooth wave. 

rms y3 


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Thr> rv - 


spherical wave limit for the far-field is 


t*AY» f “ ") 

* vi. vuv; - 


rms 


matched at 3^ feet, The expected for the sound wave is 

rms 

blended from its initial level to approach first the near-field 
limit then the far-field limit. 


The experimentally measured sound pressure of 140 db at 10 ft is 

in almost exact agreement with the p^ and about 1 db higher 

*rros 

than the expected value obtained by curve blending. It is seen 
that the expected at 93 ft is 105 db which is some 12 db 

lower than that which would be expected by considerling molecular 
absorption alone. This difference while large has even more im¬ 
portance when we consider means for extending the range of the 
SMAC Prcbe. By considering molecular absorption only, we might 
expect to be able to increase the sound level at any distance by 
increasing the source output. Figure 4.15 shows that increase 
in source power would make no increase in the sound level at 
distances beyond 10 feet. 

Further consideration of the wave form of the sound indicates 
that harmonic content becomes significant beyond a distance of 
about 2 ft and energy is transferred from the fundamental to 
higher harmonics. The decrease in the level of the fundamental 
is indicated by the heavy dashed curve. It approaches a value 


P 


rms 


fundamental - 



p__ sawtooth 

rms 


which is 2.2 db lower than the level of the sawtooth. 


When the sawtooth wave progresses to the distance at which the 
rate of decrease in level due to the finite amplitude limit is 
less than the rate of atmospheric absorption (primarily molecular) 


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the sound level will drop below the finite amplitude limit. The 
higher harmonics raise the level of the sawtooth wave above the 
fundamental. Therefore, we must apply a higher attenuation to 
the sawtooth and the attenuation of the fundamental to the funda¬ 
mental itself. By this process it is seen that the sawtooth level 
is expected to approach the fundamental and the wave shape reverts 
to sinusoidal. This action is shown to take place beyond 60 ft 
for the 22 kc signal in Fig. 4.1$. 

From these discussions it is clear that no system using high 
frequency sound can produce useful signals much beyond two or 
three-hundred feet even with unlimited acoustic power at the 
source. 

From these experimental results the serious nature of the finite 
amplitude limit is clearly apparent. In order to attain dis¬ 
tances of several thousand feet it will be necessary to reduce 
the radiated sound frequency. Reducing the frequency will 
raise the finite amplitude limit in direct proportion to the 
increase in wavelength. Lowering the frequency will also decrease 
the rate of atmospheric absorption but this appears to be a second¬ 
ary consideration. 

In reducing the frequency we are faced with the fact that the 
increase in wavelength will affect the directivity of any chosen 
antenna and thereby affect the amount of power reouirea at the 
source to create a given sound intensity on the axis of the 
radiator. 

For a first cut we may look at a frequency near 1000 cps since 
this frequency will have a wavelength still short enough to serve 
as an effective reflector for useful radar wavelengths. Let us 


Eolt Beranek and Newman Inc. 


c noose the f re .money 1140 sin -jo this wil? . '•* >- = I rt. We will 

tne-i choose arbitrarily a 10 it ciameter ai as a source. Then 
the near-field R n extends to 73 ft. Figure -‘.io snows that the 
finite amplitude limit permits a level of 100 do in onoss of 
3000 ft. Applying the atmospheric attenuation of between .01 and 
.001 db per ft indicates the sound wave amplitude would fail below 
the finite amplitude limit at some distance between 300 and 2000 ft 
as indicated by the shaded area. So it is obvious that a 114C cps 
signal can be maintained above 100 db to a distance of 1500 to 
3000 ft. 

If we assume the wave should become sawtoothed at least by a 
distance of 100 ft then the average sound intensity ir. the near¬ 
field should be approximately 145 do. This would require a sound 

*7)2 „ TO 

power level of 145 + 10 log -V- = lb4 db re 10 J watt or approx¬ 
imately 2.5 kw of acoustic power. 

Any increase in acoustic power would not increase the range but 
would serve only to cause the shockwave to be developed closer 
to the source and cause more objectionable disturbance to equip¬ 
ment and personnel. 

This intensity would be extremely ob j .-ctionaole to p . son. el even 
outside the main beam and even for relatively ,»hort quests of the 
acoustic signal. 

Such a signal in short burst woula retain the problem of matching 
the raaar and acoustic wavelength c give coherent reflections 
from the several waves of the pui_. 




Bolt Beranek and Newman Inc. 


It now appears that any acoustic signal Involving a train of 
repeated waves which arc commensurate with the longest usable 
radar wavelength will not be able to be projected much over 
1000 ft and therefore will not serve for pr -bing the atmospher 
at any useful range. We now therefore direct attention to the 
use of a single shock pulse as the only practical reflecting 
acoustic surface for long ranges. 


4.44 Shock Wave Phenomena 

A sound impulse may be considered to be made up of an infinite 
series of sine waves. If such an impulse is radiated from a 
plane piston source it will have a complicated directivity 
pattern. As an approximation we may consider this directivity 
pattern to be made up of the directivity patterns of all of the 
harmonic components of the impulse. 


For the frequency components having wavelengths which are long 
compared tc the diameter of the source the directivity pattern 
is essentially spherical. Only for wavelengths which are com¬ 
parable with or shorter than the circumference of the source 
Is there any practical gain in intensity along the axis due to 
directivity. As we have already found waves which are short 
compared with the diameter of the source (i.e., I ft long) will 
not have sufficient range. Therefore, let us consider generat¬ 
ing a single sinusoidal half-wave pulse whose wavelength Is 
equal to the diameter of the source, i.e., 10 ft or 114 cps for 
a 10 ft dish. 


For this frequency and dish size the near-field extends to 8 ft 
and the plane wave finite amplitude limit (rms of a sawtooth) 
passes through 164.7 db at 100 ft. The plane wave and spherical 






Bolt Beranek and Newman Inc. 


wave umit are assumed e^ual at 8 ft as shown in Fig. 4.17. In 
order for the wave to be substantially sawtoothed at- 100 ft the 
average intensity in the near-field should be of the order of 
175 db. A single pulse may tend to sharpen on its trailing edge 
also thereby producing a double shock or N-wave which is not de¬ 
sirable for the EMAC Probe system. To avoid such sharpening of 
the trailing edge the intensity of the wave may be dropped approx¬ 
imately 10 db; the wave would not be expected to sharpen signifi¬ 
cantly for over 1000 ft. However, if the pulse generated contained 
a fundamental of this lower magnitude and also contained higher 
harmonics so phased that the wave had a steep leading edge and 
gradual trailing edge as generated, the leading edge would sharpen 
to a shock rapidly and the trailing edge would be expected nev^r 
to sharpen. 

The higher harmonics included in the pulse for sharpening the 
leading edge would have higher directivity than the fundamental 
ar»d would remain close to the center of the beam. Thus, these 
harmonics, although more objectionable to personnel, would be 
confined to the center of the acoustic beam. 

The finite amplitude limits near the source would be those 
applicable to the higher harmonics but at large distance would 
be that applicable to the fundamental. A gradual transition 
should occur as the wave progresses. This transition has not 
been studied in detail and appears so complex that it should be 
submitted to experimental test. 

From these considerations it appears feasible to create a wave 
which will become a shock wave within a few hundred feet from 
the source and remain a sharp shock for a distance of the order 
of 10,000 ft. 


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Bolt Berpnek and Newman Inc. 

The thickness of a shock front theoretically should depend only 
- the amplitude of the overpressure not upon the frequency of 
the fundamental. Calculations of the shock front thickness-^/ 
using the equations of motion for a steady state non-isentroplc 
transition across a shock indicate that it should be of the 
order of 3 cm for a shock wave having a pressure amplitude of 
the order of 100 db. This departs widely from the experimental 
observations of shock fronts in air. Theoretical considerations 
including the effects of molecular absorption have indicated 
that the shock front should be about 15 cm thick at levels between 
120 and 100 db. This is indicated indirectly by the curves of 
Fig. 4.16 where the 1140 cps repeated shock wave begins to drop 
below the finite amplitude pressure limit at between 120 and 
100 db. In this region the shock front has grown to 1/2 wave¬ 
length of the 1140 cps wave or approximately 15 cm. Experimental 
evidence with N-waves of sonic booms bears out this conclusion. 

qp/ 

Measurements-^ of several sonic booms are summarized in Table 4.1. 


3 

4 

5 

6 

7 

8 


Boom No. 

Amplitude 
incident 
lb/sq ft 

; of 
waves 
db 

Rise time of 
steepest section 

Corresponding Thickness 

1 

.5 

122 

.6 ms 

.7* 

2 

.42 

120 

.9 ms 

1.0* 


apparently a ground wave; no shock front 


.75 

.24 


126 

116 


.7 ms 
.5 ms 


apparently a ground wave; no shock front 


.31 

.42 


118 

120 


.6 ras 


.5 ms 


. 8 * 

. 6 * 

.7* 

. 6 * 


( 


i 


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Bolt Beranek and Newman Inc 


4 


Sonic booms 3 arid 6 in Table 4.1 appeared to arrive in a nearly 
( horizontal direction since there was no visible separation be¬ 

tween the incident and reflected wave. The rise time was very 
long, several milliseconds, and included many shocklike ripples 
which are assumed to be due to the successive additions of com¬ 
ponents of the wave retarded by obstacles and inhomogeneities 
near the ground. Even for these waves the initiation of the rise 
was sharp. 

A typical N-wave signature (Boom #7 from Table 4.1) is displayed 
as oscilloscope traces at two sweep rates differing by a factor 
of 10 in the photograph of Fig. 4.18. The leading edges of the 
incident and the ground reflected waves are clearly separated. 

For the incident wave the wavefront thickness is about 0.7 ft. 

It should be noted however, that in the fast trace the initiation 
of the pressure pulse forms a noticebly sharper comer than does 
the crest of the pressure pulse. The observed sharpness in 
Fig. 4.18 appears to be limited by the passband of the recording 
? system which rolls off above 2,500 cps. Thus, the actual sharp¬ 

ness cannot be assessed from this figure. It seems likely, 
however, that the Index variation accompanying such a sound 
shock most closely approximates an index variation with one sharp 
comer and one round comer. 

For the purpose of radar reflection, the presence of one 3harp 
comer significantly increases the reflection when the index 
of refraction variation occurs over a distance in excess of a 
radar wavelength as indicated in Fig. 2.3. Thus It is reasonable 
to look for a useful radar reflection from an acoustic shock 
wave even after the wave front has broadened beyond a wavelength 
of the radar wave. Again, this premise needs experimental 
verification. 


-79- 







( 





0.01 0.1 I 10 100 

DISTANCE IN WAVELENGTHS 

FIG.4.1 EXPERIMENTAL VALUE OF SOUND PRESSURE LEVEL ON AXIS OF 
PLANE PISTON SOURCE 5 X DIAMETER 









































































VS FREQUENCY 


BOLT 8ERANEK a NEWMAN INC 



FIG. 4.5 PLOT OF a/a MAX VS. h/h m (AFTER HARRIS) 







MAX IN DECIBELS PER THOUSAND FEET 


BOLT BERANEK 8. NEWMAN INC 


0.02 



£pji00 o F 

60° 
40° 
20 ° 
0 ° 


100 !000 
FREQUENCY IN CYCLES PER SECOND 


2 3 4 5 6 8 

10000 


FIG.4.6 EXTENDED PLCT; OF MAXIMUM MOLECULAR 
ABSORPTION C' /FiCIENT a M £ X VERSUS 
FREQUENCY AT VARIOUS TEMPERATURES 













BOLT BERANEK a NEWMAN INC 




±N3Dy3d N! AJLIQiWnH 3AilV13d 













BOLT BERANEK 6 NEWMAN INC 


TIME AXIS FOR TRACES 



</> 

LlJ 

CJ> 

< 

oc 


rs 



til 

(£ 

D 

tO 

CO 

aj 

ax 

£t 


X 
H 
O 
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-J 
UJ 

> 
< 

z 

Ui 

o 

8.5 or 

o 

to 

65 I 

a: 

u. 

OJ 

1.5 o 
z 
< 
h- 
co 

0.22 o 


88 

44 

22 


/40 145 150 155 

DB RE 0.0002 MICROBAR 



FIG.4.8 SC-UND PRESSURE LEVEL IN AIR AVERAGED 
OVER THE FACE OF A PLANE CIRCULAR 
RADIATOR, 5 WAVELENGTH IN DIAMETER, 

IN A BAFFLE (DIA.4.8”, FREQUENCY 14.6 KC) 
SYSTEM GAIN ADJUSTED TO GIVE EQUAL 
TRACE HEIGHT 




BOLT BERANEK ft NEWMAN INC 



—~ = n WAVELENGTHS 

A 


FIG. 4.9 RECIPROCAL PRESSURE IN A SAWTOOTH 
ACOUSTIC WAVE VS. DISTANCE IN WAVE 
LENGTHS 












.90-- / ~P=I0 6 MICROBAR 















y 












RELATIVE SPL 


BOLT BERANEK a NEWMAN INC 






180 


BOLT BERANEK a NEWMAN i.NC 



fO 

iD 

U. 


' lu "JU 1000 3000 

DISTANCE FROM SOURCE IN WAVELENGTHS 

EXPERIMENTAL VALUES OF SOUND PRESSURE LEVEL IN THE FAR 
FIELD OF A PLANE PISTON SOURCE 5\ IN DIAMETER 



















BOLT 9ERANEK 8 NEWMAN INC 


o 



H 

LU 

UJ 

U_ 


UJ 

O 

z 

<t 

h- 

(/) 

O 


FIG.4.14 SOUND PRESSURE LEVEL EXPECTED WITH MIDWEST RESEARCH SOUND SOURCE 
CONSIDERING MOLECULAR ABSORPTION NEGLECTING FINITE AMPLITUDE LIMITS 







V 





FIG. 4.15 SOUND PRESSURE LEVELS EXPECTED WITH MIDWEST RESEARCH SOUND SOURCE 
CONSIDERING BOTH FINITE AMPLITUDE LIMITS AND MOLECULAR ABSORPTION 




























SOUND PRESSURE LEVEL FOR 1140 CPS SIGNAL .RADIATED FROfv 
DIAMETER SOURCE WITH AVERAGE SPL OF 145 DB NEAR SOURCE 





























SOUND PRESSURE LEVEL- DB RE 0.0002 MICROBAR 


BOLT BERANEK a NEWMAN INC 





5 iO ICO 1000 10,000 

DISTANCE IN FEET 


FIG. 4.17 SOJND PRESSURE LFVEI FOR 114 OPS SIGNAL RADIATED FROM 
10' DIA. SOURCE WITH AVERAGE SPL OF 175 DB NEAR SOURCE 






dCrt'f-ri 


St 















80LT BERANEK a NEWMAN INC 



FIG. 4.18 OSCILLOSCOPE TRACE OF 
SONIC BOOM SIGNATURE 
BOOM NO.7 (TABLE 4.1) 





Bolt Beranek ar.d Newman Inc. 


5. SOURCE CHARACTERISTICS FOR MAXIMUM RANGE 
5.1 Acoustic Source 

A sound source must piovide sufficient shock intensity to travel 
several thousand feet in order to be useful. The source should 
be somewhat directl/e in order to conserve source power out more 
important it should be directive in order to avoid hazards to 
operating personnel and minimize annoyance in surrounding 
communities. 

It now appears that an ideal pulse at the source should have a 
rise time which is of the order of a few milliseconds so that 
there will be a minimum of the high frequency sound components. 
The high frequency components are undesirable at the source be¬ 
cause they are more hazardous and more annoying than the very 
low frequency components. 

It appears that the pulse should have a long decay time for two 
reasons: (1) a long decay time implies a large amount of energy 
in the single pulse, (2) the long decay time, returning to atmos¬ 
pheric pressure without the creation of a negative pressure, will 
prevent the creation of a negative shock. Thus, such an acoustic 
wave can avoid the variable interference effects expected in 
radar reflections from the sonic boom N-waves. 

The source need not and should not produce a shock wave near the 
radar installation but should rely upon finite amplitude distor¬ 
tion to create the shock at a distance somewhere in the region 
between 100 and 1000 ft from the source. Such a design would 
minimize hazard and annoyance and maximize the conservation of 
energy in the vjave. The decay of the trailing edge of the pulse 
will not create a shock wave at any range if the decay is 
sufficiently gradual. 

- 80 - 










oolt 3eranek and Newman Inc. 


form a between 100 and 1000 feet from the source, the 

intensity i*jvl near the source must exceed a critical value de¬ 
termined by the initial pulse shape. The source power level will 
be determined by this intensity level, the source size and its 
directivity. On the other hand the sound intensity level outside 
the main beam must not be high enough to cause personnel hazard 
or annoyance. As mentioned in Sec. 4.44 the high frequency com¬ 
ponents of the sound pulse will be much more directive than the 
low frequency components. These high frequency components which 
are more annoying can be confined to a fairly narrow beam and 
can be directed away from the populated areas. The design of an 
acoustic source with these desired characteristics will require 
future study. 

The intensity and directivity of the sound field near the source 
will of course be greatly Influenced by the size of the sound 
source itself. If the EMAC system is to be mobile, the source 
and radar antenna both probably will be restricted to units of 
the order of 10 ft in diameter. Such a source will give some 
appreciable directivity for a 100 cps wavSjQ will be of the order 
of 10. For the higher frequency components needed to sharpen the 
leading edge, the source will be more highly directive, Q is about 
1000 for 1000 cps. Since the amount of power needed In the har¬ 
monics is small compared with that in the fundamental, and since 
the source is more directive for these components the amount of 
sound which spreads away from the center of the main beam is 
relatively small for the high harmonics and should therefore 
cause only a minor and perhaps negligible problem as regards per¬ 
sonnel exposure especially since the pulses are of short duration 
and spaced at relatively long intervals. 


- 81 - 



Bole Beranek and Newman Inc. 


i-’er a longer range installation where the sound source and radar 
antennas may be permanently located, larger source areas may be 
utilized with the added advantage of greater directivity at the 
chosen frequency or with the possibility of reducing the fundamen¬ 
tal frequency component of individual pulses. 

The personnel hazard for pulses is much less than that for contin¬ 
uous tones. No experimental results are at hand for the effects 
of lew frequency pulses but extrapolation of data from 100 cps 
indicates a probable Increase In permissible exposure levels for 
pulses repeating at 10 cps or less would be of the order of 20 ab 
or more. 


The personnel hazard for low frequency pulses should be subjected 
to experimental study. Some work Is planned at 33N In this area 
and equipment Is available for controlled experiments at the 
present time. It may be advantageous to augment this work by 
experiments directed specifically toward evaluating the effects 
of an EMAC source once a more definite specification of the system 
has been developed. 

y.2 Electromagnetic Source 

As opposed to the situation of the acoustic source, there is no 
fundamental limit to the intensity which can be propagated in 
tne radar beam (at least within the range of power capabilities 
currently availaole). Thus, the radar power can and should be 
increased as necessary to utilize the full range for which the 
acoustic signal is above the background noi3e, but need not be 
increased further. Some existing radar systems seem adequate 
for this purpose. The most-important parameter of the radar 
system for maximizing the range of the StfAC Probe is the radar 
wavelength. As discussed in Sec. 2.2 the power reflection 


- 82 - 



Bolt Beranek and Newman Inc. 

coefficient from a dielectric variation is sensitive to the ratio 
of the radar wavelength to the thickness of the dielectric varia¬ 
tion. To have an adequate reflection the radar wavelength must 
be comparable with or smaller than the shock thickness. The shock 
becomes thicker as it propagates and thus the range of the probe 
is limited substantially at the distance where the wave front 
thickness equals the radar wavelength. There exist Doppler radars 
such as the FPS-7 and FPS-20 having a wavelength of about 23 cm 
which is sufficiently long to provide adequate reflections from 
shock waves with sound pressure levels of the order of 120 or 130 db 
re 0.0002 microbar. 

Other parameters of the electromagnetic source have les3 effect on 
range and can be varied within fairly wide limits. The beam width 
can be decreased to give greater detail and higher intensity or 
can be increased to cover a larger area. The duty cycle and search¬ 
ing sequence can be modified depending on the meteorological condi¬ 
tions and atmospheric parameters of interest. 

Since the overall power loss will be very high it will be necessary 
to use such techniques as coherent integration and parametric ampli¬ 
fication to obtain maximum range. It is estimated that, under the 
most favorable conditions, an overall power loss of 239 db can be 
permitted between the transmitted and received signal at the limit 
of detectability for a system such as the FPS-7 or FPS-20. Using 
this information and the method given at- the end of Sec. 3 an estimate 
of the maximum range of an EMAC Probe system can be made. Such 
calculations will be given for a variety of atmospheric conditions 
in Sec. 6 . 7 . 


-83- 





Poit Eci’anck and Newman Inc. 


6 . 

6.1 


ACCURACY OR MEASUREMENT 


op 


ATMOS PHERIC PARAMETERS 


Wind Speed in Direction of Search 


The local speed of propagation of the acoustic wavefront is the 
vector sum of the sound speed and wind speed a(r,t) + V(r,t). 
The measured Doppler shift indicates the radial component oi 
this speed. Tnus. 



2 

c 


( a r + Vj 





where 


V( y p) 


a 

o 


frequency of returned signal 
frequency of radiated signal 

speed, of light 

radial component of sound (wind) speed 
sound speed at source 


( 6 . 1 ) 


The relations between the several variables can be seen in 
Pig. 6.1. We find 


a = a cos p and 


3 ~ sin 3 


Vt sin <t> _ V sin 0 
at a 


( 6 . 2 ) 


Also 


V = v cos 0 so that 

y» 

& r + V r = a + V cos 0 - — V sin 0 , (6.3) 


-64- 





Bolt Reranek and Newman Inc, 


'....ore* - = angle between 7 and radial direction. The V cos 0 

\J 2 o 

ion., nominates the — V sin 6 term except for 0 ^ 90 where the 

cl 

vi^r-ection oi‘ search is perpendicular to the wind direction. If 
the wind were uniform at all points, the Doppler shift vs 0 
curve would have the form shown in Fig. 6.2 


In this case, the magnitude and direction of V could be deter¬ 
mined from the shape of the curve. However, if the wind is not 
uniform, Eq. (6.3) must be used. Unless the wind is very strong 
and 0 ss 90°, only the V cos 0 term is needed. Sven in that case, 
the maximum error in V would be 10£ and this could be reduced by 
applying the correction term. The error in the Doppler shift 
from changes in the direction of propagation is thus fairly small. 
The shift also depends on the magnitude of the sound speed which 
is related to the local temperature, T, by 

a = VyKT 

where y = C /C , R = gas constant. 

c * 

If T deviates from the temperature at the source, T , then there 
is a change in f-f given by 


m m 

2/ a _ a ) _ 2 1^0 

c^ a V ~ c a o 2T 


This change in sound speed due to a temperature change would appear 
the same as a change in the radial wind speed. The value of Aa 
corresponding to various AT's is shown in Table 3*1> Sec. 3- 


- 85 - 







1 i c. 


ioll 


S~.i c temperature varies prir.ari.iy height, ..ncre corrections 

•.-.i-- eiter as the altitude of the test region is .increased. If 
the prore is pointed vertically, it will measure the change or 
temperature with altitude and the vertical component of the wind. 
V„. Since V is almost always less than 5'/sec, a vertically 
pointing probe can measure T as a function of altitude to within 
about 3°. If V is known roughly, then T as a function of height 
can be found much more accurately. Knowledge of the temperature 
at a specified altitude can then be used in computing horizontal 
components of the wind at the corresponding altitude as discussed 
in Sec. 6.2. A horizontal wind which is uniform in direction is a 
good assumption when considering altitudes which are high compared 
with influencing obstacles on the ground. This assumption will be 
used in computing the wind components. 


6.2 Wind Direction 

Complete determination of the wind direction requires a determi¬ 
nation of three components of the wind velocity. In practice, the 

vertical component, V , is much smaller than the other two and can 

z 

be neglected. Under some atmospheric conditions the vertical 
component is far from negligible but in such cases the vertical 
component is confined to rather local areas and examination of 
these areas in relation to surroundings can yield vertical velocity 
calibration data. 


6.21 Single Probe Methods 

If the wind is assumed to vary slowly with distance and time, then 
measurement of the radial wind in two directions can give infor¬ 
mation on two components of the wind. Consider the following 
measurements made on two nearby regions at low elevations as shown 
in Fig. 6.3. 


( 



-86- 


f 




Bolt Berancn a.)a Newman Inc. 


.....i.ig the w..ui has the sane components V . V , V at (l) and 
(i), we na\ ..ensure the Doppler shift at (l) and (2). If the 
te. ..•e return is the same at the two locations then the Doppler 
rhL.'cs give directly the radial wind velocities, Vr»(i)' V rfp) * 
Prom Fig. 6.3 these can be seen to be 

V r( i, = V y cos 6 sin || + V y cos 9 cos + V 2 sin 9 

(6.4) 

V r ( 2 ) = -V v cos S sin | + y y cos £ cos ^ *r V 2 sir, e 
where c and <5 are defined in Fig. 6.3 

The term V„ sin ? can be dropped since both V and £ are small, 
z z 

cos £ can be set equal to one giving 


V r(l) = V x sin I + V y 005 I 

V r(2) =* V x sin I + V y 005 I ' 

Solving Eq. (6.4*) gives 

v , V r(lj - V r(2) v = V r UJ_ ^ Jrj2l 
2 sin | y 2 cos | 


(6.4«) 


The error inherent in this method can be seen by considering that 
there is an uncertainty 6 in each radial velocity measurement. 
Then 


-87- 





i-i'-.t stra-.o 


■i- ro 


-- hhVs 2rror ( V - sory/i 


Inis in graphed •‘o Fig. 6.4. 


(fco) 


For 0 very small, V can be determined very poorly as would be 
expeeteu since both probings are essentially measuring V . In- 
creasing 0 increases the accuracy of wind direction measurements, 
but decreases the probability that the wind and temperature are 
the crime at both points of probing. Adding additional regions o? 
measurement can provide more information on T and V to reduce the 
uncertainty as these parameters change from one point to the next. 

6.22 Multiple Probe Methods 

inis method uses several prcDes to sample one region rather than 
one probe to sample several regions. The wind components are 
obtained in the same manner as with one probe. The advantage of 
this method over the single probe method is that it is not affected 
by spatial variations of wind and temperature. However, it does 
not seem that this advantage compensates for the additional com¬ 
plexity and cost required to erect and coordinate two or more probe 

systems. Various technioues of this type are discussed in the MRI 
9/ 

Reports." 

6.3 Turbulence 

6.31 Detection and Intensity 

One effect of turbulence on the acoustic -wavefront will be to 
cause some parts to move faster or slower than others. Thus, 
different parts of the wavefront will have different Doppler 
shifts. 


If all parts of the wavefront had the same speed relative to the 
radar, the returned signal would have a single frequency and 


-88- 



Bolt Beranek and Newman Inc. 


would cive a definite Doppler shift. However, if this is not 
the ease, the returned signal will have a spread of frequencies. 
The Doppler shift is measured by comparing the phase difference 
between transmitted and received signals as a function of time. 
This phase difference will have a form similar to that shown in 
Fig. 6.5. 

The frequency spectrum of this curve then can provide information 
on velocities and turbulence. A possible frequency spectrum Is 
shown in Fig. 6.6. 


Trie location of the maximum gives the mean radial velocity while 
the width of the maximum gives the rms fluctuation in radial 
velocity. These fluctuations result from wind and temperature 
inhomogeneiuies and are related as follows: 


Af 


2 

c 


[AV r + 


**) - I + fl 


a o ] = 


2a. 


u 


rms 


( 6 . 6 ) 


where AV, AT = amplitude of velocity, temperature fluctuations 
throughout the reflecting region, and u. is defined in Eq. (3.3°)* 

6.32 Localization 

The measurement of Af determines the largest variation in radial 
velocity occurring in the echoing region of the wavefront. It 
would be very difficult to localize the turbulence to a smaller 
region than this. It may be expedient, however, to use more than 
one radar frequency in order to be able to obtain extended range 
with the lower frequency and fine definition of close wind struc¬ 
ture with the short radar waves. 


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Bolt Beranek and Newman Inc. 


o.S3 Structural Definition 

i.icrc art. several ways that the scale of the turbulence can be 
measures. The simplest method uses the fact that the width of 
the echoing region increases with range and is fairly well known. 

The Af for each region measures the full intensity of turbulence 
with a scale smaller than the region but only part of the inten¬ 
sity of larger scale turbulence. If the intensity of turbulence 
is plotted against the size of the echoing region, a curve like 
Fig.' 6.7 is obtained. 

Since there is no increase in turbulence intensity above L = L , 

max 

the maximum scale of the turbulence is I. . It will be much more 

max 

difficult to determine the minimum scale of the turbulence. One 
possible method uses the results of Sec. 3* The reflected power 
at large ranges decreases as SPL/R^ because of spherical divergence 
and beam spreading. However, if the v?avefront is rough on the scale 
of the radar wavelength, the radar reflection will be almost iso¬ 
tropic and the beam will not spread with increasing R. In this 
case the reflected power will decrease as SPL/R . In this case, 
the existence of turbulence having a scale comparable with A e can 
be determined. 

The scale of the turbulence discussed above relates to the size of 
individual turbulent fluctuations or eddies and does not necessarily 
relate to the size of a turbulent region. The size of a turbulent 
region must be determined in a different manner. If the turbulent 
intensity is known for all echoing regions within a large volume, 
contour lines of equal intensity can be drawn which will show the 
size and shape of regions of strong turbulence, ihis method will 
work well for turbulent volumes larger than several echoing regions. 
Smaller patcnes of turbulence might be localized by using measure¬ 
ments from overlapping echoing regions but since these regions do 


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Lolt Beranek and Newman Inc. 


not have o-.arp counaaries, ohe precision of this method will 
require experimental evaluation. 

6.4 Possibility of Differentiation Between Inhomogeneities 
of Various Kinds 

As seen in Eq. (6.6) temperature and wind fluctuations affect the 
Doppler shift if the same manner. Observations in the atmosphere 
show that these fluctuations are of the same order of magnitude. 

While it will be very difficult to distinguish between temper¬ 
ature and wind fluctuation experimentally, this will not be a 
major problem. Variations in wind and temperature are related 
theoretically through the equations of atmospheric dynamics. 

Thus, experimental knowledge of Af can provide information on 
both AV and AT. The accuracy of this method will probably have 
to be determined experimentally. 

6.5 Temperature Discontinuities 

Temperature discontinuities or sharp temperature gradients will 

reflect both sound and radar and can be detected in several ways. 

Consider the discontinuity shown in Fig- 6.8. At points A, B, C, 

where the discontinuity is perpendicular to the radar beam, the 

radar signal will be reflected and will return to the probe. Ihis 

2a 

signal will not have a Doppler shift near —~ and may be difficult 

V 

to detect. Sound reflected ail along the discontinuity will re¬ 
turn to the probe but will not give much information on the shape 
of the discontinuity. It may also be masked by sound reflected 
from other objects. The reflected radar and sound waves will give 
the information that there is a discontinuity which can be investi¬ 
gated with standard J3»iAC probe techniques. The transmitted sound 
wave will be speeded up (if ? 2 > T-^) and this will appear as an 


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Loi t Lerane.'c and Newman Inc. 


• r.. y» c* **i -i * 1 


.. c . *c 

- A * 


.v.re:i. ?r.ij \.ilJ suffice to determine the location and 
ude of the temperature discontinuity. 


<..c Hu.r.iclity Changes 

humidity changes serve to alter the attenuation coefficient of 
the sound waves and the dielectric constant of the air. A change 
in attenuation coefficient will considerably alter the range of 
the MAC Probe. Since changes in the wind alter the range in 
some directions more than others while humidity changes alter the 
range in all directions, a change in the : verage range probably 
corresponds to a humidity change and can be used to detect and 
measure these changes. 

Changes in the dielectric constant of air affect the returned 
signal much less than does a change in attenuation coefficient 
and will not be very useful for humidity measurements. 

Changes in liquid water content should be examined by means of 
humidity and water vapor absorption of sound. 

6.7 Maximum Range of EMAC Probe 

The maximum range of the probe depends on the characteristics of 
the acoustic system, the radar system, and the atmosphere. We can 
control the characteristics of the acoustic and radar system but 
cannot control those of the atmosphere. In this section we will 
choose some operating parameters for the MAC Probe system and 
calculate the maximum range under several atmospheric conditions. 

The radar system will be characterized by a radar wavelength of 
23 cm., a radar antenna diameter of 10’, and a maximum permitted 


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Bolt Beranek and Newman Inc. 


difference of 239 ub between transmitted and received power 
as discussed in Section 

The output of the acoustic system will be chosen as a single 
pulse but will be considered to propagate as an acoustic signal 
with a fundamental frequency of 114 cps and a SPL near the source 
of 175 db. A sound source., 10’ in diameter is assumed as a plane 
circular radiator. This sound field is discussed in Section 4.44 
and shown in Pig. 4.17. 

The atmospheric parameters for which we will take several values 
are the steady transverse wind speed component (V sin <f >), and the 
turbulent wind speed (AV). The fractional radar power reflected 
at any range is found by using Figs. 4.17, 2.7, and 2.3 in com¬ 
bination. The received power is then found using the method of 
Section 3, page 48. We will assume a turbulence scale of s = 

100*. The maximum range is found by equating the received power 
level to the radiated power level minus 239 db. 

The maximum range for given values of V sin $ and AV varies with 
the amount of atmospheric attenuation the sound signal encounters. 
This attenuation may vary by a factor of 10 at any given frequency 
depending upon temperature and humidity as detailed in Section 4.3. 
Values for the maximum range calculated for several values of V 
sin <t> and AV are presented in Table 6.1a and 6.1b. Table 6.1a 
represents conditions of low atmospheric attenuation and Table 
6.1b represents conditions of high atmospheric attenuation. 


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Bolt Beranek and Newman Inc. 


Ay 

V sin^0^ 

0 

l’/3ec 

lO’/sec 

0 

50,000’ 

15,000’ 

10,000’ 

10’/sec 

500» 

10,000’ 

10,000’ 

lOO’/sec 

50’ 

50’ 

9,000’ 


Table 6.1a 


V sin 

0 

l’/3ec 

lO’/sec 

0 

20,000* 

5,000* 

4,000* 

lO’/sec 

500’ 

3,000’ 

4,000* 

lOO’/sec 

50* 

50* 

3,000* 


Table 6.1b 


It is apparent that where a transverse wind occurs there is need 
for turbulence in order that a usable amount of the reflected 
radar signal be returned to the radar antenna. Without such 
turbulence, specular reflection directs the main signal away 
from the antenna. Fortunately, where high winds exist, turbu¬ 
lence is usually encountered and in general the turbulence will 
be of greater magnitude when the wind velocities are high. Cer¬ 
tainly large turbulence will exist In regions where there are 
large wind gradients which are probably the regions of greatest 
interest. 



















WAVEF 



ACOUSTIC VELOCITY ALONG RADAR RAY 





ujs ro- 







bolt beranek a mewman inc 



FIG.6.3 MEASUREMENT OF HORIZONTAL WIND 
COMPONENTS WITH SINGLE PROBE 


{ 




BOLT BERANEK & NEWMAN INC 



FIG. 6.4 RELATIVE ERROR IN HORIZONTAL WIND COMPONENTS VS. 
ANGLE BETWEEN PROBING DIRECTIONS 









BOLT BERANEK 8 NEWMAN INC 



L = DIAMETER OF ECHOING REGION 


FIG.6.7 SPREAD IN DOPPLER SHIFT VS. 
DIAMETER OF ECHOING REGION 






i 


Bolt Beranek and Newman Inc. 

•7 T5DPT TUTH«mr r»vTyr*n-rium* y 

( . rnDuxi'iinnni riArrirv.xnc.iYi mj oio 

Four phases of experimental and developmental study are proposed 
which may be undertaken in succession: (I) An experimental study 
of radar reflection from 3onic booms using suitable existing 
Doppler radar Installations. (II) An experimental study of a 
number of simple, impulsive sound sources and a theoretical 
design study for optimizing the most favorable one as an EMAC 
component. (Ill) Construction and acoustical test of the sound 
source designed in Phase II. (IV) An experimental study using 
the source of Phase III in conjunction with a suitable radar 
system. This phase Is intended to demonstrate the practical 
range and weather limitations to a first approximation and to 
reveal the nature of the more important refinements which should 
be incorporated Into a working EMAC System. 

Phase I 

Phase I is designed to demonstrate the feasibility of obtaining 
usable Doppler radar returns from snock waves in air. 

It is suggested that a suitable radar system be operated so as to 
provide substantially normal incidence upon the ground reflected 
sonic boom produced by an aircraft passing directly overhead as 
indicated schematically in Fig. 7 . 1 . *i*ere is the possibility 
of obtaining radar reflections from ground-reflected boom and 
also from the high altitude boom, however, these reflections will 
be easily separated because of range differences. There is also 
the problem of double reflections from the two shock fronts of 
the sonic boom N-wave. The reflections from the bow and tail 
waves will be added and probably will not be resolvable because 
they are generally separated only bj ^ distance of the order of 
100 ft. This addition will involve variable amounts of phase 
cancellation depending upon the exact distance between the two 


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Bolt Beranek and Newman Inc 


- .-ks of the N=wave. Thus, the returned signals may vary widely 
in amplitude because of this interference phenomenon. At some 
ranges, however, (because the distance between the shock fronts * 
is continually expanding) the tv/o signals should add in phase 
and give four times the reflected power of a single shock. At 
these ranges the velocity of the wave should be determinable by 
Doppler techniques. The variation In intensity which is antici¬ 
pated by this interference process should prove valuable In 
determining the lower limit of sensitivity of the system. 

Study of the returns from both the ground reflection and the 
high altitude booms should provide a measure of the diminution 
of radar reflection with height and with two related sound 
intensities at the same height. 

The actual experiment which is contemplated is the observations 
of sonic booms created by supersonic aircraft provided by the 
Air Force. As an example it might be possible to use one or 
both of the two radar stations at North and South Truro on Cape 
Cod for such observations. It is anticipated that the PPS-7 
and the FPS-6 systems at the ADC installation at North Truro, 
and the FPS-20 and FPS-6 systems at the Mitre Corp. installa¬ 
tion at South Truro could be operated by experienced government 
personnel under the direction of the Air Force and suitable 
recordings made which can be correlated In time direction and range. 

Acoustic measurements would be made simultaneously. These measure¬ 
ments would be made near the ground at two or three positions 
along the ground zero flight path to establish the value of the 
shock over-pressure and provide a detailed analysis of N-wave 
signature. Several shocks should be observed at various times 





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Bolt Beranek and Newman Inc. 

during tho day i:i order to determine as far as practicable the 
effects of weather upon the shock wave and upon the observable 
radar reflection. 

Ph ase II 

* II is designed to utilize the results of Phase I in a 
tical study of sources of controlled shock waves which 
be adapted to an EMAC Probe ground installation. 

r.i f \ corns although readily available for the initial experiments 
; / sc I obviously have serious limitations as a tool for weather 

v.'.tion. Their expense is prohibitive, their direction of 
trav'- • is not optimized with respect to the radar, and the charac¬ 
teristics of an N-wave are probably not ideal because of the double 
shock and the resulting uncontrolled interference between the two 
reflected pulses. 

Several sound sources should be investigated including: 

1. Yachting cannon 

2. Dynamite 

3. Mild explosives 

4. Internal combustion devices 

5. Compressed air discharge 

The last of these appears, at the outset, to offer the greatest 
promise because of the much closer control of the significant 
parameters such as over-pressure, volume change, rise time, decay 
time, and discharge products. 

Specifically a theoretical study program should be undertaken to 
determine in detail the control parameters of such a source and 
to determine the necessary power and physical dimensions which 


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v*c •:_! optimise the- useful range. As a starting point for this 
theoretical analysis acoustic measurements should be made of the 
shock wave signatures of •» limited number of simple impulsive 
sources near the ground. 


Phase II I 

Phase III is directed towai- the production of an experimental 
sound source applicable for use in conjunction with a suitable 
radar installation. This phase depends largely upon the outcome 
of Phases I and II, 


The cost of the source obviously will depend upon its mode of 
operation and final size as determined by Phase II. It is 
expected that a usable source could be constructed from the 
developments of Phases I and II which could be tested for its acous¬ 
tic characteristics by ground measurements on an open range such 
as Bedford or Logan Airport. Ground measurements of the acoustic 
pulse should be made over distances, hopefully up to one mile 
from the source depending upon the clear range which can be made 
available. 

The operating parameters of the source should be varied by steps 
during these experimental measurements in order to obtain optimum 
values for pulse shaping and for maximizing range. Such tests 
might involve a few weeks of performance in order to cover a 
range of operating parameters and to encounter at least a moderate 
amount of variation in atmospheric conditions. The acoustic source 
parameters should also be adjusted to minimize personnel hazard 
and annoyance without reducing the range significantly. 



Bolt Beranek and Newman Inc. 


Phase IV 

Phase IV Is intended to demonstrate the joint operation of a 
suitable radar system and the sound source developed under 
Phase III. 

The sound source developed under Phase III should be operated 
with a suitable radar system. Measurements of the acoustic sig¬ 
nal along the ground should be made simultaneously with some of 
the near-horizontal radar observations. 

Measurements of acoustic wave signatures at elevated heights by 
means of balloon-supported microphones should also be conducted 
for some of the non-hcrizontal sound projections. Measurements 
at heights beyond those for which cable connections are practical 
might also be considered with radio-link systems. 






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Bolt Beranek and Newman Inc* 


8. CuNCLUSIONS 

1. The use of a high frequency acoustic beam is the major 
limiting factor in the range of the EMAC Probe system. 

2. For long range, 10,000 ft or more, the acoustic signal 
should have a frequency of less than 500 cps. 

3. The use of a long wave train for obtaining reinforcement 

of the radar reflection involves serious problems which outweigh 
its advantages. 

a) Such a long train will require coherent matching 
between the radar wavelengths and the sound wavelengths where- 
ever the reflection is to be reinforced by this process. 
Therefore, as the wave passes through areas vihere the ground 
velocity of the wave is altered the radar frequency must be 
altered simultaneously. Circuitry to enable such frequency 
tracking is complex and valuable radar search time will be used 
in order to provide a wavelength matching adjustment. 

b) In turbulent and Inhomogeneous areas, sound wavelength 
will vary and may be expected to change within the length of the 
wave train, thereby restricting the length of the useful beam. 

c) If the acoustic wavelength is increased by reducing the 
acoustic frequency as necessary for long range propagation, the 
radar wave will require a corresponding increase, and the radar 
beam can no longer be maintained as narrow as is necessary for 
detailed probing with any practical size of radar antenna. 


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Bolt Beranek and Newman Inc. 

a) At the low frequencies required for long range trans¬ 
mission, the length of a wave train itself would preclude 
detailed probing of small regions of interest. 

4. As an alternative for the multiple wave train, the use of a 
single shock wave front as a radar reflection surface has many 
advantages. 

a) The single shock provides a thin reflection surface 
which is well defined and thereby provides the best or possibly 
optimum condition for the radar reflection. 

b) More power can be carried by a single shock than can 
be carried by a train of sound waves. A shock wave can be 
launched as a portion of a sine wave and thereby result in little 
annoyance to personnel in the vicinity of the launching site, 
even though the sound pressure may be extremely high near the 
source. 


c) The single sinusoidal pulse can be made as long as is 
consistent with the requirements for directing the sound in 
desired directions while shading critical areas that may be 
affected by the intense sounds. The single pulse, though gen¬ 
erated nearly sinusoidal in shape will deform and become a 
shock ivave as the wave progresses provided only that its ini¬ 
tial amplitude is sufficiently high. The single shock will 
remain sharp for a distance approximately n times as far as a 
train of n shock waves having the same length as a single shock. 



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Bolt Beranek and Newman Inc. 

d) When the sound wave surface is carried down stream by 
the wind the specular reflection of the radar from the sound 
surface will be directed away from the radar antenna. Turbulence 
and homogeneities in the air will serve to roughen the spherical 
wave front surface and cause scattering of the radar beam. 

The effectiveness of this scattering mechanism for returning 
radar power to the antenna is far greater for a single shock 
wave than for a train of waves since the latter would have 
inherent coherence in the direction of the specular reflection 
and would cause a high retension of reflected energy in that 
direction even with scattering irregularities. 

5. A sound source for developing single shock pulses appears 
to be relatively simple. A chamber which can be filled with 
air and opened explosively should be tried as the actual source. 
This might be placed at the focus of the parabolic reflector in 
order to obtain the advantage of directivity. 

6. The propagation of a single acoustic pulse through the air 
should be studied by a simple experiment. The proposed experi¬ 
ment should include as a minimum the generation of an explosive 
signal having high energy at frequencies as low as 100 cps and 
this pulse should be tracked with Doppler radar to determine 
the magnitude of the signal and the potential range using for 
example a 400 megacycle signal and perhaps also a higher 
frequency for comparison purposes. It would be desirable 
simultaneously to make acoustic measurements of the wave along 
the ground at elevations as high as practicable as a check upon 
the theoretical analysis which is presented in this report. 


Bolt Beranek and Newman inc. 


it io that the radar wavelength be larger than 

tne thickness oi* the acoustic shock wave front for good reflec- 
tiv/fi. A preliminary experiment should be carried out using 
sonic booms to determine the practical thickness of shock fronts 
with small values of overpressure for a range of atmospheric 
conditions. These experiments should include simultaneous 
observation of the amount of radar reflected from the measured 
booms. Such experiments should materially aid in the evaluation 
of the requirements of an acoustic source for an EMAC Probe 
system. 


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Bolt Beranek and Newman Inc. 


REFERENCES 


1. Atlas, David, "Radar Detection of the Sea Breeze," 

J. of Meteoroi., 17, No. 3, pp. 244-258, June I960. 

2. Atlas, David, "Possible Key to the Dilemma of Meteorological 
'Angel* Echoes," J. of Meteoroi., JL7, No. 2, pp 95-103, 

April, 196c. 

3. Atlas, David, "Radar Studies of 'Angels'", Session IV, 

Radar Studies of Meteorological "Angel" Echoes , J. of 
Atmos, and ‘Ferres, Fhysicsj l5, pp. £62-2877 1959. 

4. Atlas, David, "Meteorological 'Angel' Echoes," J. of Meteoroi., 
16, No. 1, pp. 6-11, 1959. 

5. Atlas, David, "Indirect Probing Techniques," BULL, of the 
Am. Meteoroi. Soc., 43, No. 9, pp. 457-466, 1962. 

6. U. S. Pat. No. 2.-539,593, 2,823,365; patents issued to 
Robert H. Rines, a member of the staff of Bolt Beranek 
and Newman Inc. 

7. Smith, P. L., Jr., "Remote Measurement of Wind Velocity 

by the Electromagnetic Acoustic Probe," I. System Analysis, 
Conf. Proc. 5th Nat. Conv. on Military Electronics, Wash., 

D. C., Midwest Research Institute, Report No. 419, pp. 48-53, 

1961 , 

8. Fetter, R. W., "Remote Measurement of Wind Velocity by the 
Electromagnetic Acoustic Frobe," II. Experimental System, 

Conf. Proc., 5th Nat. Conv. on Military Electronics, Wash., 

D. C., Midwest Research Institute, Report No. 420, 

pp. 54-59, 1961. 

9. Fetter, R. V/., P. L. Smith, Jr., B. L. Jones, H. F. Schick, 
and R. M. Stewart, Jr., "Investigation of Techniques for 
Remote Measurement of Atmospheric Wind Fields," Phase II: 
Analysis, Report No. 2, Midwest Research Institute, 

Oct. 1961 - Feb. 1962. 

10. Fetter, R. W., P. L. Smith Jr., and 3. L. Jones, 

"Investigation of Techniques for Remote Measurement of 
Atmospheric Wind Fields," Phase III: Design of Experiments, 
Report No. 3, Midwest Research Institute, Feb. 1962 - 
June 1962. 


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Bolt Beranek and Newman Inc 


11. Kerr, Donald S.> Ed., Propagation of Short Radio Waves , 
Radiation Lab. Series, “Vox. 13, Mc&raw-Hill Book Co., Inc., 
Appendix B., 1951. 

12. Friend, Albert W., "Theory and Practice of Tropospheric 
Sounding by Radar," Proc. Inst. Radio Engr., pp. 116-137, 
1949. 

13. Jones, B. L. and P. C. Patton, IRE Trans, on Antennas and 
Propagation , AP-8 , pp. 418-423, I960'. 

14. Harris, C., H andbook of N oise Co ntrol , McGraw Hill Book Co., 

Inc., Chap. 3, 1951. ~ 

15. Ref. 11, pg. 46. 

16. Chernov, L. A., Wave Propagation in a Random Medium , English 
Edition, McGraw Hill Book Co., l9t>0. 

17. Golitsyn, G. S., A. S. Gurvicn and V. I. Tatarskii, 
"Investigation cf the Frequency Spectra of Amplitude and 
Phase Difference Fluctuations of Sound Waves in a Turbulent 
Atmosphere," Soviet Acoustics, 6 , No. 2, pp. 185-194, i960. 

18. Ref. 16, pg. 83. 

19. Ref. 16, pp. 84-107. 

20. Silver, Samuel, Ed., Microwave Antenna Theory and Design, 
Radiation Lab. Series, Vol. 12, McGraw-Hill Book Co., Inc. 
p. 188, 1949. 

21. Ref. 16, pp. 125-146. 

22 . Ridenour, Louis N., Ed., Radar System Engineering, Radiation 
Lab. Series, Vol. 1, McGraw-Hill Book 0o7, Inc., d. 20,. 
1947. 

23. Ref. 22, pg. 271. 

24. "Atmospheric Physics and Sound Propagation," prepared at 
the Dept, of Phys., The Penn. State Univ., under Signal 
Corps Contract W-3o-D39-SC-32001, Sept. 1, 1950. 


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Bolt Beranek and Newman Inc. 


25. Allen, C. H., "Finite Amplitude Distortion," thesis. Dept, 
of Phys., The Penn. State Univ., 1950. 

26 . Nybcrg, W. L. and D. Mintzer, "Review of Sound Propagation 
in the Lower Atmosphere," WADC Tech. Report 54-602, 

May, 1955. 

27. Ref. 26 , pp. 19-22. 

28 . "Investigation of Acoustic Signaling Over Water in Fog," 

BBN Final Report Phase 2, USCG Contract No. Tcg-40854, 

CC- 43 , 458 -A, p. 64, Jan. i 960 . 

29. Rudnick, I., "On the Attenuation of High Amplitude Waves of 

Stable Form Propagated in Korns," J. Acoust. Soc. Am., 30, 
339, 1958. — 

30. Laird, Donald T., "Spherical Sound Waves of-Finite Amplitude, 
thesis. The Penn. State Univ., 1955. 

31. Becker, R., "Shockwave and Detonation," Zeit. ftir Phys. 8, 

pp. 321-362, 1921. “ 

32. Pearsons, Karl S., BBN Quarterly Progress Report No. 5, 
Contract No. NASr- 58 , July 1962 -Oct. 1962 . 

33. Harris, C. M., "Absorption of Sound in Air in the Audio- 
Frequency Range," J. Acoust. Soc. Am., 35, 11, 1963.