MODELING OF SOUND PROPAGATION IN THE SEA
MODELIROVANIE RASPROSTRANENIYA ZVUKA V MORE
MO.llEJlHPOBAHHE PACnpOCTPAHEHH5I 3BYKA B MOPE
THE LIBRARY
MODELING OF
SOUND PROPAGATION
IN THE SEA
A. N. Barkhatov
Gorki State University
Gorki, USSR
Translated from Russian by
James S. Wood
@CONSULTANTS BUREAU •
NEW YORK-LONDON • 1971
ISBN 978-1-4684-1583-4
ISBN 978-1-4684-1581-0 (eBook)
DOI 10.1007/978-1-4684-1581-0
The original Russian text, published by Gidrometeorologicheskoe Izdatel'stvo in 1968, has been corrected by the author for the present edition. The
English translation is published under an agreement with Mezhdunarodnaya
Kniga, the Soviet book export agency.
BapxaTOB
A~eKCaHnp
HHKO~aeBHq
MO)l.EJlHPOBAHHE PACnpOCTPAHEHHl1 3BYKA B MOPE
Library of Congress Catalog Card N umber 74-136985
SBN 306 -10855-0
© 1971 Consultants Bureau, New York
Sofcover reprint of the hardcover 1st edition 1971
A Division of Plenum Publishing Corporation
227 West 17th Street, New York, N.Y. 10011
United Kingdom edition published by Consultants Bureau, London
A Division of Plenum Publishing Company, Ltd.
Donington Rouse, 30 Norfolk Street, London, W.C. 2, England
All rights reserved
No part of this publication may be reproduced in any
form without written permission from the publisher
PREFACE
The book is concerned with the application of modeling techniques and procedures to
the investigation of sound propagation in the sea.
The modeling method affords a means for studying the laws governing the sound fields
in the sea and in other, similar media under controlled laboratory conditions and can be used
in underwater acoustics as a coroHary to field experiments. The method has a number of
advantages, principal of which are the relative simplicity and low cost of model tests by comparison with fuH-scale tests under oceanic conditions, the high accuracy of acoustical measurements, excellent reproducibility of the measurement results, and the capability of rapidly
varying the experimental conditions, which, unlike the conditions of field experiments, are
under complete control.
For the modeling of sound propagation in the sea the latter is treated, depending on the
problem to be solved, either as a volume-homogeneous medium or as a medium possessing
regular and randomly-distributed inhomogeneities.
We direct our primary attention in the book to the modeling of layered-inhomogeneous
media, but we also discuss separate problems bearing on the study of sound propagation in
the sea.
It is demonstrated in examples how modeling is employed to investigate the sound field
in the ocean for certain typical vertical distributions of the velocity of sound in the ocean.
In some cases the results of model experiments are compared with the theory of sound
propagation in layered-inhomogeneous media as developed over the last twenty years in a
multitude of studies by Soviet and foreign scientists, most notably in the fundamental research
of L. M. Brekhovskikh. The consistency exhibited between the experimental and theoretical
data testify to the reliability of the modeling method and, at the same time, serves as confirmation of the theory. The most valuable asset of the method, however, is the fact that it
can be used in situations devoid of a theoretical solution.
In the first chapter of the book we consider the similarity conditions for sound propagation in an oceanic medium. The second chapter is devoted to the techniques of modeling experiments. In the third chapter we discuss the methods for modeling certain inhomogeneous media
and the sound fields in those media. In view of the fact that the behavior of different kinds of
waves is governed by many common principles, acoustical modeling can also prove very useful
for the investigation of radio wave propagation in the earth's atmosphere, with the latter treated
as a layered-inhomogeneous medium characterized by a refractive index that varies with height.
This problem is discussed in the Appendix.
v
vi
PREFACE
In the presentation of the material the author relies heavily on his own work carried out
in the Acoustics Department of the N. 1. Lobachevskii Gorky state University from 1951 through
1966 on the modeling of sound propagation in the sea. The author takes this opportunity to express his appreciation to the members of the Acoustics Department and the students of the
Radiophysics Department of Gorky State University, who at various times have assisted with the
experimental work.
The author is indebted to V. A. Zverev, I. D. Ivanov, Yu. M. Zhidko, and B. N. Gershman
for taking time to review portions of the manuscript at various stages of its preparation, as
weIl as for valuable consultation and suggestions.
The author acknowledges that the book is not without certain unavoidable shortcomings.
The deficiencies present in the book are attributable in some measure to the meager state of
the art of some particular problems in underwater acoustical modeling. Typical of these problems is the modeling of ocean ground soils and bottom relief; the surface wave state of the
sea, internal waves, media with a horizontal sound velocity gradient, etc.
All comments regarding the book will be most gratefully received by the author.
The author will be pleased if the book, its shortcomings and deficiencies notwithstanding,
proves of benefit to underwater acousticians engaged in research and to students preoccupied
with the fundamentals of underwater acoustics.
CONTENTS
Chapter 1. The Similarity Problem in Modeling of the Oceanic Medium • • • • • . • • •
1. Description of the Method and Fundamental Modeling Relations. • . • • • . . •
2. Some Characteristics of the Modeling of Inhomogeneous Media. • • • • . • • .
1
1
5
Chapter 2. Model Experimental Procedure . . . • • • • • . . • • • • • . . • • . . . . . . . . •
3. Basic Components of the Underwater Acoustical Equipment . • . . . . • • • . •
4. The Anechoic Tank • • • • . • • • • . • . • • . . . . • . • • . • • . • • . . • • . • . • • •
5. Models of the Bottom and Slrface of the Sea. • . • . • • • • • . . . • • • . • . . • •
6. Principal Electroacoustic Equipment . . . • • . • . . • • • • • • . • • • . • • • • • •
7. Scanning of the Sound Field • • . • • . . • . . • . • • • • • • • • • • . • • • • . • . • • .
8. Instruments for Measuring the Velocity of Sound in the Model Medium. • • •
9
9
10
13
19
27
29
Chapter 3. Modeling of Sound Propagation in Inhomogeneous Media. • . • • . • • . • • •
9. Methods for the Modeling of Layered-Inhomogeneous Media. • . • • • • • . • •
10. Modeling of aSound Channel • • . . •• • . • . • • • • • • . • • • • • • . • . • • • • • •
11. Modeling of Antiwaveguide Sound Propagation. • . • . . • . • • • . . • • • • • • • •
12. Modeling of the Near Field of aSound Velocity Discontinuity Layer . . . • • .
13. Modeling of Media with a Vertical-Horizontal Sound Velocity Gradient. • • •
14. Modeling of Sound Propagation in the Sea in the Presence of Rough Free
33
33
36
44
49
53
Surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
15.
16.
17.
18.
19.
11
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
55
Modeling of Media with Random Inhomogeneities. • • • . • • • . • • . . • . . • • •
Model Experiments with Internal Waves . • • • . • • • • . • . • • . • • . . • • • • •
Sound Scattering by Bodies in Water • . • . • • . • . • • • . . • . . • . • • . • . • • •
Analog Modeling of Wave Fields in Inhomogeneous Media. . . . • . • . • • . • .
On the Applicability of Ray Representations in Underwater Acoustics . • • • •
64
67
69
72
73
Appendix. Acoustical Modeling of Radio Wave Propagation in the Earth's
Atmosphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Propagation of Radio Waves in the Troposphere . • . . • • • • • • • • • . . • . • • • •
Propagation of Radio Waves in the Ionosphere. • • • . • • • • • • • . • • • • • • • • • •
79
80
83
Literature Cited . • . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
87
vii
111
•
•
•
CHAPTER 1
THE SIMILARITY PROBLEM
IN MODELING OF THE OCEANIC MEDIUM
§ 1.
Description of the Method and Fundamental
Modeling Relations
The modeling of sound propagation in the sea is effected under laboratory conditions by
means of a physically identical process. In this process the particular effect under investigation is studied directly in the model, and the modeling problem is reduced to a change of sc ale
of the space-time variables.
It is required to employ physical modeling in situations invol ving a complex investigated
process. Just as a number of simplifying assumptions are introduced in the theoretical analysis of the problem, so in the model experiment is it necessary to isolate and reproduce under
laboratory conditions only the most significant factors governing the effect as it occurs in nature. Model experiments make it possible to test and refine the theory, thereby facilitating the
application of the theory in practice. For this reason the modeling method is used, for example, in geophysics for the investigation of seismic wave propagation processes in the earth's
crust and mantle (see, e.g., [27, 54, 63]).
The applicability of the modeling method to the study of the principles governing the sound
field in the oceans and seas is also dictated by the complexity of the conditions under which
sound is propagated in those bodies, due to the inhomogeneities of the medium.
Besides the local density and temperature inhomogeneities, which have a random character, the sound field in the oceanic medium is strongly affected by the regular variation of the
temperature, salinity, and hydrostatic pressure with depth, as well as the variation of the
physical properties of the medium in the horizontal direction in certain regions of the world
ocean. Consequently, the investigation of the laws of sound propagation in natural bodies of
water in general requires consideration of the vertical and sometimes the horizontal gradient
of the velocity of sound, as well as various statistical processes (volume reverberation of
sound; sound scattering by ocean surface waves, the bottom, sound-scattering layers, local
obstacles, etc.). The inhomogeneities of the medium give rise to sound refraction and fluctuations of the amplitude and phase of the sound wave. Therefore, together with the investigation
of sound propagation under natural conditions, where the sound field is acted upon by a composite
of many factors, it is also instructive to conduct laboratory investigations in which the influence
of each one of the variegated inhomogeneities of the natural medium can be studied separately.
In connection with the widespread ingress into scientific research of problem-solving
procedures involving electronic computers, the latter afford an alternative to physical simula1
2
THE SIMILARITY PROBLE M
[CH.1
tion. However, there are problems in which the modeling or simulation method outranks the
method of machine computation. The model equipment is particularly well suited to the rapid
verification of theory of individual hypotheses brought forth in the development of theory.
The author, in particular, has conducted a test under model conditions on the applicability
of the geometric and wave theories for the investigation of sound propagation in an oceanic
medium with a given depth distribution of the velocity of sound and has assessed the possibility
of the physical interpretation of the field in the shadow zone by ray representations. There are
also well-known model experiments in which the theory of sound wave scattering by an uneven
surface having a specified profile, by elementary bodies (spheres, cylinders), and in other situations has been corroborated.
On the other hand, computer solutions present a difficult matter and are often realizahle
only by a program written for more idealized conditions than those set up in model experiments. For instance, it is not always possible on a computer to program the details of the
vertical sound velocity distribution curve, the singular features of the boundary surfaces or
local scatterers, the influence of statistical volume inhomogeneities, etc.
In such situations the application of the modeling method yields information concerning
the sound field far more rapidly than the computer, and the data obtained on the model have
the added virtue that they enahle one to obtain a physically transparent picture of the analogous
effect in the oceanic medium. The excellent visualization of the results of model experiments
makes modeling a workahle means for gaining insight into the effects that one should look for
during the propagation of sound in the sea, even when the values of the experimentally measured
variables are not as accurate as they should be.
Despite its indisputable advantages, the modeling method has not gained much acceptance
in underwater acoustics, a fact that must be blamed on a certain delinquency in the general development of the method in this particular area of science and engineering. Modeling has
usually been employed for the investigation of phenomena observed preeminently in volumehomogeneous media. Only in very recent years has work been initiated on the study under
conditions of sound propagation in media having volume-distributed random inhomogeneities
and in layered-inhomogeneous media.
From the foregoing account of the modeling method we are in a position to portray its
status in underwater acoustics. Clearly, model experiments must be coordinated with fullscale investigations in nature by a unified research program under conditions dictated by the
specific problem. It is also a good idea to parallel the model study with theoretical work in
underwater acoustics. Then the experiments performed on the model setup will afford a firstclass verification of the theory.
It is essential in modeling to preserve the similarity of the model to the effect under
investigation. It is customary to adopt as the measure of similarity a certain dimensionless
variable or a function thereof, which remains constant under conversion to the model (the
scaling invariant). In hydrodynamics, for example, the dynamic similarity of the motion of a
viscous fluid in tubes or of solids inside and on the surface of a liquid is described in terms
of aseries of dimensionless parameters (the Reynolds number, Strouhal number, Froude number, etc.) [36,56].
In acoustics similarity may be characterized by the parameter wt or by the parameter
kl, where w is the cyclic frequency, k is the wave number, and l is a characteristic linear dimension (length of a sound-scattering body, wave height on the surface of a liquid, etc.). The
constancy of these parameters is preserved both in a homogeneous medium and in a medium
with random inhomogeneities (quasi-homogeneous medium). In the latter case it is required
§l]
METHOD AND FUNDAMENT AL MODELING RELATIONS
3
to use the mean-square amplitude or radius of correlation of the inhomogeneities as the characteristic linear dimension.
In layered-inhomogeneous media the parameter kl is not sufficient as a characteristic
of the field, insofar as it fails to account for the curvature of the rays along which the particles of the medium are moving during the propagation of sound therein.
The similarity conditions for the sound field in layered-inhomogeneous media, whose
properties depend only on the vertical coordinate z, may be deduced from the two-dimensional wave equation in the velocity potential cp(x, z), where the coordinate x is measured along
the horizontal axis in the direction of wave propagation; in the case of harmonic oscillations
the equation has the form
21t
]2
ßl'+ [ Ton(z) 'f'=o.
(1.1)
Here n(z) = colc(z) is the refractive index, c(z) is the velocity of sound at depth z, Co is
the velocity of sound at z = 0, and A 0 is the sound wavelength at c = co. It is assumed that there
is zero steady motion of the medium relative to the acoustic transducers.
Suppose that a layered-inhomogeneous liquid medium described by the velocity potential
cp is simulated by an analogous medium through a proportional M-fold scaling-down of all
linear dimensions (we refer to the number M as the scaling factor). The refractive index (or
velocity of sound) and sound frequency in this case must be varied so as to maintain similar
conditions of sound propagation in the two media.
Let us transform Eq. (1.1), performing a change of coordinates according to the relations
x=Mx'; z=Mz'.
(1.2)
The sound wavelength must be varied analogously:
)'0= M),~.
Equation (1.1) is transformed to the expression
i::;-
ß''f' + [ 21t n (Mz')
]2'f' = 0,
(1.3)
where A' denotes the Laplacian on the primed coordinates.
On the other hand, the wave equation for the model medium has the form
ß''f' + [~: n' (z')Y'f' =0,
(1.4)
where n'(z) = c~/c'(z') is the refractive index for the model medium and c~ and c'(z') are the
velocities of sound on the surface and at depth z', respectively, for the same medium.
By the stipulation of the problem the last two equations must be identical. Comparing
them, we find the relation
n (z) = n' (z') ,
which is equivalent to the following equation:
c(z)
c' (z')
Co
Co
--=--,-
(1.5)
THE SIMILARITY PROBLE M
4
[eH. 1
For the modeling of a medium it is required to know the law of variation of the velocity
of sound with depth.
Let us consider several layered-inhomogeneous media with different laws governing the
depth variation of the velocity of sound:
a) The velocity of sound depends linearlyon the vertical coordinate:
C(z) = Co
c' (z') =c~
+ bz;
+ b'z'.
According to (1.5).
b'
b
Co
Co
- , =M-.
(1.6)
This equation may be represented in the form
(1.7)
which contains and sound wavelengths and relative gradients of the velocity of sound:
,
b'
a=-,
Co
dc'(z')
d'
.
Z
b) The velocity of sound is a quadratic function of the vertical coordinate:
+ bjz + b2z2;
C' (z') = c~ + b;z' + b; (Z')2.
c (z) = Co
In this case we use (1.5) to find the following two conditions that must be met for modeling:
c) In the general case when the velocity of sound is apower function of the form
n
c(z)= ~bkZk; bo=co ;
k=O
n
C' (z') =
~ b~ (Z')k; b~ = c~ ,
k=O
the given media are similar if the coefficients of the terms in different powers of z are related as follows:
(1.8)
In order to simplify the plotting of the ray pattern and the calculation of the sound field
in the ray approximation it is customary to replace the smooth curve representing the sound
velocity function c(z) by a broken-line curve consisting of segments corresponding to horizontal
layers of the medium with a constant vertical sound velocity gradient. Thus approximated, the
medium is modeled on the basis of the foregoing discussion with regard for the one relation
(1. 7), which may also be written in the form
(1.9)
§2]
MODELING OF INHOMOGENEOUS MEDIA
5
where l. a. and 1 denote the characteristic length. relative sound velocity gradient. and sound
frequency in the modeled medium; the same notation primed refers to the model medium.
Consequently. the following dimensionless parameter must be used as the invariant of
simulation of layered-inhomogeneous media in the case of a constant sound velocity gradient:
(1.10)
B=al,
where 1 is a characteristic length (distance between transducers. depth. sound wavelength. etc.*
The sound field in the medium can also be characterized by the parameter alk or the
parameter acl1, where k = 21f1 A. 1 is the sound frequency, and c is the velocity of sound.
The parameter B uniquely characterizes the degree of inhomogeneity of the unconfined
infinite layered-inhomogeneous medium. In the analogous medium bounded by an uneven surface or containing isolated bodies or randomly distributed volume inhomogeneities acoustical
similarity is described by the parameters Band kl. where 1 is a suitable characteristic dimension of the inhomogeneity.
Uniform and rectilinear motion of the sound source (or receiver) relative to the medium
does not violate the form of the invariant (1.10). because the wave equation (1.1) does not undergo any modification.
With motion of the transmitter a Doppler frequency shift is observed; it is equal to [36]
1:11=10
v
-cos6
(c eos 6
c'v
1 --
)'
where v is the velocity of the sound source relative to the medium and e is the angle between
the direction of the velocity vector
and the wave vector k . In order for the relative frequency shift b.111 0 to remain constant in modeling. it is necessary that (v/c) cos () = const hold.
a condition which must also be observed during motion of the sound reciever relative to the
medium.
v
§ 2.
Some Characteristics of the Modeling
of Inhomogeneous Media
Inasmuch as all the linear dimensions undergo a considerable reduction in the conversion
from real objects to the model (M» 1). it follows from (1.9) that the simulation frequency l'
must be many times the frequency 1 of sound propagating the sea. This gene rates the problem
of the legitimacy of transferring the results of investigation of asound field with frequency l'
to wave processes occurring at the frequency 1. All the relations fundamental to the modeling
problem are derived from the wave equation (1.1). which, in turn. is a consequence of the
linearized equations of hydrodynamics and the assumption of adiabaticity on the part of the
acoustical process. Hence. the validity of the given assumption at both frequencies is in the
final analysis a sufficient criterion of the compatibility of the wave equations (1.3) and (1.4)
and, consequently. the fulfillment of the similarity relation.
*In a medium in which the vertical sound velocity profile is approximated by a broken line each
segment of the latter corresponds to different values of the sound velocity gradient a1. a2' ....
ak"'" where
ak f'
M I ~ a;
= - = - = - = ••• = l' a, a2
ak
=-.
f
For the modeling of such a medium, therefore, the quantities Bi' ~ •...• Bk •.•. corresponding
to individual segments of the broken line are invariants, i.e., an equation of the type Bk;::;
alt l' ;::;a kl must hold.
6
THE SIMILARITY PROBLE M
[CH.1
We are weIl aware that a wave process is adiabatic if the sound wavelength A is greater
than the thermal conduction wavelength At. Suppose that on a certain surface, due to the transmission of a plane sound wave through it with a cyclic frequency w the temperature varies with
time according to the law T = T 0 exp (iwt). The temperature oscillations are propagated from
the center of perturbation in the form of rapidly oscillating thermal waves with a wavelength
At = 271'(2 xw- 1)1/2 = 2 (7I'Xf- 1)1/2, where X is the thermal conductivity, which has dimensions
cm 2 • sec- 1 [36].
We see from a comparison of the corresponding equations that the thermal wavelength
varies inverse ly as the square root of the frequency, whereas the sound wavelength varies inversely as the frequency. At a frequency above the critical value fcr = c 2(471'x)-1, which is determined from the condition A = At, the adiabaticity of the sound transmission is violated. Conversely, at f < fcr the inequality A > At is always maintained, i.e., sound propagates adiabatically.
For water the critical frequency has an order of magnitude of 1012 cps. In underwater
acoustics, therefore, the adiabaticity of sound transmission is guaranteed by a wide margin of
safety in all experiments performed either in nature or in model situations.
Modeling is complicated to a certain extent by the allowance for sound attenuation in the
medium. As a matter of fact, sound attenuation in sea water is caused by absorption and
scattering. With an increase in frequency the attenuation increases, where, as shown by oceanic measurements, thecharacter of the frequency dependence of the attenuation factor differs·
in different frequency intervals and depends also on the region of the world ocean in which the
measurements are performed. According to [99], in the range from 20 cps to 60 kc the attenuation ß, expressed in dB per km, is determined by the following empirical formula in an
underwater sound channel:
ß=O.036l/',
(2.1)
where f is the frequency in kilocycles.
This formula is nonunique in underwater acoustics.
In model tests performed in laboratory tanks the scattering of sound by volume inhomogeneities of the medium is negligibly small (provided the inhomogeneities are not specifically created), and the cause of attenuation is predominantly the viscous absorption of
sound.* The sound attenuation factor due to viscosity is proportional to the frequency squared
[36].
Consequently, the attenuation of sound in sea water differs from that in the water of the
laboratory tank. llie to the disparity of the frequency dependence of the sound attenuation in
the given media the modeling of sound attenuation in the sea is quite impracticable. It is
necessary instead to ac count for and eliminate this attenuation in the model medium. This
is done by calculating beforehand the sound absorption in the laboratory medium at the modeling frequency (tabulated data may be used for the attenuation) and incorporating corrections
into the experimental values of the sound pressure to eliminate the sound attenuation. Once
the corrected values of the sound pressure have been referred to the distances covered in the
sea by means of the similarity relation, new corrections are introduced into those values to
ac count for the attenuation of sound in the oceanic medium at the other (lower) frequency.
* The relaxation absorption of sound in the liquid contained in the laboratory tank (water,
aqueous solutions of sodium chloride, and ethyl alcohol) is negligible by comparison with
viscous absorption in the frequency band normally used for model experiments (100-3000 kc).
§2]
MODELING OF INHOMOGENEOUS MEDIA
7
We now give a sampie estimate of the sound attenuation. Let the model measurements
be performed in water at a frequency j' = 1 Me. According to [17], for example, the sound
attenuation is equal to CI" = 30 . 10- 17 ([,)2 ern-i. Hence we find the reduction of the intensity
level of a plane sound wave at a distance r' = 5 m from the source: .0.b = 20 log exp(CI"r') """
1.3 dB.* For a scaling factor M = 104 these experimental conditions correspond to the propagation of sound in the sea at a frequency j = 0.1 kc over a distance r =50 km. The sound attenuation at that distance, according to (2.1), is equal tO.0.ß = 0.036j 3/2 r = 0.06 dB.
In the case cited the discrepancy between the sound attenuation in the tank and in the
oceanic medium is appreciable and needs to be taken into account.
We also wish to bring to attention another characteristic feature of model experiments, in that all the dimensionless variables characterizing the medium and the acoustic
transducers remain invariant in the modeling situation. This applies to the directivity characteristics of the transducers, their angles of rotation, the beam angle, the coefficients of sound
reflection from the bottom and surface of the water, etc.
If the directivity characteristic of the transmitter is to be held constant in model experiments, its dimensions must be scaled down in accordance with the increase in the acoustic frequency. In this case, however, a proportional reduction of the dimensions of the transmitter
(or receiver) housing is not always feasible, in view of design considerations. This fact can inject adefinite artificiality into the setup of model experiments, a fact that must not be overlooked.
* The calculation of the sound attenuation in the sea by the indicated formula yields values that
are too small for the attenuation by comparison with the experimental data.
CHAPTER2
MODEL EXPERIMENTAL PROCEDURE
§ 3.
Basic Components of the Underwater
Acoustical Equipment
The model equipment is designed for the performance of acoustical measurements under
conditions approximating those under which sound is propagated in the ocean. Le., in a medium
that contains regular and random inhomogeneities of a definite type and that is bounded below
by a bottom and above by an uneven sea surface. Inside the medium there can also be individual
bodies having diverse configurations and functioning as sound scatterers.
These conditions are created artificaHy in the underwater acoustical tank and must be
amenable to control throughout the entire experiment.
In light of the foregoing the modeling of sound propagation in the sea comprises a physical
experiment entailing the preparation of a medium with assigned properties. as weH as hydrological and acoustical measurements. In undertaking model experiments. the experimenter
must have access to materials and means for the simulation of an inhomogeneous medium of the
oceanic type (a supply of liquids characterized by various acoustical properties. equipment for
the heating and cooling of separate portions of the water in the tank. etc.), and he must have
measuring and testing instruments for determining the parameters of the medium (thermometers, salinometers, devices for measuring the velocity ofsound in the liquid. etc.) and the
requis ite electroacoustical apparatus.
A medium invested with specific properties 1s created in an experimental. usually anechoic tank. over which is mounted a coordinate positioning device, which permits the electroacoustic transducers (transmitter and receiver) to be positioned at any desired point in the
medium and their coordinates to be measured.
For model experiments it is also necessary to have an adequately broad base of models
of various ocean soils and an uneven sea surface, mechanisms for the excitation of waves on the
surface of the water and internal waves, etc.
Certain effects observed during the propagation of sound in the sea. such as the scattering of sound by isolated bodies or limited portions of the uneven surface, the formation of a
sound field in a water layer whose thickness is commensurate with or less than the acoustic
wavelength, etc., can be investigated with relative precision in the model equipment in a volume-homogeneous medium. This simplifies the experiment considerably.
For the performance of electroacoustical measurements the following basic instruments
are required: a generator of continuous or pulsed high-frequency electrical oscillations. elec9
10
MODEL EXPERIMENTAL PROCEDURE
[CH.2
troacoustic transducers, an amplifier for the signals received by the sound receiver and transformed into electrical pulses, and sound field indicators (cathode-ray oscilloscopes and loop
oscillographs, level recorders, etc.).
In the ensuing sections we describe the individual components of the underwater acoustical model equipment.
§ 4 ..
The Anechoic Tank
The core of the model equipment is the experimental tank. The size and shape of the
tank are chosen to meet the objective of the investigation. Long tanks (greater than 1000 times
the sound wavelength) are suitable for the observation of effects that depend on the distance between the transmitter and receiver. Tanks of this type are required, in particular, for the investigation of fields in sound channels and secondary irradiation zones. In other situations (for
example, in the investigation of near fields or acoustic shadow zones, in which the field falls off
rapidly with distance from the source) it is convenient to use a smaller tank, as this facilitates
the installation of equipment in the tank and reduces the quantity of liquid required for the experiment. Normally the tank has a rectangular parallelepiped configuration.
In special cases the tank can be made with a circular cylindrical shape, as for example
in the determination of the angular characteristics of the reflection coefficient [71], or the
side walls of the tank can be made to slope downward so as to diminish the interference produced by signal reflection from the walls [13].
The tanks are constructed from such materials as sheet steel, reinforced concrete, ceramic-line masonry, wood, or plastic (vinyl or plexiglas). A good working tank is one whose walls
are made of sheet plexiglas [15].* Transparent tank walls afford the possibility of observing
whatever objects are placed in the liquid.
A schematic diagram of an experimental tank equipped for the modeling of a medium
with a vertical sound velocity gradient is shown in Fig. 1.
For the elimination of signals reflected from the tank walls the sound field is generally
measured in the pulsed mode, in which case the difference between the arrival times of the
direct signal and the wall-reflected signal at the receiver must be greater than the signal
width. In this way the useful signal and the wall-reflected signal are separated on the oscilloscope screen with respect to their arrival times.
On the other hand, the wider the signal, the steadier the sound field will be in the tank.
Consequently, the optimum signal width is chosen in the experiment as a compromise between
the two indicated requirements.
Also, in order to abate the influence of the walls on the field of the direct signal the tanks
are lined with a coating that will ensure sound absorption and the scattering of sound in düferent directions. The value of the absorbent lining of the walls becomes particularly prominent during operation with sound sources having a wide directivity pattern, as in this case
there is a sharp increase in the tendency for wall-reflected signals to be picked up by the receiver.
The various methods of damping the tank walls are based on a diminution of the sound
reflection coefficient by:
* The walls are cemented and joined with screws, and for greater strength the tank is housed
in a metal frame or is reinforced externally with wooded braces or props.
§4]
THE ANECHOIC TANK
11
Fig. 1. Diagram of the experimental tank. 1)
Liquid-filled tank; 2) coordinate positioning device with carriage; 3) transmitter; 4) sound receiver; 5) motor with reduction gear for movement of the carriage along the tank; 6) high-frequency oscillator; 7) sound field indicating device; 8) wave vibrator; 9) heater; 10) cooler.
1) causing it to impinge on a medium that strongly absorbs sound in a certain frequency
band, or on a medium characterized by resonance absorption at several discrete frequencies;
2) improving the acoustic matching between the water and tank walls;
3) placing scatters of various configurations on the walls, where they transform the
specularly reflected signal into a diffuse field, which is easily discriminated from the direct
acoustic signal.
In order to increase the efficiency of the damping system it is advisable to combine sound
absorption in the tank walls with scattering.
We now look in closer detail at some of the techniques of damping the tank, where the
efficiency of the technique depends on the acoustic frequency.
The simplest damping is effected by a layer of absorbing liquid (such as castor oil),
which is separated from the water by a thin layer of water-compatible rubber (i.e., rubber
whose acoustic impedance is the same as that ofwater). The absorptive properties of the liquid
layer are enhanced with the immersion of a pure metal grid into the layer [90]. This type of absorber lowers the level of the reflection coefficient (relative to a steel wall) 20 dB at low ultrasonic frequencies (around 10 kc).
For the damping of sound at higher ultrasonic frequencies (ab out 1 Mc) it is possible to
use a coating of water-compatible (butyl) rubber. According to [21]. such a rubber layer 6 mm
thick reduces the reflection coefficient 30 dB at grazing angles above 30°.
For the damping of laboratory tanks and underwater acoustical reservoirs resonance absorbers are sometimes used, consisting of a layer of absorbing rubber with internal air pockets
whose dimens ions have resonance values at the appropriate acoustic frequencies [53].
In [101] a damping technique is described which is based on the tendency of a layer of air
bubbles to lower the sound reflection from a metal surface when the layer is situated between
the surface and sound source. Felting made from horsehair and impregnated with rubber affords an equivalent alternative to a bubble layer in that the rubber coating the hairs contains
air bubbles. The efficiency of this absorber depends on its water impermeability . It has been
demonstrated experimentally that the coating can be used effectively for five eight-hour operating cycles without additional drying. This type of coating 5 cm thick reduces the sound reflec-
12
MODEL EXPERIMENT AL PROCEDURE
[eH. 2
tion from the wall by about 20 dB in the frequency band from 100 to 400 kc at various grazing
angles (30° to 90,,), and the scattering level in the direction of detection does not exceed the
average scattering level in the other directions.
The acoustic matching of the absorber and water is improved if the contact surface between them is not flat, but is composed of wedges, spikes, or blinds.
The damping afforded by rubber wedges containing air pockets, which increase the
compliance of the rubber, has been applied in [102] at 10 kc. The wedges had a length of
7.5 cm and were attached perpendicularly to the tank wall on a damping panel coated with
several layers of the same rubber.
In [103] the reflection of sound from the walls was reduced by means of spiked rubber
mats of the ordinary household variety (the spikes had a height of 9.5 mm and a spacing of
7.5 mm). Strips cut from the mat (about 10 cm in width) were packed in clusters so that the
spikes would sit vertically. The clusters, which had a height equal to the thickness of the
water layer, were attached to the tank walls and lowered the echo signal intensity from the
walls by about 25 to 30 dB in the frequency band from 500 to 1000 kc.
In [43] damping of the walls by means of blind absorbers in the form of vertical metal
panels with blinds stretched over them is described. The blinds comprise rubber strips with a
length equal to the height of the panel and with a width that is greater as the sound wavelength
is increased (for operation at an acoustic frequency from 250 to 1000 kc the width of the strip
was 6 cm, and the spacing between them was 4 or 5 cm). The most efficient type of rubber
for damping was decided experimentally.
It is desirable in the construction of the panel to provide a louver mechanism permitting
rotation of the blinds about a verticalaxis during the experiment, so as to set their plane at
various angles relative to the damped walls. Rubber is cemented into the spaces between the
blinds on the panel in order to increase the absorption.
In a blind absorber of simplified construction [15] the vertical rubber strips were attached to the panel at a constant angle of 45° relative to its plane, and the panel was mounted
in the tank so that the acute angles between the blinds and the tank walls faced the incident
sound. It was found experimentally that blind damping of the walls provides approximately a
25-dB reduction of the sound intensity in the direction of specular reflection in the frequency
band from 500 to 1000 kc at grazing angles of 10-15 ° relative to the side wall, and about a 40dB reduction at grazing angles from 15 to 40°. At lower frequencies (about 250 kc) the attenuation is lower, amounting to 20-25 dB for sound incident on the wall at grazing angles between
10 and 40°. Inasmuch as the spacing between the blinds remains much greater than the sound
wavelengths for sound at all the frequencies cited and can in this sense be regarded as equivalent, the indicated reduction in the attenuation is clearly attributable to the frequency dependence of the acoustical properties of the rubber.
Blind absorbers are used similarly to deaden the front and back walls and bottom of the
tank.
The action of blind absorbers is based on the absorption of sound in rubber, as well as on
scattering by the blinds themselves. \Vhile strongly attenuating the neId reflected from the
side wall in the specular direction, the blinds scatter sound in the back direction. As the author's observations have shown, blinds are poorly suited to the damping of a tank for the investigation of echo-ranging signals from bodies submerged in the water, because those signals
are masked by the neId scattered by tre blinds.
For acoustical measurements the tank is furnished with a coordinate positioning device,
which permits the transmitter and sound receiver to be placed at any desired point of the tank
§5]
MODELS OF THE BarTOM AND SURF ACE OF THE SEA
13
and their position to be measured with a given error. We now describe briefly one of the
positioning devices used by the author [15]. The base of the positioning device is the topside
of a platform situated over the tank (Fig. 1). A metal frame is mounted on the platform, its
longitudinal members along the tank consisting of Duralumin tubes 4 or 5 cm in diameter.
Along the tubes runs a carriage consisting of two longitudinal and two trans verse metal bars.
The carriage rests on four slide bars (two on each side of the tank), which move on ball bearings along the guide tubes of the frame. The displacement of the carriage along the tank is
read from a scale inscribed on the longitudinal members of the frame. The carriage functions
as a movable bridge with an attachment for translating and setting a vertical column in the
form of a tube, through which there is a rod with the acoustic transducer mounted on its lower
end. A worm gear is used to translate the bridge along and across the carriage. The transducer holder is moved in the vertical direction by hand or by means of a motor mounted on
the column. The upper part of the tube is fitted with a knob, which can be rotated to turn the
transducer about the vertical axis. The angle of rotation is determined from a graduated
circle.
Some holders are constructed so as to permit in addition rotation of the transducer upward and downward about a horizontal axis. This is necessary for the transmission of the
sound beam in different directions in the vertical plane. The receiver holder permits rotation
through 360 in the vertical plane for the purpose of ascertaining the direction of arrival of the
signal at the receiver.
0
The carriage can be used for displacement of both the receiver and the transmitter. If
it is not required to move the transmitter in three dimensions, it is better to be constrained
by a more rigid mode of attachment permitting only the depth of the transmitter or its position
of rotation in the vertical plane to be varied.
The displacement of the carriage with the acoustic receiver along the tank is effected
either manually or automatically. In the latter event the rate of motion of the carriage is
varied by means of areduction gear provided with a centrifugal regulator.
The direction of rotationof the motor is changed during motion of the carriage by a system of two relays and contacts, which are closed and opened by the carriage itself as it reaches
the ends of the tank.
§ 5.
Models of the Bottom and Surface of the Sea
Bot tom Mo d e 1 s • By covering the bottom of the tank with specially selected materials
it is possible to investigate the propagation of sound in water for various types of bottom.
The coefficient of sound reflection from the ocean bottom depends on the composition of
the bottom soU. In deciding upon a suitable material for the simulation of a particular bottom,
therefore, it is necessary to know beforehand the sound reflection coefficient of the various
materials.
A method for determining the modulus of the reflection coefficient is presented in [39].
Suppose that a transmitting source S and receiver Aare situated in water a certain distance
apart at the same depth (Fig. 2). If the surface of the water is far enough from the acoustic
transducers, the sound receiver will pick up only two rays, the direct ray and the one reflected
from the bottom. This will happen as long as the bottom is perfectly smooth and the sound
transmitter and receiver are point objects.
Let Pi be the pressure amplitude of the direct wave at the point of reception, Pr the pressure amplitude of the reflected wave at the same point, and l/! the phase difference at that point
between the direct and reflected waves.
MODEL EXPERIMENTAL PROCEDURE
14
S~A
Then the total pressure amplitude at the reception
point is equal to
Po = (p7
8
Fig. 2. Calculation of the coefficient of sound reflection
from the bottom.
[CH.2
+ P; + 2p/p, cos 0/)'''.
The attenuation of the direct and reflected rays
due to spherical divergence may be considered to be
identical, because the lengths of the rays SA and SB + BA
are scarcely different. Hence the reflection coefficient
is equal to V = Pr !Pi, and
Po = Pi (1
+ 2V cos ~ + V2)'/' .
(5.1)
In experiments the phase difference between the dire.ct and reflected waves is varied by
changing the depth of -immersion of one of the transducers. Equation (5.1) shows that the
resultant pressure Po runs through extremal values in this case. For the maximum (M) and
minimum (m) pressure amplitudes we have
M=Pi(l +V);
m=P/(l-V).
From this we find the following equation for the reflection coefficient:
(5.2)
This formula is in need of modification in the event of operation with a directional transmitter.
Let SO be the direction of the axis of the directivity characteristic of the transmitter
(Fig.2). Then the amplitude of the resultant pressure at point Ais equal to
Po = p' [B; «(1.1)
+ 2V B (11 1) B (112) COS 'f + V2B~ (112)] '(,
s
s
(5.3)
where p' is the pressure amplitude on the transmitter axis at the distance SA, Bs(a) $1 is
the directivity characteristic of the transmitter, and Bs(O) = 1.
On the basis of the foregoing we deduce the following equation for the reflection coeffi-
eient:
(5.4)
The method just described can also be used to determine the reflection coefficient for
normal wave incidence on the bottom.
In this case an alternate method is more practicable. Let us assume that a directional
sound source transmits asound beam vertically downward toward the bottom. In line with the
transmitter we set up a sound receiver and measure the amplitude of the sound pressure P2 in
the bottom-reflected ray. In order to preclude the divergence of the incident and reflected rays
and to account only for the variation of the press ure in reflection we measure the sound pressure amplitude Pt on the beam axis at a point far from the transmitter, at a distance equal to
twice the distance from the transmitter to the bottom. Then the reflection coefficient is equal
to
(5.5)
§5)
MODELS OF THE BarTOM AND SURFACE OF THE SEA
15
TABLE 1
Frequency. kc
460
2000
Cerarnic cell
on concrete
Dural 2.4 rnrn
thick
0.52-0.65
0.57-0.60
0.90-1.0
0.91-0.93
Plasticine. of thickness:
3 rn rn
0.40
0.49-0.55
I
6 mrn
-
0.5
Some values obtained in the laboratory of the Acoustics Department of Gorky state University for the modulus of the reflection coefficient at two frequencies for normal incidence on materials used to simulate the ocean bottom are presented in Table 1.
The numerical values shown in Table 1 suggest the possibility of modeling a bottom with
arefleetion coefficient that varies over relatively wide limits. In particular, metal is suitable
for the modeling of shale and rock bottom, while plasticine is better for silt-type bottom soils.
The reflection coefficient from different bottom models depends on the acoustic frequency
and grazing angle of the ray. For example, the reflection coefficient at 1 Me for a plasticine
bottom model has a minimum value of 0.35 for a grazing angle of about 20°, increasing to 0.8 at
an angle of 6° and to 0.5 at 40°. The reflection coefficient for the same model in the frequency
band from 250 to 500 kc varies from 0.8 to 0.4 as the grazing angle is varied from 10 to 40°.
The experimentally observed spread of the values for the reflection coefficient is attributable to the irregularities of the bottom. Thus, suppose that two rays arrive at the receiver
after being reflected from small areas located at some distance apart and slanted relative to
one another. In this case the resultant sound press ure amplitude in the reflected wave is
Pr = (pi
+ p~ + 2P1P2
COS 'f)'!2 ,
where Pi and P2 are the sound pressure amplitudes in rays 1 and 2, cp = (27T / lI.)6l, and 6l is
the difference in length between rays 1 and 2.
With variation of the relative slope of the small areas the reflected wave amplitude also
changes, inducing a spread in the values of the reflection coefficient. Actually not two, but
several rays arrive at the reception point, having been reflected from different portions of the
bottom. The interference of these rays further complicates the field pattern and increases the
spread of the measured values of the reflection coefficient.
We note in conclusion that it is a simple matter to model a sloping ocean bottom in the
tank by orienting the bottom model at a specific angle relative to the horizontal.
Mod e 1 i ng of a Rough Se a Bur fac e. The following techniques are used to
simulate the ocean surface: 1) corrugated surface models are built to simulate the rough
surface of the sea in "frozen" form; 2) a wave state is produced on the surface of the water,
representing an approximate scaled-down replica of the wave state of the ocean surface.
Each technique has the advantages and disadvantages.
The main shortcoming of the second technique is the impossibility of stimulating the
wave-generation conditions on the sea such as to produce the unstationary character of the
sea wave state. Moreover, the surface wave state causes a displacement of the surface layer
of water, thereby gradually changing the depth distribution of the velocity of sound in the medium. As an example, Fig. 3 shows the dynamics of the variation of the temperature distribution
with the depth of the tank under the action of traveling surface waves with a wavelength of about
10 cm and height (trough to crest) of 1 cm; curve 1 corresponds to the vertical initial distribu-
16
MODEL EXPERIMENTAL PROCEDURE
o
10
20
30
[CH.2
tion of the temperature, curves 2 and 3 to the temperature distribution recorded 10 min and 1 h, respectively, after the initiation of the wave process [15].
The observations of the time variation of the vertical
gradient of the temperature and, hence, of the velocity
of sound under the influence of surface waves shows
that the temperature distribution in the tank is sustained with 0.1 error only for a few minutes after the
initiation of the wave process on the surface of the water
in the tank. This limits the optimum duration of each
experiment.
0
40
Z, cm
Fig. 3. Dynamics of the variation
of the temperature distribution in
the tank under the action of traveling surface waves.
Fixed models of the uneven surface placed on the
water surface, as in the first modeling technique, are
devoid of the above shortcoming.
On the other hand, an advantage of the second modeling technique is that the typical boundary conditions on the surface in the propagation of sound in the sea are automatically fulfilled.
In the first modeling technique those conditions, on the contrary, are met only more or
less approximately. It has been shown experimentally that the boundary formed by water and
cork [35] or foam plastic* [15] comes closest in this properties to a water-air boundary on
which the sound pressure is equal to zero. However, due to the absorption of sound in these
materials the phase of the sound signal reflected from them differs somewhat from the phase of
the signal reflected from the interface with air. Therefore, the use of cork or foam plastic
models of the uneven free surface of the water is admissible only as long as one determines
just the modulus of the sound pressure without concern for the phase relations.
In [37] a model of an uneven surface is also described; it is made of thin brass foU (0.1
mm thick) in the form of a box, which is then filled with air. The wall of the model is transparent to the ultrasonic waves used for modeling, thus meeting the required conditions on the
water-air interface.
Foam plastic is easy to work with and one can cut it in the shape of an uneven surface
with a specified profile, the spatial variation of the unevenness being either periodic or following a random law. Figure 4 illustrates a modelof a waveform surface of foam plastic wi th a
trochoidal profile.
The simplest technique for creating the profile of a surface with randomly distributed
irregularities is to use an actual record of a random process corresponding to the required
distribution. One can, for example, model the surface on the basis of the ocean wave state
as recorded on a loop oscillograph. In this case the profile of the ocean wave state is plotted
on a reduced scale (with regard for the scaling factor) on the two opposite sides of the piece
of material from which the model is to be built. The model is then cut from the plotted profile. It will be two-dimensional, because the irregularities of the surface vary along the model,
but remain constant across it. The modeling of an uneven surface by means of records of the
ocean wave state is described in [1].
For modeling of a surface with irregularities fitting a normal distribution a pattern is
made of some statistical process that fits such a distribution. One can, for example, investigate
* Foam plastic represents a hard foam of plastic material containing cellular pores. Under
heat treatment a thin film is formed on the surface of foam plastic, closing off the interior
pores.
§5]
MODELS OF THE BOTTOM AND SURF ACE OF THE SEA
17
Fig. 4. Vertical section and general view of a waveform
surface with a trochoidal profile. A) Wave height; A)
spatial period.
~_-_o
Fig. 5. Kinematic diagram of a vibrator for
the excitation ofwaves on thesurface of water.
1) Slat; 2) motor; 3) crankshaft mechanism; 4)
water-filled tank.
experimentally the surface formed by a plane saw cut through a piece of foam plastic. In this
case the integral distribution of the surface irregularities of the foam plastic is very c10se to
the theoretical curve representing the integral Rayleigh distribution function calculated from
the formula for a normal distribution using the experimental values of the variance and height
of the irregularities [18].
A similar type of uneven surface can be made from cork [97]. Rectangular strips of sheet
cork of various thicknesses and length equal to 1he width of the model are cemented tightly together transversely to a thick sheet of plywood, which serves as the base of the model. The
spatial frequency of the strips of various thicknesses on the model must correspond to irregularities with a normal distribution. The profile of the model has a stepped configuration with
the minimum step height equal to the smallest thickness of the cork layer (about 1 mm).
In the second technique of modeling rough ocean surface, as mentioned above, waves are
excited on the water.
Nearly trochoidal waves are generated on the water by means of a vibrator, a kinematic
diagram of which is shown in Fig. 5 [15]. The waves are generated by a slat set in vibration
18
MODEL EXPERIMENTAL PROCEDURE
[CH.2
in a vertical direction by me ans of an electric motor and a crankshaft mechanism. The depth
of immersion of the slat can be regulated and controls the height of the waves generated; by
changing the rpm of the motor it is possible to obtain varying wavelengths. The author used
this technique to generate waves with wavelengths from a few centimeters to several tens of
centimeters and heights from 5 to 50 mm. The waves propagated along the tank, the shortest
ones decaying out before reaching the end of the tank and the others becoming reflected from
the far end wall to produce, in combination with the incident wave, a standing wave. When a
wave absorber in the form of an artificial shoal gently sloping relative to the direction of the
incident wave was placed at the end of the tank, a traveling wave was produced in the latter. *
The waveform can be recorded on a lOQP oscillograph in conjunction with a bridge network.
One of the arms of the bridge is a resistance half-submerged ;.n the water when the surface of
the latter is quiet. In this situation the bridge is balanced, but when the surface is wavy the
submersion of the resistance to different depths throws the bridge out of balance, and this effect is recorded on the oscillograph. The period of oscillation of the wave is found from the
oscillogram, and this can be used, ifthe wave propagation velocity is known, to determine the
surface wavelength.
The profile and length of the wave on the surface can also be measured by photographing
the wave through the transparent wall of the tank. Strips of graph paper are cemented to the
tank wall in order to determine the height and length of the waves.
It is impossible to excite sufficiently short periodic waves that will propagate on the
water by means of a wave vibrator. For example, sea waves with a wavelength of 5 m, given
a scaling factorof 103, are simulated by capillary waves with a length of 5 mm and height less
than 0.5 mm.
According to [36], the velocity v of sm all-amplitude capillary waves is related to the wavelength A. as follows:
,.~
v= V~'
where ö is the coefficient of surface tension of the water.
Waves of length A. are excited by a vibrator operating at a frequency
Capillary waves decay in propagation with an attenuation
2~(21tf) '{,
a=
(pö2)1/3
'
where 1] is the viscosity of the water.
From the above formulas we find that waves of Iength 5 mm can be excited with a vibrator operating at a frequency of 60 cps. An estimate shows that the amplitude of the wave decays
by l/e over a distance less than the wavelength itself. This means that waves of the given
Iength (or shorter) do not propagate on the surface of water and cannot be used to simulate the
waveform surface of the sea.t
* An artificial shoal can be fabricated from a plate of suitable material with a piece of coarse
woolen cloth cemented onto it.
tThe wave vibrator can excite capillary waves of fairly large amplitude, which decay more
slowly and are well-suited to model experiments.
§6]
PRINCIPAL ELECTROACOUSTIC EQUIPMENT
19
On the other hand, an irregular wave state is fairly easy to simulate by blowing a stream
of air from a fan or blowers over the surface of the water in the tank.
In [76] a modeling arrangement is described as used to investigate the distortion of a
sound signal after a single reflection from the wavy surface of the water. Four airstreams 50
cm in diameter lasting 15 min were used to create a nearly steady wave state on the surface
of the water in a tank with an area of 3.6 x 1.2 m 2• An insulated wire ahout 1.5 mm in diameter
placed near the surface of the water served as an indicator of the wave height on the water.
The wave height variation was determined in a bridge network from the variation of the capacitance between the wire and the water.
I ce Co ver Mo dei s. An integral part of the problem of modeling the surface of the
ocean is a consideration of the possibility of modeling ice covers under lahoratory conditions.
The successful solution of this problem would be a vital factor in the study of under-ice sound
propagation.
The ice formed on the surface of saline sea water has a complex structure and highly
characteristic irregularities on its undersurface.
It is impossible at the present time to come up with a single material that will even
approximately simulate the typical characteristic of sea ice on a reduced scale. For this
reason ice must be simulated in the laboratory by real ice.
Preliminary experiments conducted by the author in collaboration with V. V. Lavrov and
V. I. Efimov in the refrigeration tank of the Arctic and Antarctic Institute* have indicated an
encouraging outlook for the modeling method as a means of studying under-ice sound propagation in certain Arctic regions.
Thus, in 1 to 1.5 hat an air temperature of -7°C it is possible to form an ice layer 2.5
to 3 mm thick with an undersurface structure typical of annual ice. With a scaling factor M =
103 this ice cover is equivalent in nature to ice 2.5-3 mm thick of the type observed in the
polar seas in the northern and southern hemispheres [25].
During the experiment the thickness of the ice increases rather quickly. Ice with a
thickness of 3.5 to 4 mm has a more stahle behavior.
§ 6.
Principal Electroacoustic Equipment
Electroacoustic Transducers. Model experiments require a set oftransmitters having different natural frequencies. The usual band of ultrasonic frequencies used
for modeling extends from 100 kc to 3 Mc. The band is limited ahove by the sharpincrease
in the absorption of ultrasound in water and aqueous solutions in the megacycle range, and it
is limited below by the physical size of the experimental tank, the difficul ties of deadening the
latter, and the complications involved in the formation of a highly directional sound beam in
the tank at low ultrasonic frequencies.
For the generation of sound beams with varying angular widths, from very narrow (less
than ±5 ° subtended by the principal lobe of the directivity pattern) to such widths as simulate
a point sound source, it is necessary to have transmitters of different sizes and shapes. Disk
(piston-type) transmitters, which have a symmetric directivity pattern ahout their axis, are
the type most often used in model experiments. In cases where it is required to transmit a
wider beam in the vertical than in the horizontal plane transmitters in the shape of a rectangular strip or section of a hollow cylinder cut parallel to its own axis are used [15].
* The author is indebted to V. V. Bogorodskii for affording hirn the opportunity of conducting
this study.
MODEL EXPERIMENT AL PROCEDURE
20
8
7
3
I
I
I
I
:
L __
rli
2
n
L----4!
Fig. 6. Diagram of the
mounting of a piezoelectric plate. 1) Piezoelectric element; 2)
holder casing; 3) stub
or plastic connecting
tube; 4) neutral wire; 5)
hot wire; 6) mounting
stub; 7) brass tube; 8)
coax from electrical
oscillator.
3
[CH.2
The materials used for the fabrication of electroacoustic
transducers are quartz and polarized ceramics (barium titanate,
lead zirconate-titanate, e tc.). The transducers a:re embedded
in a metalcasing or a plexiglas mounting. In the latter event
the holder with the piezoelectric element mounted inside are
coated with a thin layer of plastic dissolved in dichloroethane.
Once the cement has been dried out the transmitter has a good
water seal. For a better seal it can also be coated with a layer
of epoxy resin. The attachment of a piezoelectric plate in the
holder is illustrated in Fig. 6.
It is difficult by me ans of disk transmitters to produce
a wave that will adequately approximate a plane or spherical
wave. The need for such an approximation arises in experiments set up to test theories developed from the postulated existence of a plane or spherical wave in the medium (as, for example, in the study of sound scattering by obstac1es).
This requirement is fulfilled by a spherical transmitter.
It consists of a hollow sphere made of piezoelectric ceramic
(Fig. 7), the silvered surfacesof which (inside and outside)
function as electrodes. A wire is soldered to the inner surface
of the sphere and runs through a narrow brass tube to the
amplifier. The other conductor is the tube casing, which is connected to the outer surface of the s phe re . The ceramic sphere
and its junction with the conducting tube are coated with layers
of neoprene (synthetic chloroprene rubber) or some other hermetic material in order to seal them against water. If the hollow
sphere is made with sufficient precision the transmitter generates a spherical wave and at large distances an approximately
plane wave.
A spherical transmitter of the type indicated [931 generates
apressure with an amplitude of 2.1 • 104 J.!bar (at 230 kc) at a distance of 1 m from the transmitter. A similar field level obviously could not be attached by any other type of small spherical
transmitter.
Another kind of point transmitter is a short piece of metal
wire (say, about 10 cm in length) with a conical tip, which is
attached to the center of the transmitting surface of a piezoelectric disk, whose oscillations are sent through the end of the wire
Fig. 7. Diagram of a
into the water (Fig. 8) [1031. In order for sound to be transspherical transmitter.
mitted exclusively by the end of the wire its lateral surface is
1) Hollow piezoelectric
coated with an acoustically compliant ,material (special grades
sphere; 2) hot wire; 3)
of glue' containing microscopic air bubbles, latex doped with
brass tube; 4) water
manganese dioxide and hydrogen peroxide, plastic tubes with
an air layer), which acts simultaneously to seal the wire against
seal.
water. The metal wire comes into contact with the foH, which
is grounded to the transmitter casing. For the elimination of vibration of the holder the piezoelectric disk and foH are clamped in rings of sponge rubber. The electrical voltage is delivered
to the piezoelectric element along a cable through a coaxial joint. This type of transducer can
also operate as a point sound receiver.
PRINCIPAL ELECTROACOUSTIC EQUIPMENT
§6)
4
21
The transducers used for sound receivers must perturb the
sound field as little as possible, particularly when the receiver is
situated between the transmitter and a reflecting body. This requirement is met by an acoustic element whose dimensions are
small relative to the wavelength. Such an element has an almost
uniform directivity pattern at frequencies elose to its natural frequency. However, a uniform directivity pattern is not always
desirable in the receiver. For example, in the measurement of the
secondary field of a reflector it is advisable to diminish the sensitivity of the receiver in the back direction so as to eliminate the
reception of the direct signal from the transmitter or of signals
reflected from the walls.
The reception of sound can be realized with receiving units
of various configurations, ineluding disk transducers, radially
polarized cylindrical probes, miniature probes (described in [55]),
etc.
Fig. 8. Diagram of a
point-source transmitter. 1) Metal
wire; 2) piezoelectric
disk; 3) transmitting
tip of wire; 4) acoustic and water insulation; 5) foU; 6) rubber
mousse; 7) coaxial
terminal.
The mounting of a cylindrical probe is illustrated in Fig. 9.
The probe is coated for water insulation with a thin layer of plastic
and in order to enhance the sensitivity of the receiver it is necessary to leave an air cavity inside the cylinder. The diameter of the
tube to which the sensitive element is fastened must be made as small
as possible in order to prevent distortion of the sound field. The
holder tube is flexible, so as to permit measurement of the sound
field beneath objects floating on the water or immersed in it.
Following are the dimensions of one of the cylindrical receivers used by the author, as a typical example: outside diameter
and length of the cylinder, 3 mm; wall thickness, about 0.5 mm;
diameter of the brass holder tube, 1.5 mm; the length of the latter tube is dictated by the depth
of the tank.
A disk-shaped receiver with a natural frequency considerably higher than the frequency
of the received sound has a uniform frequency characteristic over a sufficiently wide frequency
band. It averages the sound field over its surface only in one plane, thus permitting measurements of the sound pressure with greater accuracy than receivers having other shapes. If the
receiver has a sharp directivity pattern, it can be used to determine the direction of arrival
of rays at the point of observation.
Eie c tri ca I Ci r c u i t s • The electrical part of the model eqaipment is capable of
generating high-frequency pulses of nearly rectangular shape and various widths (for instance,
from 5 to 2000 fJ.sec), depending on their carrier frequency, and of realizing the continuous radiation of sound. The advantage of pulsed operation is the feasibility of conducting measurements in an undamped (nonanechoic) or partially damped tank. However, this complicates the
receiving-transmitting apparatus, and instead of acoustic tone signals waves in a certain frequency band are propagated in the water. The main sections of the electrical circuit are preferably assembled from standard radio equipment, with only some of the units being additionally
constructed.
The transmitted sound pulse and received signal are viewed on the display screen of a
cathode-ray oscilloscope, where any portion of the signal can be discriminated at will by means
of a gate pulse. It is important to incorporate the capability of receiving the sound signal at
isolated points of the tank and of automatically recording the sound pressure distribution at
different levels.
MODEL EXPERIMENTAL PROCEDURE
22
[eH. 2
A pulse signal arriving at the input of the sound
receiver has a frequency band f 0 ± Af, where f 0 is
the pulse carrier frequency. If the pulsewidth is T,
the bandwidth is about 2 I T. This implies that sufficiently
wide pulses will simulate almost continuous radiation
with an equivalent frequency f 0 (for example, if f 0 =
500 kc and T = 100 Jl sec, then 2Af I f 0 = 4%). In this
case it is permissible to compare the measured values
of the sound pressure with the theoretical values for the
stationary frequency f o.
Fig. 9. Diagram of the mounting
of a cylindrical probe. 1) Plastic
layer; 2) holder tube; 3) piezoelectric element.
The electrical equipment used in different model
investigations tends to vary somewhat, especially in the
receiving section. The methods used to indicate the
sound field and process the measurement results also
differ, depending on the problem to be solved. We shall
delve further into this later in the discussion of individual problems.
Here we consider one version of the electroacoustical section of a model arrangement [3, 6] (Block
diagram 1). The trigger-pulse generator 1 produces
CJ-0-G [J-[J-[J-[J-[J
I
I
8
Block diagram 1.
video pulses of rectangular shape with an amplitude varying between 0 and 50 V. The frequency
of the pulse signals is chosen so that by the time the next sound pulse is transmitted the field of
the preceding will have died out throughout the entire volume of the tank and so that the brightness of the electrical pulse will be adequate for viewing on the oscilloscope screen. For example, the transmission time of a sound signal over the maximum path in a tank of length 1 =
7.5 m is equal to t = 211 c = 10 msec. This means that the repetition frequency of the signals
must not begreater than j = (1/t)=100 cps. Inparticular, the signal repetition frequency
should be made equal to 50 pulseslsecond with synchronization from an external circuit. These
pulse are used to modulate the output of the high-frequency oscillator 2. As a result pulses
with a specified duration, carrier frequency, and repetition rate are generated. These pulses
are sent to a resonance power amplifier. The amplified pulses are tapped from the circuit
by a transformer coupling and continue on to the ultrasonic transmitter 3, where they are
transmitted into acoustic pulses.
After transmission through the medium 8 the sound pulses are picked up by the sound
receiver 4, where they are converted into electrical signals and sent to the input of the resonance amplifier 5. The pass band of the amplifier is chosen to meet the condition of obtaining
undistorted rectangular pulses. If the pulsewidth is T, the amplifier pass band must be at
least 2 IT. It is also important in this case to filter out extraneous signals induced in the amplifier input.
§6]
PRINCIPAL ELECTROACOUSTIC EQUIPMENT
23
The main sources of electrical noise are radio stations, leakage from the line (50 cps),
and ambient industrial discharges. For the elimination of radio station interference it is advisable to generate ultrasound at a frequency that does not coincide with the broadcast frequency. An important factor in the reduction of electrical interference is reliable grounding
of the metallic parts of the tank, POsitioning device, and other metal structures.
From the amplifier the signal is transmitted to too cathode-ray oscilloscope 6. In order
to view the shape of the pulse on the display screen the oscilloscope sweep must be triggered
later than the instant of pulse transmission to the transmitter with a delay equal to the pulse
travers al time through the medium. This is accomplished by delivering a delayed-sync pulse
to the oscilloscope. The delay circuit 7 is actuated by the trigger-pulse generator 9.
Rather than investigate the sound field at individual points of the medium, one can employ the method of automatie sound recording. The method is based on the use of a selectorpulse generator for the discrimination of the direct sound pulse as it arrives at asound pressure pickup (gating system).
A block diagram of the receiving seetion of the system for automatie recording of the
sound pressure is shown in Block diagram 2.
13--0
I
8-CJ-G-G-G-G-G
I
I
o
G
Block diagram 2.
The sound signal picked up by the receiver 4 and amplified in the resonance amplifier
5 is transmitted to the selective amplifier 10. Arriving at the input of the latter are the timeseparated voltages from the direct sound pulse and from the pulse obtained by sound reflection from the bottom and walls of the tank. Also, the high-frequency generator induces an electrical pulse.
The selective amplifier is designed to separate the required pulse from the mixed signal
described above and then to amplify it. The amplifier basically comprises a single-stage resonance amplification circuit. The input tube of the amplifier is set to operate in a mode such
as to keep the amplifier tube shorted. A negative rectangular pulse is transmitted from the
selector-pulse generator 11 (Block diagram 2) to the grid of the first tube, where it blocks the
tube. As a result the second lamp is thrown into the normal amplifying mode and transmits
the signal delivered to its grid. By proper selection of the time of transmission of the selector
pulse to the amplifier it is possible to segregate any pulse from the sound signal, including the
primary sound pulse.
The discriminated and amplified signal is sent to a monitor cathode-ray oscilloscope or
to a recording circuit. The horizontal sweep of the oscilloscope is initially disengaged, leaving
a bright vertical line on the screen, where the width of the line is proportional to the amplitude
of the discriminated pulse.
The selector-pulse generator 11 consists of two trigger circuits. A schematic of the
generator is shown in Fig. 10. The tube T 1 operates in the amplifier-limiter mode on the grid
MODEL EXPERIMENTAL PROCEDURE
24
6S5S
6N7
6N8
[CH.2
6Zh8
+300V.-~----~------~--------~----~----~
20x
0.5
0.25
1
Ql
[0
0; I-C=:J,---,
AO
6900
0-;
4.7K
2500
20K
200K
Fig. 10. Schematic of the selector-pulse generator. AO) Audio oscillator.
and plate current. The sinusoidal voltage from the audio oscillator (AO) is delivered to the tube
input, where it synchronizes the operator of the selector-pulse generator and ultrasonic pulse
generator. Rectangular pulses are taken from the plate of the tube and transmitted through a
differentiating circuit to the first trigger circuit, which is built on T 2• The pulses taken from
the cathode of the first trigger circuit are also differentiated, and the positive peaks obtained
from the back edge of these pulses are used to actuate the second trigger circuit (on tube Ta),
which develops negative selector pulses. The pulsewidth of the first trigger circuit is varied
by means of the resistances R 1 and R 2• i.e., the firing time of the second trigger c ircuit relative to the first is varied.
For the separation of only the direct sound pulse from the mixed set of voltages the
selective amplifier must be opened in time intervals equal to the propagation time of the pulse
from the transmitter to the receiver and closed again as soon as the pulse has been transmitted.
Consequently, the electroacoustical seetion does not trans mit any other pulses. * The
propagation time of the pulse from the transmitter to the receiver is a linear function of the
separation of the two transducers. Therefore, the delay time of the opening of the amplifier
relative to the pulse-generation time of the generator must be varied linearly. This is done
automatically as the contact of the carriage holding the receiver slides along a linear resistance
(potentiometer) R1 mounted on an insulated slat oriented along the tank. Grounded contacts are
placed at definite intervals along the potentiometer. where they come into contact with the
sliding contact of the carriage. The resistance R 2 is used for correction. The second trigger
circuit is capable of developing either wide or narrow selector pulses (for example, from 200
to 5000 p.sec). A wide selector pulse is used for display on the oscilloscope of the total signal
picked up by the receiver; a narrow selector pulse is used to discriminate only one ultrasonic
pulse. A cathode follower (tube T~ is used for power amplification of the selector pulses.
The pulse signal selected by the selective amplifier is delivered to the peak detector 12
(Block diagram 2) with a relatively large time constant, say, 0.05 sec. If the motion of the re-
* The gating system can be tuned in principle for the transmission of any des ired sound pulses,
for example those reflected from the bottom.
§ 6]
PRINCIPAL ELECTROACOUSTIC EQUIPMENT
25
6.2x
24.6M
750
(a)
IH
(b)
Fig. 11. Schematic of the detector circuit (a) and balanced circuit (b).
ceiver along the tank is slow enough (say, 5 to 10 ern/sec) the voltage on the detector load has
time to follow the amplitude variation of the received high-frequency pulse. The positive voltage
taken from the detector load can also be transmitted to the cathode follower 13. The load of the
latter develops a voltage proportional to the amplitude of the selected high-frequency pulse.
This voltage is sent to a balanced circuit, which is connected to a dial milliammeter.
The circuit is balanced so that without a signal at its input the milliammeter shows no readings.
With the arrival of pulses at the detector input the instrument indicates a current whose amplitude is proportional, within the limits of the linear characteristic of the instrument, to the amplitude variation of the pulse transmitted to the input of the peak detector.
In place of the dial instrument one can connect the loop oscillograph 14. Then for a constant speed on the part of the carriage with the receiver along the tank a graph is produced on the
film showing the sound press ure in the tank as a function of the distance from the transmitter.
The detector circuit (a) and balanced circuit (b) used to record the sound pressure on a
loop oscillograph (as weH as on a logarithmic level recorder) are shown in Fig. 11.
Following are some of the shortcomings of the given circuit: 1) the narrow dynamic
recording range of the oscillograph; 2) nonlinearity of the detector for small signals; 3) the
impossibility of recording rapidly varying processes (greater than 10 cps) due to the relatively
large time constant of the detector.
A logarithmic level recorder is used to obtain arecord of the sound field in decibels.
The measured voltage must he converted into an alternating voltage with a frequency within the
limits of the operating range of the recorder and an amplitude that varies as the signal amplitude. This type of transformation is realized by the modulator* 15 with a master oscillator 16
in the form of a symmetrie multivibrator operating at a carrier frequency of 10 kc. In order
to eliminate the transformation nonlinearity inherent in the circuit for small signals a minimallimiter 17 is introduced. Then the output voltage is delivered to the input of the recorder
18 and is recorded on log scale.
In situations where the sound field in the tank is characterized by a choppy spatial pattern
it is impossible to record the field on a recorder invested with a large time constant. This ob-
* Present-day recorders (such as the N-1l0 tape unit) have a buHt-in modulator.
26
MODEL EXPERIMENT AL PROCEDURE
[CH.2
100V
5
From amplifier
output
4:=====:JF ro m dela y -'-'---'-----'-_~
circuit
L...i
Fig. 12. Semiautomatic sound pressure indicating
circuit. 1) Cathode-ray oscilloscope; 2) sighting
arm; 3) knob; 4) linear potentiometer; 5) loop oscillograph; 6) sliding contact; 7) stationary contacts; 8) battery.
jective can be met by the use of an apparatus that allows rapidly varying processes to be recorded by means of a camera attachment on an oscilloscope.
A typical recording device is illustrated in Block diagram 3, which incorporates the
following nomenclature: 1) sound pressure receiver; 2) amplifier; 3) selector-pulse generator; 4) selective amplifier; 5) audio oscillator (for synchronization); 6) monitor oscilloscope;
7) recording oscilloscope; 8) camera attachment.
====8-0-9-0- 0- 0
0-0
Block diagram 3.
The objective of the camera attachment projects the image from the oscilloscope screen
into film, which is moved by an electric motor. The rate of motion of the film must be greater,
the faster the variation of the sound pressure from one point to the next in the tank. For example, the sound pressure in an illuminated zone, where darkening of the illuminance maxima
and minima is observed, has been recorded by the author at a film speed of 30 cm/min. In a
region of monotonic variation of the intensity the recording speed can be reduced. During recording the horizontal sweep of the oscilloscope is turned off, and the pulse transmitted by the
amplifier is sent to the vertical plates. At the time of arrival of the pulse the illumination pulse
is also sent to the oscilloscope. As a result a bright vertical line is recorded on the film, its
width being proportional to the amplitude of the received signal •. If necessary, the amplification
can be readjusted during recording; this makes it possible to record signals that vary sharply
in amplitude.
In order to shorten the process of recording the sound field as a function of the horizontal
distance from the transmitter a system has also been developed for the semiautomatic indication of the sound field; the basic principle of the system involves the transfer of the pulse amplitude from the oscilloscope screen to the film of a loop oscillograph (Fig. 12).
§7]
SCANNING OF THE SOUND FIELD
27
With uniform displacement of the carriage and receiver along the tank the pulse amplitude varies on the oscilloscope screen 1. By tracing the variation of the investigated pulse it
is possible to align the sighting arm 2 manually with the vertex of the pulse by means of the
knob 3, which is mounted on the axis of the linear potentiometer 4 (if the pulse has a large
width, its middle portion can be traced). The magnitude of the constant voltage taken from
the· contacts a and b of the potentiometer in this case is proportional to the pulse amplitude on
the oscilloscope screen. This voltage is transmitted to the input of one loop of the oscillograph
5 and is fixed on themoving film. At the same time, distance marks are recorded on the film
by means of the contact 6 mounted on the moving carriage and the equally spaced contacts 7
mounted on one of the guide tubes along which the carriage moves. During motion of the carriage the contact 6 comes successively into contact with a contact 7, whereupon the voltage
from the battery 8 is delivered to the other loop input, and a mark is produced on the film,
corresponding to the position of the given contact 7.
The resistance R 1 prevents the low-resistance input of the oscillograph from shunting
the variable potentiometer 4. The resistance R 2 functions as a current limiter.
Concurrently with the recording of the distribution in the water of the sound pressure
amplitude it is possible to record the pressure phase as weIl. An arrangement for recording
the amplitude and phase of the signal on the screen of a dual-beam oscilloscope is shown in
Block diagram 4 [103], in which the nomenclature is interpreted as follows: 1) receiving transducer; 2) preamplifier; 3) attenuator; 4, 5) amplifiers; 6) pulse generator; 7) sweep generator;
8) synchronizing signal from transmitter; 9) dual-beam oscilloscope.
The phase difference between the transmitted and received signals is determined from
time marks on the screen. For recording of the phase the vertical sweep is synchronized
with the transmitted signal.
8-0-~-[J-rn
I
~-~
--[J
9
I
~
Block diagram 4.
§ 7.
Scanning of the Sound Field
The sound field records made in adefinite direction along or across the tank or depthwise in it do not yet portray the overall pattern of the spatial distribution of the sound pressure in the water. Visualization of the field is therefore desirable. One method of spatial
"imagery" of the sound field in water during model experiments is scanning. This makes it
possible to obtain the over-all field pattern in a vertical plane along or across the tank on a
single record.
The scanning mechanism is mounted on a supporting platform near the edge of the tank
or on a carriage moving on rails along the tank, depending on whether the transmitter or receiver is the scanning transducer. The main part of the mechanism is a metal plate, which
moves up or down on smooth guide slots along the vertical shaft of the carriage together with
the transducer mounted on it (in Wood's experiments [103] the displacement period was approximately one third of a second).
28
MODEL EXPERIMENTAL PROCEDURE
[CH.2
The mechanism is set in motion by a rotating arm, the length of which is chosen so that
twice its amplitude in the vertical plane will be equal to the depth of the water. The motor used
to drive the plate also controls the motion of the brush contact of a potentiometer fed by a
battery. The voltage taken from the brush is amplified and transmitted to the vertical plates
of a single-beam oscilloscope, whereupon the light spot on the oscilloscope screen moves up
or down in sync and in phase with the motion of the transducer in the water. The brightness
of the spot is modulated preliminarily by the amplified signal from the receiver output.
Thus, a brightness variation of the light spot as it moves up and down on the oscilloscope
screen is caused by the sound intensity variation du ring the motion of the scanning transducer
between the surface of the water and bottom of the tank. If the carriage with the scanning mechanism moves along the tank, and if film is run slowly and evenly in front of the screen in the
direction perpendicular to the line formed by the light spots, a continuous image of the sound
pressure distribution in a vertical cross section along the tank is obtained on the film, dark
spots corresponding to regions of reduced intensity. The photographic re cord of the field can
be calibrated in intensity by comparing it with the brightness pattern obtained during modulation of the light spot with a stepped attenuator at a known decibel level.
The procedure described above has been used in [103] to obtain sound field patterns in a
homogeneous medium simulating a shallow sea and to analyze their dependence on the directivity of the transducers, the separation of the transmitter and receiver, the depth of the water
layer, the sound wavelength, the acoustical properties of the bottom, and the temperature of the
water in the tank. It was confirmed in the experiments that a steel, concrete, or glass bottom
may be regarded as equivalent to a shale bottom and weIl reflect incident sound in the frequency
range from 500 to 1000 kc; a rubber coating on these materials endows them with reflecting
properties similar to those of a silty sandy bottom. Moreover, in [103] observations were
nade on the modulation of the received signal amplitude by waves on the surface of the water
(the wavelengths fell in the interval from 5 to 25 cm) and demonstrated qualitatively that a vertical temperature gradient in the water affects the sound pressure distribution in it.
It is possible with a model arrangement similar to the one described above to observe
other phenomena that occur during the propagation of sound in a shallow sea. In this case
modeling is particularly useful in that the propagation of sound in shallow seas depends on a
number of sometimes elusive factors, the most important of which are the state of the sea
surface, the acoustical properties of the bottom soil, the shape and curvature of the bottom
relief, the ratio of the sea depth to the sound wavelength, and the volume inhomogeneities of
the medium.
The complete theoretical solution of the propagation in a shallow sea presents considerable difficulties, so that the existing acoustical theories, which necessarily treat the sound field
in the water layer under idealized conditions can only account for the principal effects that take
place in a shallow sea (see, e.g., [29, 49]). An experimental procedure based on modeli ng of
the shallow sea, on the other hand, permits a detailed investigation of the spatial pattern of the
sound field under prescribed conditions, which are easily controlled during the model experiment.
The modeling method enables one to investigate the propagation of sound in a shallow sea
under complex, yet realistic conditions, which are difficult to take into account in theory.
One such problem is the investigation of the structure of the field in water (formation of
various normal modes) as a function of the sound absorption in the bottom soil and the formation therein of shear waves from the inclination and irregularities of the bottom, etc. [77].
§S]
INSTRUMENTS FOR MEASURING SOUND VELOCITY
-
3
,....----l2
§ 8.
29
Instruments for Measuring
the Velocity of Sound
in the Model Medium
In order to determine the distribution of the velocity
of sound in a liquid it is required to measure that quantity
at several points of the medium.
Fig. 13. Block diagram of the
sing-around velocimeter. 1)
Pulse generator; 2, 4) transducers; 3) medium; 5) amplifier,;
6) frequency meter.
For measurements in nature a so-called sing-around
velocimeter is used [24]. A block diagram of the veloCimeter is shown in Fig. 13. A signal from the shaping
stage I, which is of the trigger type and gene rates short
pulses with steep fronts, is delivered to the transmitter 2.
The pressure pulses are sent through the medium 3 in a
period of time equal to llc, where l is the constant baseline and c is the velocity of sound in the medium.
The position of the transducers is precisely checked, and their holders are firmly attached
to the base, which is made of a material that has almost no thermal expansion. The pulse received by the transducer 4 is converted into an electrical pulse, which is amplified and synchronizes the pulse-shaping circuit.
The pulse repetition frequency 1, which is measured and recorded by the frequency meter
6, is found from the expression for the total delay time:
~=t
+~
7
e
c"
(S.I)
where t e is the sum of the electrical delays and time losses due to noise.
The constants t e and l are found during calibration of the instrument by measurement of
the quantity j in a liquid in which the velocity of sound is known. For this it is convenient to
use distilled water, performing the measurements at several temperatures and, acco rdingly ,
different values of c.
According to (S.I) the velocity of sound in the investigated liquid is calculated from the
equation
l
c=-'-l--
-=-- t e
I
For operation under laboratory conditions there is a miniaturized version of the velocimeter. In lieu of an instrument for direct measurement of the velocity of sound the latter
can be determined indirectly by the measurement of other variables on which the velocity of
sound depends.
If a vertical gradient of the velocity of sound is created by the presence of a vertical
temperature gradient in fresh water, the problem reduces to a measurement of the water temperature* at various depths and, from that, a calculation of the velocity of sound.
* For measurements of the water temperature three types of temperature gauge can be used
with varying degrees of accuracy: a laboratory mercury thermometer (from which the readings are taken through the transparent wall of the tank), a thermocouple, or athermistor.
MODEL EXPERIMENTAL PROCEDURE
30
[CH.2
c, rn/sec
1750
1100
1650
1600
1.370n
Fig. 14. Relation between the velocity of
sound and optical refractive index in an
aqueous solution of sodium chloride at a
temperature of 20°C.
c, rn/ser;;
1600
1500
11;00
1300
1200
Fig. 15. Relation between the velocity of sound and optical refractive index in an aqueous solution of ethyl
alcohol at several temperatures.
This can be realized, for example, with the aid of a simple empirical formula [58] for the
velocity of sound in fresh water:
c = 1410
+ 4.21 t - 0.037t
2,
(8.2)
which in the temperature range from 15 to 25°C gives values for the velocity of sound that concur with the experimental correct to within 1 rn/sec [74].
In situations where a vertical gradient of the velocity of sound in the medium is created
by aqueous solutions of certain substances the distribution of the velocity of sound in the liquid
is determined by measuring the concentration of the solution at various points of the medium
and calculating the velocity of sound from it.
§8]
INSTRUMENTS FOR MEASURING SOUND VELOCITY
31
Next we discuss an optical method for determining the concentration of the solution. By
means of a pipette graduated with scale divisions or horizontally placed clamped capillaries
mounted in the tank wall sampies of the liquid are taken at various levels. The quantity of liquid
taken for the sampie must be small (2 or 3 drops) so as not to have to take depth averages.
The optical refractive index in the liquid (n) at a given temperature is determined on an optical
refractometer, in which a constant temperature is maintained by athermostat. The solution
concentration and ultimately the velocity of sound at the given level are then found from the appropriate tables or graphs.
Graphs of the velocity of sound versus the refractive index of light in aqueous solutions of
sodium chloride and ethyl alcohol are shown in Figs. 14 and 15, respectively.
The salinity at various levels can be measured directly by me ans of electric salinometers
[47], the operation of which is based on a bridge circuit measurement of the electrical resistance
of the solution, which depends on the concentration. This method is sufficiently accurate for
solutions with a salt concentration of no more than 10-15%, but is almost totally unsuitable for
near-saturation solutions. Also, it is sometimes necessary in the equipment for the modeling
of layered-inhomogeneous media to measure the salinity of the solution from zero to saturation.
This limits the possibility of using electrical salinometers in model experiments.
A highly visual pattern of the concentration distribution of a salt solution can be obtained
(in a glass-walled tank) by the use of minute spherical aerometers of various weights, calibrated for a definite solution concentration. The small spheres are blown from glass, and to
each is welded a small piece of glass to serve as ballast. For calibration in a salt solution of
known concentration the glass is carefully ground off. Immersed in a tank with an inhomogeneous salt solution, the spheres are situated in layers of corresponding concentrations.
CHAPTER 3
MODELING OF SOUND PROPAGATION
IN INHOMOGENEOUS MEDIA
The present chapter is devoted to the modeling of some of the physica1 effects observed
in connection with the propagation of sound in the sea, which is regarded as a medium containing various types of inhomogeneities, inc1uding regu1arly and random1y distributed vo1ume inhomogeneities, irregu1arities of the free surface of the sea (waves), interna1 waves, and scattering bodies.
§ 9.
Methods for the Modeling of Layered-
Inhomogeneous Media
Layered-inhomogeneous media characterized by a variety of 1aws governing the variation by depth of the velocity of sound can be produced in the tank by a number of methods [15].
T he rm a1 Ac t ion Me th 0 d. This method is used for the modeling of a medium invested with a negative vertical gradient of the temperature and, hence, of the velocity of sound,
because in this type of medium the warmer liquid 1ayers, being lighter, take a stab1e position
above the heavier cooler 1ayers.
Two elements, a heater and a cooler, are used to simu1ate the temperature gradient in
water.
A cooler in the form of an array of thin tubes or a flat box is p1aced on the bottom of the
tank. The ends of the tubes are solde red into supp1y and drainage cans, which are connected
in turn to the tap and a sink, respective1y. Cold water (at about 4°C) is circu1ated through the
cooler for coupling of the water in the tank.
The heater, which is designed for uniform heating of the water surface, is suspended a
few centimeters above the surface of the water, but be10w the coordinate positioning device.
It consists of a sheet of metal (a1uminum or brass) with its edges bent downward. On the 10wer
side of the sheet, facing the water, several e1ectric heating coils are mounted on roller insu1ators. The heating of the e1ectric furnace is regu1ated with a rheostat. Aspace is 1eft
between the side walls of the tank and the edges of the heater, wide enough for the insertion
of thermometers, liquid samp1ers, etc. into the tank. In order to allow movement of the vertical sound receiver holder a narrow groove is made in the midd1e of the heater a10ng the
entire 1ength of the tank, and during experiments the groove is c1osed.
The required mode of heating of the coils (i.e., the princip1e of time variation of the electrical vo1tage supplied to the coils) is determined experimentally. For example, with simu133
34
MODELING OF SOUND PROPAGATION IN INHOMOGENEOUS MEDIA
[CH.3
taneous heating of the surface of the water by an electric furnace at an average power of 2
kW1m 2 and cooling of the bottom with running water in a layer of water 50 cm deep for 20 to
40 hof continuous operation the author was ahle to build up astahle, depth-constant negative
temperature gradient on the order of 0.1 to 1 deg/cm. Measurements of the temperature of the
water in horizontal planes at various depths showed that the resulting horizontal temperature
gradient turned out to be two or three orders of magnitude smaller than the vertical gradient and
could therefore be disregarded.
Under oceanic conditions, due to the movement of the water, a quasi-homogeneous surface
layer with a constant mean temperature is formed. For the modeling of such a layer in the experimental tank an abrupt drop in the air temperature in the laboratory is set up over the medium with the negative temperature, in combination with the blowing of air over the water surface
by means of fans. This technique can be used to generate a surface isothermal layer from 2
to 10 cm thick. Beneath this layer there remains a negative temperature gradient that differs
only slightly from the original one.
Diffus ion Method. The diffusion method for the simulation of media characterized
by a velocity of sound smoothly varying by depth is based on the dependence of the latter on the
concentration of aqueous solutions of certain substances.
Various interdiffusing liquids can be used to model the medium according to the type of
sound velocity distribution specified. The author's experiments have shown that the best agents
for this purpose in terms of their physical properties are water and aqueous solutions of sodium chloride (tahle salt) and ethyl alcohol. A possible substitute for the latter is methyl alcohol,
provided suitable precautions are taken to guard against its injurious effects in handling.
Some of the applications of the diffusion method of modeling certain layered-inhomogeneous media with typical oceanic vertical sound veloeity profiles are diseussed in the ensuing
sections. Variations on the method allow eonsiderahle expansion of the types of media that
can be modeled in this fashion.
At this point we diseuss the problem of the experimental error in the modeling of layeredinhomogeneous media in an underwater acoustical tank. The degree of reliahility of the results
depends on the aeeuraey of the aeoustieal measurements per se, the validity of the model in
terms of the preseribed sound veloeity profile, and the aeeuraey with whieh the veloeity of
sound is measured in the medium. The reliahility also depends on the stability of the sound
veloeity gradient during the experiment.
The aecuracy of the aeoustical measurements under the model eonditions ean be signifieantly greater than that of analogous measurements performed directly in the sea. The
increased accuracy promotes favorable conditions surrounding operati~>ns on the stationary
laboratory equipment (the absence of sea swells, independence of meteorological conditions,
etc.). Consequently, the accuracy of the eleetroacoustical measurements in the model equipment depends only on the quality of the instrumentation used and the precision of the mechanical part of the equipment (the positioning device, earriage, rotating mechanisms, etc.).
The validity of the model, i.e., the degree of similarity of the model medium to the true
medium is determined largely by the experienee and resourcefulness of the experimenter. The
methods described below for the modeling of inhomogeneous media afford working guidelines.
The accuracy of measurements of the velocity of sound in the medium depends on the technique by which they are carried out. In the direct method of measurement the accuracy is
limited by the accuracY of the sound velocity measuring instrument (veloeimeter), while in
indirect methods it is limited by the accuracy with which the variables are measured from
whieh the velocity of sound is to be calculated.
LAYERED-INHOMOGENEOUS MEDIA
§9)
1480 1500
1600
I.
35
IlOoe, rn/sec
i
i
10
20
zcrn'
Fig. 16. Stability of a layered-inhomogeneous medium modeled
by the diffusion methode
Thus, if the layered-inhomogeneous medium is modeled
by the thermal method, the error in the measurement of the
velocity of sound depends on the temperature measurement error. According to (8.2) the error in this case is
ßc=4.21M - O.074tM c:::: 4M.
8
Zcrn
/
Fig. 17. Stability of a
layered-inhomogeneous
medium modeled by the
thermal methode
Suppose that the temperature error is ßt = 0.1°. Then the
error with which the velocity of sound is calculated according
to Eq. (8.2) amounts to LlC = 0.4 rn/sec. Actually the error will
be greater, because Eq. (8.2) itself expresses the values of the
velocity of sound correct to within 1 rn/sec (at a temperature
of about 20°C).
By using more precise equations or tables of the temperature dependence of the velocity of sound in water, one can increase the accuracy with which the velocity is calculated from
the measured temperature.
In the diffusion modeling method the velocity of sound is
determined from graphs (or tables) relating the velocity of sound
in the medium to the optical refractive index in the same medium.
If the error in the measurement of the refractive index in the laboratory is .6.n = 0.0001, we find
from the graphs (Figs. 14 and 15) that the given variation of the refractive index corresponds to
asound velocity variation of about f::,c = 1-2 m/ sec (except in the vicinity of extremal values of
the graphical curves). Thisvalue dictates the maximum error in the determinationofthe velocity of sound.
We conclude with a consideration of the stability of the sound velocity gradient during
the experimental period.
Experiments have shown thai the properties of a medium modeled by the diffusion technique using solutions of a liquid with varying concentrations change very slowly with time. Consequently, the medium may be thought of as essentially invariant for several hours of operation. All that is required is to exercise caution in the movement of the acoustic transducers,
liquid sampiers, and other objects through the medium so as not to disturb its stratified structure. The stability of the medium in this case is illustrated in Fig. 16, which shows the vertical
distribution of the velocity of sound prior to the beginning of operation (dots) and after six hours
(crosses).
36
MODELING OF SOUND PROPAGATION IN INHOMOGENEOUS MEDIA
[CH.3
A layered-inhomogeneous medium modeled by the thermal method is less stable with time.
It exhibits a gradual temperature equalization and a comparatively rapid decrease of the sound
velocity gradient, as shown in Fig. 17: 1) just before the experiment; 2) after 30 mine
§ 10.
Modeling of aSound Channel
A medium of the waveguide type (sound channel), which functions as an energy-focusing
system in a certain bounded layer, is modeled in the experimental tank by interdiffusion between
layers of liquids that differ in their composition or concentration of dissolved substances,
where these factors are judiciously selected for each particular vertical distribution of the
velocity of sound.
Medium with aSound Velocity Minimum at the Liquid Surface
(Su r fa c e Sou nd C ha nne 1) • For the modeling of a medium in which the vetocity of
sound increases monotonically with depth one proceeds from a specified ocean depth and average value of the sound velocity gradient. The following is a sample application of the modeling
methode
Let it be required in a layer of thickness ~z = 20 cm to obtain (in accordance with the
modeling relation) an average relative sound velocity gradient of the order a = 0.1 m-t. From
the equation = (l/cO>(~c/~z) we calculate ~c/co = 0.02. Let us assurne that the water temperature at the surface is 15°C. Then Co = 1464.8 rn/sec, and ~c ~ 29 rn/sec, while the velocity
of sound at the bottom must be roughly equal to 1494 rn/sec. This velocity corresponds to a
2.5% concentration of salto
a
For the formation of a medium having the indicated parameters a layer of salt solution
with a concentration of 2.5% and thickness of 10 cm is deposited on the bottom of the tank; over
this layer is carefully poured a layer of water of the same thickness. In order to prevent motion of the liquids and any disturbance of the stratification of the medium the liquids must be
poured very slowly (the preparation of the medium is a several-hour process).
Gradually between the solution and pure water diffusion causes the formation of a transition layer (a layer of sudden change in the density of the liquid), in which the salinity gradient
can be computed by means of the diffusion equation if the medium can be regarded as layeredinhomogeneous and as having a salinity gradient in the direction of the z axis. The diffusion
equation has the form
as
iJ2S
IJt=D(jZ2'
where S is the salinity, D is the diffusion coefficient, and t is the diffusion time.
Inasmuch as the velocity of sound c is linearly dependent on the salinity, the diffusion
equation may be represented as follows:
From this we deduce [62] the equation
ac)
(Tz
max= 2Y'ltDt '
Cl -C2
(10.1)
which relates the maximum velocity gradient in the transition layer and the diffusion time.
Here Cl and c2 denote the velocity of sound in the solution and in the water, respectively.
MODELING OF ASOUND CHANNEL
HO]
37
TABLE 2
sr.
5
10
20
Cl' rn/sec
0.968
0.988
1.014
1521. 8
1578.8
1692.8
1500 _ _ _ _ _..:..16;:Oo.:::O'--_ _ _ _-'-'i1700 c, rn/sec
7440
.~---..--_.:..:;.:::.,,--
2
4
6
8
22"70 N aCI solution
10
z, crn
Fig. 18. Modeling of a surface sound channel.
The values of the variables entering into Eq. (10.1) for a temperature of 15°C (c2 = 1464.8
rn/sec) are shown in Table 2 [67].
As time passes the transition layer becomes wider, a nearly linear distribution of the
salinity and, hence, of the velocity of sound being set up in its middle portion. The buildup
process of this distribution lasts about Jive to seven days. In order to accelerate the formation
of the layered medium it is necessary between the lower solution and the water to create another layer of solution with an intermediate concentration.
If it is required in the tank to obtain a stronger positive gradient of the same type, a more
concentrated salt solution is poured onto the bottom. For example, an average sound velocity
gradient on the order of 2 or 3 rn-i is produced by the formation of three liquid layers in the
tank: on the bottom a saturated salt solution about 10 cm thick, over that a layer of solution
with a concentration of 8-10% (2 cm thick), and still higher a water layer about 3 cm thick
(Fig. 18, the broken line 1). Mter two or three days the sound velocity distribution in the center of the layer becomes almost linear (Fig. 18, curve 2).
Medium with aSound Velocity Minimum below the Liquid Surface
(Underwater Sound Channel). Amedium characterized by asound velocity minimum
below ~he surface is modeled with water and solutions of sodium chloride and ethyl alcohol. The
technique for the preparation of the medium is similar to that described above; three or four
liquid layers are created in the tank, where the bottom layer, as the heaviest, is a salt solution
of high concentration, over which is poured a layer of water and, above that, the lightest layer
of ethyl alcohol solution.
In order to accelerate the preparation of the medium with a smoothly varying velocity
of sound one can place an extra layer of salt solution of lower concentration between the bottom
layer of solution and the water layer (Fig. 19, line 1). The proper choice of water layer thickness is vital in this instance, because the sound field must be measured while the intermediate
layer of water is still preserved between the alcohol and salt solutions, ensuring a minimum
velocity of sound at that depth (Fig. 19, curve 2).
After a certain passage of time some of the alcohol evaporates, whereupon a thin (about
1 cm) homogeneous layer of solution of constant concentration is formed just beneath the surface.
38
MODELING OF SOUND PROPAGATION IN INHOMOGENEOUS MEDIA
1450
1550
1750 C
[CH.3
m sec
30 "/0 Ethyl alcohol solution
8
12
16
18
22"/0 N aCl solution
20
Z, cm
Fig. 19. Modeling of an underwater sound channel.
The medium can be "corrected" by pouring additional alcohol solution on the surface at
a somewhat higher concentration than the layer formed at the surface. Asound velocity gradient close to the original is established at the surface after 10 to 12 h. If the alcohol concentration at the surface turns out to be larger than the concentration at which the veloc ity of
sound in the alcohol solution has a maximum (see Fig. 15), a thin layer will be formed below
the liquid surface with asound velocity maximum, i.e., two sound channels will coexist, one
near the surface and one at a deeper level.
T he F ie 1 d in aSound C hanne 1. We now consider the sound field in surface
and underwater sound channels under model conditions [10].
The waveform of the acoustic signal in the channel, as in other media, depends on the
shape and width of the pressure pulses radiated into the medium.*
The pulsewidth T is estimated by comparison with the width AT of a signal at the point
of observation. The total width of the signal is equal to the difference between the arrival
times at the observation point of the first and last pulses over the shortest and longest physical
paths. Short pulses arriving at the observation point by different paths are separated according to their arrival times on the screen of the cathode-ray oscilloscope or are omy partially
superimposed. Long pulses, on the other hand, always overlap at the reception point; consequently, the field is portrayed on the oscilloscope screen by a complex pulse whose middle
portion, correspondingto the steady process, is flat. The sound pressure amplitude is read
from the height of this flat portion. The theory of stationary processes is applicable to long
pulses. The graph of the sound pressure as a function of the distance from the transmitter for
long pulses represents a single continuous curve corresponding to the variation of the amplitude of the flat portion of the signal on the oscilloscope screen.
Working with short pulses, on the other hand, one can trace the variation of each pulse
in the signal.
In order to study the sound field both in the near zone of the su,rface channel and at relatively large distances from the transmitter it is required that several complete cycles of the
rays grazing the lower boundary of the channel fit within the length of the tank.
Horizontal sections of the sound field along the tank in a surface sound channel (the vertical profile of the velocity of sound is shown in Fig. 20, curve 1) disclose alternating maxima
and minima of the intensity. Figure 21 shows the sound intensity level as a function of the dis* Only in a good anechoic tank, as mentioned earlier. 1s it possible to work with continuous
sound.
MODELING OF ASOUND CHANNEL
UO]
0
1450
1650
1550
39
1700
T750c, rn/see
4
8
12
16
20
Z,em
Fig. 20. Vertical profile of the velocity of sound in asound
channel. 1) Surface channel; 2) underwater channel.
dB
': t
2~5~------~~~----~'0~0--~~5~0-1 r. ern
Fig. 21. Lengthwise horizontal section of
the sound field in a surface channel.
1550
1600
1650
1700 c. rn/see
.O~,-~~----~~----~~----~~----~~
2
4
8
z, ern
Fig. 22. Vertical profile of the velocity of sound in a
surface sound channel with an absorbing layer.
tance from the transmitter for pulses long enough to be equivalent to continuous radiation.
Here the acoustic frequency is 480 kc, the width of the transmitter directivity pattern at the
principallobe is ±22.5°, and the depth of immersion of the transmitter and receiver is 3 cm.
The distance from the transmitter is plotted on the horizontal axis on log scale.
The sound intensity maxima in Fig. 21 correspond to caustics, the loci of which are found
from the ray pattern (in the figure they are marked by vertical dashed lines). The difference
between the intensity levels on the caustic and the adjacent shadow in the surface channel does
not exceed 10-15 dB.
According to the theory ([20], p. 457), the focusing factor on the caustic increases with
distance from the sound source as r 1 / 3• Consequently, the sound intensity at the maxima must
obey the law I ..... r- 2 • r 1/3 ..... r- 5 / 3• This dependence is represented by the dashed Une in Fig. 21.
The author in particular has investigated the case when the channel is contiguous with a
surface absorbing layer (Fig. 22), which is modeled by means of castor oil, which has asound
absorption coefficient much larger than that of water. Measurements of the sound pressure revealed that the surface absorbing layer increases the rate of decay of the sound intensity in the
channel as a function of the distance from the transmitter.
40
MODELING OF SOUND PROPAGATION IN INHOMOGENEOUS MEDIA
dB~
[eH. 3
~I__r-1
-_...J
20
•
_
-
-I
1
1;1
I
1
--...,
-
I
-
-
I
_
I
O·I~
SO
100
200
300
400 r • cm
Fig.23. Lengthwise horizontal section of the sound
field in an underwater sound channel. The transmitter
parameters are the same as in Fig. 21; the depth of the
transmitter and receiver is 5 cm.
0'r4~SO~~/~50~O~~~__~~~~S~
2
JO
14
18
Z. cm
Fig. 24. Vertical profile of the
velocity of sound in an underwater channel.
The field in an underwater sound channel as depicted by the ray pattern consists of a near "illuminated"
zone in which the sound intensity decays on the average
as r- 2, plus alternating shadow and illumination zones.
If the transmitter is situated on the axis of the channel,
the main component of the sound energy is concentrated
about the axis, and the shadow and illumination zones are
not as sharply delineated as when the transmitter is placed
above the channel axis.
The results of measurements of the field in an underwater channel are shown in Fig. 23 (the vertical profile of the velocity of sound is represented by curve 2 of
Fig. 20) for the case when the transmitter and receiver
are located above the channel axis. The dashed line corresponds to a cylindrical law of decay of the sound intensity •
The positions of the caustics, calculated from ray theory (the locations ofthe caustics are
denoted by vertical dashed lines in Fig. 23), agree satisfactorily with the experimentally determined regions of maximum intensity, and the intensity drop at adj acent caustics and the
shadow attains 20 to 30 dB.
If the sound transmitter is situated near the axis of the underwater sound channel, a
large number of signals arriving by different paths become superimposed at a point of observation also on the axis. In this case the average intensity level of the sound field is computed with sufficient accuracy by energy (incoherent) addition of the signals.
Along with the investigation of the sound intensity it is instructive to study the shape of the
acoustic signal at various distances from the transmitter. The distortion of the signal serves,
in particular. as a measure of the reliability of the information carried by the channel. It is
important to discriminate between the signalform variation induced by the arrival of several
pulses of varying intensities at the observation point and the actual pulse distortion due to nonlinear effects in the medium or unequal attenuation of the spectral components of the pulse. In
the event of a narrowband transmitted pulse with smaU amplitude there is no possibility of nonlinear distortion of the pulse, and the only cause of variation in 'the signalform at various distances from the transmitter is the propagation of multiple rays.
The structure of the signal. as noted above. also depends on the width of the component
pulses. With the transmission of relatively long pulses (pulsewidth T greater than the difference L1 T between the arrival times of the individual pulses at the observation point) the variation of the signalform is caused by the superposition of pulses arriving at the observation point
in disparate phases. Sufficiently short pulses (T < L1T). on the other hand, are not super-
§l0]
MODELING OF ASOUND CHANNEL
41
a
b
t= --
c
+
d
Fig. 25. Signals received in an underwater channel at various distances from the transmitter at
an acoustic frequency of 2.75 Mc for the case of
short 5-psec pulses (left) and longer 50- to 100psec pulses (right). a) h = z = 6 cm, r = 15 cm, Ti =
5 psec, T2 = 100 psec; b) h = z = 6 cm, r = 23.5
cm, Ti = 5 psec, T2 = 70 psec; c) h = 2 cm, z =
4 cm, r = 73 cm (shadow zone), Ti = 5 psec, T2=
50 psec; d) h = z = 5 cm, r = 176 cm (near the
third caustic), T1 = 5 psec, T2 = 50 psec.
imposed, and the signalform can be analyzed by inspection of the variation of each pulse in the
signal independently of the others. For an approximate calculation of the propagation time of
lt pulse along a ray the medium through which the ray passes is partitioned by depth into several horizontal layers such that the velocity gradient in each layer may be assumed roughly
constant. The total transit time of the pulse along the ray from the source to the receiver is
"
t= ~ Ißtkl,
k=l
where [20]
Here Zk-l and Zk are the coordinates of the upper and lower boundaries of the k-th layer,
respectively, Ck-l and Ck are the velocities of sound at the boundaries, and Bk-l and Bk are the
grazing angles at the boundaries.
42
MODELING OF SOUND PROPAGATION IN INHOMOGENEOUS MEDIA
[CH.3
o 1500 1600 1l00e. rn/sec
16
Fig. 26. Vertical profile of the velocity of sound (a) and
ray pattern (b) in an underwater channel.
For small grazing angles (8 ::::: 1) in a medium endowed with a weak sound velocity gradient the transit time is expressed by the approximate equation
As an example we consider the signalform in a channel with its axis situated below the
liquid surface (Fig. 24) with the transmission of pulses having various widths. Close to the
transmitter (within the limits of one reflection of the beam from the surface) two pulses arrive at the receiver, the direct pulse and one singly reflected from the surface (Figs. 25a
and 25b) , the difference between the arrival times of these pulses depending on the depth of the
transmitter (h) and receiver (z) and their mutual separation (r).
With an increase in the width T of the pulses they interfere with one another, causing
the signalform to be distorted, particularly at points of the field where the pulses coalesce to
form a single pulse, which decays monotonically with distance from the boundary of the shadow,
maintaining a rectangular shape (Fig. 25c). The shadow terminates in the transition to a caustic;
in the transition region between the shadow and the caustic the first decaying pulse is followed
on the oscilloscope screen by a second pulse, which grows abruptly as the caustic is approached.
At the observation point the first pulse arrives along a ray that undergoes an initial refraction above the channel axis, the second along a ray that is refracted below the axis (Fig. 26).
If the receiver is placed farther from the transmitter the shape of the received signal becomes
more complex (Fig. 25d). A large separation between the transmitter and receiver contains
several caustics and shadow zones. Consequently, the field is represented by a complex signal
partially having a diffraction origin and partially formed by rays arriving at the observation
point.
A discriminatory analysis of the pulses constituting the signal enables one to investigate
the behavior of the field in the shadow zone, on the caustic, and in the vicinity of the latter.
In the beginning of the shadow zone, as mentioned, the predominant part of the signal is the
first (diffraction) pulse, which becomes small in the region oftransition to the caustic. The structure of the sound field in an underwater channel is made explicit by the ray pattern (Fig. 26).
The solid curve in Fig. 27 [65] illustrates the theoretical variation of the intensity level according to a calculation of the field of the first normal mode [20].
Here the width of the sound pulse is 20 p.sec, the carrier frequency if 400 kc, and the depths
of the transmitter and receiver, respectively, are 0.5 and 6 cm; the vertical velocity profile cor-
§10]
MODELING OF ASOUND CHANNEL
43
I:ml
1.0.
dB
0.8
0.6
0.4
0.2
Fig. 27. Distribution of the sound
field in the first shadow zone in an
underwater channel.
Fig. 28. Sound pressure near a caustic.
res ponds to Fig. 26. The distance from the shadow boundary is plotted on the horizontal axis.
The experimental points on the left side of Fig. 27 correspond to the amplitude of the first
pulse. The dashed curve is drawn through the experimental points corresponding to the amplitude of the second pulse.
The sound pressure on the caustic and in the immediately contiguous zone, where it is
permissible to neglect the amplitude of the first pulse relative to the second, is shown in Fig.
28 for the case of the vertical sound velocity distribution of Fig. 24. The width of the acoustic
pulse is 8 Jlsec, the carrier frequency is 2.75 Mc, the width of the directivity pattern of the
transmitter at the principallobe is :1:60°, and the depths of the transmitter and receiver are
6 and 2 cm. The sound field was generated by a transmitter situated on the channel axis. The
sound receiver was placed above the axis.
In the graph the distance from the observation point to the caustic (u) is plotted on the
horizontal axis, negative values of the distance corresponding to the transition region from
the shadow to the caustic; the ratio I p !Pm I of the absolute values of the sound pressure to
its maximum value is plotted on the vertical axis [10].
The theoretical curve of Fig. 28 (dashed) was calculated by means of the Airy function
([20], p. 432). The position of the caustic (u = 0) is determined from the following condition
on it:
--'fQ...=J!2....=~=O.62927.
'f'm
Pm
Um
Here <Po, Po, and Vo are the values of the wave potential, pressure, and Airy function on
the caustic; <Pm, Pm, and Vm are the maximum values of the same variables.
It is apparent from Fig. 28 that in the shadow zone, where the pressure amplitude varies
monotonically, the experimental points provide a good fit to the theoretical curve. Beyond the
caustic (u > 0) the experimental and theoretical periods of the spatial pressure oscillation almost cOincide, but the values of the pressure differ somewhat. It is evident in the graph that
the error of the pressure measurements at the maxima is about 10%.
The results obtained in the present section indicate that the modeling method is weH
suited to the investigation of the sound field in a channel. It permits the qualitative analysis
44
MODELING OF SOUND PROPAGATION IN INHOMOGENEOUS MEDIA
(300
r---;;..oC..._ 3
10
20
Z, cm
Fig. 29. Types of vertical sound velocity
profiles for the antiwaveguide propagation of sound. 1) Medium with an almost
constant negative velocity gradient; 2) analogous medium with a surface quasihomogeneous layer; 3) medium with a velocity maximum at a certain depth.
§ 11.
[eH. 3
of the structure of the field in the channel and
the derivation of average laws describing the
decay of the sound intensity, where these things
are made possible by the stringent demands imposed on the electroacoustical instrumentation.
For the detailed investigation of individual portions of the field (shadow zone, region of caustics) it is necessary to use sensitive detection
equipment. The author's own measurements
have shown that the dynamic range of the equipme nt must be at least 30-40 dB, and in the case
of diffraction field measurements in the shadow
zone it must be even higher.
On the other hand, the structure of the
field can be studied more in detail, the smaller
the dimensions of the sound receiver. Inasmuch as the sensitivity of the receiver decreases with its physical volume, it is essential in this case to augment the gain and quality
of the signal amplification in the receivingtransmitting section of the electroacoustical
system.
Modeling of Antiwaveguide Sound Propagation
For certain types of vertical sound velocity profile conditions are established in the
oceanic medium such that the sound energy diminishes along the path of wave propagation
and, beginning at adefinite distance, a zone of attenuated sound is formed, which is commonly
referred to as a shadow zone. This type of sound propagation may be called antiwaveguide
propagation.
Examples of a vertical velocity distribution for which antiwaveguide propagation is observed are shown in Fig. 29.
The types of vertical velocity profile represented by curves 1 and2 can be modeled in
water by the thermal action method (§ 9).
The diffusion method is used to simulate a medium with asound velocity maximum at
some depth (Fig. 29, curve 3). In such a medium both asound channel at the surface and the
antiwaveguide propagation of sound below it are observed simultaneously.
The Field in Media with a Negative Velocity Gradient. The field in
media of this type can be separated into two parts [4, 5, 7]: 1) the near region, in which the
sound intensity oscillates with distance from the transmitter; 2) the far region with a monotonie decrease of the intensity.
A horizontal seetion of the field in a medium with a vertical velocity profile according
to curve 2 of Fig. 29 is shown in Fig. 30. The results of the measurements are represented on
the graph of dots, through which are drawn the dashed curves 1-3, indicating the variation of
the signal as a function of the distance from the sound source.* Here the frequency is 250 kc,
the width of the directivity pattern of the transmitter at the principal lobe is ±45°, and the sound
receiver is omnidirectional'
* The measurements were performed in the Acoustics Department of Gorky State University ..
45
ANTIWAVEGUIDE SOUND PROPAGATION
§11]
dB ,
30
1
1'\
~
I'
~/\I'"
....
I' -
I, I
20
f)~ ,
I,i'i--t,'
\~ ~ 1/
tI
\1
,
,
\
,
\\
1 -4
1cf' I.........
\~
i",2
\
/
I
V
\
I
I,{ \ lt ~ t\ 11 \I
'\ 'I A
10
o
A
'01
/
I' ,.
b,,
'eilt'
-
'\
/~ 3
....
-fJ
\
31\
1
1'\' \
r
\ - , -I\..J... \
I
I \'
,
11,
,\.j---I-.JJ..:_~ ...
I I , 1 ci ~ l? I
•
_ .....- ~;:! r -l
11 ,\'f'/\)/
11
\ / ~
\
~,
11
'U
'J
~
,2
L -_ _ _ _ _ _ _ _ _ _ _ _~~_ _ _ _ _ _ _ _ _ _ _ _ _ _~,~--'~·~~lL-----~I----\~o----------~,--
2
3
I; r.
m
Fig. 30. Horizontal distribution of the sound field for antiwaveguide sound propagation.
In the near region the sound intensity decays on the average as r- 2 (it is depicted in the
figure by the appropriate dashed curve). The boundary for the near region may be chosen
conditionally as the point at which the intensity maximum farthest from the source is located.
The soundattenuation in the far zone depends on the thickness of the homogeneous surface layer, velocity gradient, and frequency. It is shown in ([20], p. 473) that the sound attenuation per unit length depends on the dimensionless parameter
1
3a )'/.
s=3
koHo( ~
,
where k o = 2rrf/co, Co is the velocity of sound at the surface, and Ho is the thickness of the
surface layer.
The smaller the parameter s, the greater will be the attenuation in the far zone, and the
closer the boundary of the near region will come to the transmitter.
At a sufficient distance from the transmitter (r » sH~ i\) the sound field in the far
region is determined primarily by the least damped first normal mode. This region is called
the shadow or effective shadow zone. In it the sound intensity falls off exponentially with distance from the source.
The greatest attenuation in the shadow takes place when the parameter s = 0 (i.e., when
Ho = 0).
In this case the ray tangent to the surface of the water marks the boundary of the geometric shadow zone (Fig. 52b).
The intens ity of the diffraction field in the shadow zone varies according to the law
where r' = PQ is the distance of the observation point from the shadow boundary and 10 is the
field intensity at the boundary.
46
MODELING OF SOUND PROPAGATION IN INHOMOGENEOUS MEDIA
[eH. 3
Then the attenuation of the sound intensity level in the shadow zone at a distance r' is
equal to [20]
I
ßß = 10 log~
= - 6.66qr' ,
where
t
q = (9a 2 ko
(11.1)
3 •
The field in the effective shadow zone for s > 0 is calculated analogously with the aid of
the first normal mode. The field calculated by this method is represented by the line segment
T in Fig. 30. The author's measurements [4, 5] have shown that the mean attenuation of the
diffraction field in the shadow zone as determined under model conditions closely fits the
theoretical.
A medium with a large value for the parameter s (greater than one) is simulated by a
suitable increase in the thickness of the surface layer, vertical sound velocity gradient in the
inhomogeneous medium, or the acoustic frequency. Since the possible increase in the surface
layer thickness and velocity gradient is limited in practical model experiments, the best approach is to increase the frequency.
For large values of s the lengths of the near region of the field and of the transition region to the far field are greater. Therefore, experiments for the observation of the effective
shadow zone must be performed in a long tank in this case. ,
In model situations it is a simple matter to obtain a vertical field distribution permitting
the observation of a gradual decrease with distance in the number of normal modes that contribute to the formation of the field. At small distances from the transmitter (r < SH~/A.O>,
owing to the presence of a large number of unattenuated normal modes, the vertical distribution of the field presents a highly dissected pattern. Ray representations are applicable in
this region.
At large distances from the transmitter (r» SH~ IA. 0> the vertical distribution of the
field becomes more uniform, indicating a gradual attenuation of the higher-order modes with
distance.
The diffraetion field is not the only one observed in the shadow zone during sound propagation in the sea, beeause the shadow is irradiated by sound scattered from water waves, the
bottom, thermal microinhomogeneities of the medium, turbulence, sound-seattering layers, ete.
The intensity of the volume seattering of sound at a distanee much greater than the linear
dimensions of the scattering zone in the ease of an isotropie medium is ealeulated within the
framework of statistieal aeousties ([60], p. 52). An estimate of the exeess of the seattered
field level in the shadow zone over the diffraetion field has been made in [50] on the assumption that the autoeorrelation funetion of the thermal inhomogeneities has the form exp(-lld),
where l is the distance between two points and d is the average seale of the thermal inhomogeneities.
In [51] the possible magnitude of the field has been estimated in the shadow zone due to
scattering of the direet sound by a layer existing in the illuminated zone and containing
stochastically distributed inhomogeneities of the eharaeter of the Epstein symmetrie layer
[79]. The ealeulations show that, depending on the degree of "oblateness" of the inhomogeneities, the sharpness of their boundaries, and the aize of the domain that they oeeupy, a
rather high level can result on the part of the refleeted field from the inhomogeneities relative
to the diffraetion field.
§11]
ANTIWAVEGUIDE SOUND PROPAGATION
47
Under laboratory conditions the formation of volume scattering zones is difficult in a
tank containing a nonmoving layered-inhomogeneous medium. Such zones occur only with the
motion of various objects (thermometer, acoustic transducers) through the water, wave formation of the surface, or nonuniform heating of isolated portions of the liquid volume.
It is possible by the artificial creation of thermal inhomogeneities in the sonically illuminated zone of the lower part of the tank to observe in the shadow zone a scattered field
that sometimes exceeds the intensity of the diffraction field considerably. The amplitude of
the signal scattered by thermal inhomogeneities fluctuates, unlike the bottom-reflected signal.
This characteristic of the scattered signals facilitates their recognition.
Under model conditions it is also easy to observe the illumination of the shadow with
sound reflected from the bottom of the tank. The distance from the boundary of the shadow to
the zone in which the scattered signal becomes prevalent is determined from a comparison of
the diffraction and bottom-scattered signals. The discriminatory analysis of the diffraction
and bottom signals in the shadow zone is abetted by the fact that, given a sufficiently small
width, these signals are resolved on the screen of the display oscilloscope according to their
arrival times.
Different components of the total signal in the shadow zone are illustrated in Fig. 30; the
direct signal is represented by the solid curves; sound reflected from a silty bottom (plasticine
model) one, two, and three times is represented by the dashed curves 1, 2, and 3, respectively.
Figure 30 shows that in the shadow zone the direct signal is comparable with the bottom-reflected signals only in the initial portion, the scattered field becoming significantly
greater than the diffraction field farther on.
In the case in question the bottom of the tank was covered with a flat horizontal layer of
plastic to simulate a silty soil.
The cited example indicates the applicability of the modeling method to the investigation
of bottom-scattered sound.
We note that the acoustic shadow zone in the earth's atmosphere with a stratüied temperature distribution over a plane boundary has been investigated under model conditions in
[95]. Applying a method similar to that described above, the authors of that paper conducted
a theoretical and experimental investigation of sound propagation over an absorbing ground
in the atmosphere with the latter characterized by a constant temperature gradient.
T he F ield in a Med ium wi th aSound Ve I 0 c ity Max imum below the
Li q u i d S u r f ace. For the modeling of a med ium invested with this type of vertical velocity profile water and an ethyl alcohol solution are used, where the concentration of the
latter exceeds the value S' for which fue velocity of sound in the alcohol solution is a maximum. At first, as in the case described above, there is a sharp boundary between the layers
of these liquids. Then, as a result of diffusion between them, a transition layer is formed in
which the alcohol concentration varies monotonically from a maximum at the surface to zero
at the bottom. At the same time, a vertical sound velocity distribution is created such that
the velocity of sound is maximized at the depth corresponding to the concentration S' (an example of the initial and later distribution of the velocity of sound is depicted by curves 1 and 2
of Fig. 31).
The sound field in a medium with a velocity maximum below the surface (Fig. 29, curve 3)
has a complex structure, as indicated by the schematic ray pattern (Fig. 32).
This pattern is augmented by model measurements [11]. Suppose that the sound field is
generated by a moderately directional transmitter (l{! 0 = ±60" sending pulses with a width of
150 to 200 fJ-sec and carrier frequency from 450 to 2000 kc into the given medium.
48
MODELING OF SOUND PROPAGATION IN INHOMOGENEOUS MEDIA
[eH. 3
14r-5:::...O_--r----,,.....,.,'Sr:5.=..O_-.-_-'-1'6""50"-----,,---_':.;7.50 c. m / sec
62"/0 Ethyl alcohol solution
8
'2
16
20
Z, Gm
Fig. 31. Modeling of a medium with asound
veloeity maximum below the surfaee of the
liquid •
.------.----c
z
Fig. 32. Sehematie ray pattern in a medium with asound
veloeity maximum.
It follows from an analysis of the results of the aeoustical measurements that near the
transmitter is an illuminated zone with alternating sound intensity maxima and minima typieal
of wave interferenee. Then. beginning at the level of the transmitter and below it is a region in
whieh the sound level falls off abruptly with inereasing distanee. The onset of this region eorresponds to the beginning of the shadow ZOne (the hatehed region in Fig. 32). In the given instanee. unlike the ease of a medium with a monotonie deerease of the veloeity with depth. the
shadow zone with its exponential intensity reduetion does not extend infinitely, but ehanges
gradually into a field in whieh the intensity falls off far more slowly.
Experimental curves of the variation of the sound intensity level along the tank in the
shadow zone (curve 1) and in the farther zone, where the intensity level elosely follows an r..,'!
law (eurve 2). are shown in Fig. 33. The graph eorresponds to the vertical sound velocity
profile according to Fig. 29 (eurve 3). Here the acoustic frequency is 2 Mc, and the depths of
the transmitter and receiver are 4.5 and 6.5 cm.
Above the level of the velocity maximum a surface sound channel is formed, in which the
field intensity falls off on the average as r- 1•
The transformation of the shadow zOne into a zOne of intensity variation according to a
spherieal law is qualitatively explained by the transfer of sound energy from the surfaee ehannel
into the energy-poor region below the channel. This type of energy transfer is observed in the
sea during the long-range propagation of sound [69].
NEAR FIELD OF ASOUND VELOCITY DISCONTINUITY LAYER
§l2]
The text of the present section confirms the
fact that the modeling of layered-inhomogeneous
media on the basis of thermal action on the medium
or on diffusion between different liquids is weH
suited to the investigation of the antiwaveguide
propagation of sound. As far as the electroacoustical apparatus used for modeling is concerned,
its dynamic range must be at least 60 dB in order
to perform simultaneous measurements of the field
both in the illuminated zone and in the shadow zone.
d13
40
30
20
10
°2~0~--~--~~~~~----
40
60 80 fOO f20 140 r. crn
Fig. 33. Horizontal distribution of
the sound field in a medium with a
sound velocity maximum below the
liquid surface.
Z, crn
10
o
49
f600
"OOe, rn/sec
-10
Fig.34. Vertical profile of the velocity of sound.
§ 12.
Modeling of the Near Field
of aSound Velocity Discontinuity
Layer
A region of sudden variation of the velocity
of sound in the sea is called a discontinuity layer or
transition layer. We shall say that the discontinuity
is positive if the velocity of sound in it increases
with depth, and negative if the velocity decreases
with depth.
With the incidence of sound on the discontinuity layer there is partial reflection and refraction of the sound beam.
Under certain conditions the sound beam undergoes total internal reflection in the discontinuity
layer, whereupon the region on the other side of the
layer is unilluminated (screened).
The effects observed in connection with the
incidence of sound on the discontinuity layer can be
investigated in models.
In the laboratory tank a discontinuity layer is formed between two layers of aqueous solutions of different concentrations of a suitable agent (or a layer of water and a layer of aqueous
solution), where one layer is poured over the other.
The Field in the Reflected Wave. Suppose that as a result of interdiffusion
between a sodium chloride salt solution and water a medium is created such that the velocity
of sound in it varies with height as in Fig. 34. For the measurement of the sound pressure
in the reflected wave [8] the sound beam generated by a directional transmitter in the homogeneous medium impinges on the ßlightly diffuse boundary between the homogeneous and inhomogeneous media. The measurements are carried out in the continuous or pulsed transmission mode. The sound field in the reflected wave is measured with a nondirectional sound
receiver. The screen of an oscilloscope serves as the indicating device. In order to preclude
the possibility of reception of the direct signal a vertical acoustically nontransmissive plane
baffle is inserted between the transmitter and receiver, where it limits the width of the sound
beam contributing to the formation of the caustic in the reflected beam and thus el iminates
almost completely the influence of the side lobes of the transmitter directivity pattern.
MODELING OF SOUND PROPAGATION IN INHOMOGENEOUS MEDIA
50
Itmlneg
1
,.
o
Fig. 35. Sound field in the
wave reflect ed from a medium with asound velocity increasing by depth.
[CH.3
The results of measurements of the sound pressure
in the reflected wave are represented by the dots and solid
curve in Fig. 35, in which the ratio I P/Pm I (where Pm is the
maximum pressure in the reflected wave) is shown as a function of the distance between the point of observation and the
caustic (u). Here the acoustic frequency is 2.75 Mc, and the
transmitter and receiver are situated at heights of 4 and 11.5
cm, respectively, above the boundary of the inhomogeneous
medium. The dashed curve in the figure corresponds to the
press ure distribution calculated by means of an improved
ray theory ([20], p. 432) on the assumption that the velocity
distribution in the medium is approximately expressed by
the law
where Co = 1465 m/sec and a = 2 m-l.
Using the data on the distribution of the sound field in the reflected beam at various levels, one can readily find the configuration and position of the caustic, which agrees weH with
their values calculated by ray theory.
The Field in the Refracted Wave. Inthe discontinuity layerthere occurs
refraction broadening of the ray tube and, associated with this, attenuation of the sound intensity. The investigation of the field of the sound wave transmitted through the discontinuity
layer can be conducted in a number of instances on the basis of ray representations. This
possibility is readily confirmed by application of the modeling method [9].
The dependence of the velocity of sound in a discontinuity layer on the vertical coordinate z is weH approximated by an equation analogous to that describing the velocity distribution of radio waves in an Epstein transition layer [79].
Figure 36 shows the variation of the velocity of sound with depth. According to the Epstein
equation,
C(Z)=C 1
[ 1 + ( c2c~ -1 ) ( e- Pz +1
)-1 ]-1 /2
,
where
Here cl and c2 denote the respective velocities of sound in the homogeneous media below
and above the transition layer. In this case cl and c2 are equal to the velocity of sou:OO in the
salt solution and in the water, respectively. Curves 1 and 2 in Fig. 36 correspond to diffusion
times of 5 and 72 h.
The variation of the velocity of sound with depth in the transition layer takes place primarily at a depth z between the limits -11' /p < Z < 11' /p, and the effective thickness of the layer
may therefore be assumed equal to l = 211'/p.
§l21
51
NEAR FJELD OF ASOUND VELOCITY DIseONTINUITY LAYER
Z, cm
With regard for the value of the diffusion coefficient D = 1.014 cm 2/day [671,
for example, in the case of a transition layer
between a saturated salt solution (ci = 1730
m/sec) and water (c2 = 1465 m/sec) at a temperature of 15 oe the thickness of the layer
has the following dependence on the diffusion
time:
•
o~--~===-~~~~~~~--­
-1
-2
-3
A diagram of the experimental setup
for measurement of the sound pressure in
the beams incident on and refracted in the
layer is illustrated in Fig. 37. In this case
the transmitter is located below the transition layer, and sound is incident on the discontinuity layer in the direction of decreasing
velocity of sound.
-4
-5
11,50
1500
1600
Fig.36. Variation of the velocity of sound
with depth in the transition layer.
c
The sound field at various depths on
either side of the layer is investigated by
moving the hydrophone in the horizontal
direction.•. The resulting data are used to
plot a graph of the sound pressure versus
the horizontal distance from the transmitter.
b
J
a
The sound reflection coefficient of the
layer is estimated from the equation for the
modulus of the coefficient of reflection from
an Epstein transition layer ([20], p. 160);
in the case under investigation the equation
may be written in the form
Fig. 37. Diagram of the experimental setup
for investigation of the transmission of a
sound beam through a transition layer. 1)
Experimental tank, including salt solution
(a), water (b). and scale- reading device (c);
2) pulse generator; 3) transmitter; 4) hydrophone with moving carriage; 5) receiver
amplifier; 6) oscilloscope.
W/=
[1tN (sin 0\ - 11 sIn2 °1 + ~ )]
sinh [1tN( sln 0\ + Vsln2 °1 + 2:
sinh
j
C
)]
,
where N = 1/11., (}i is the grazing angle of the central ray of the beam on the boundary of the
layer, and .6.c = ci - c2'
In the experiment the sound attenuation is determined by measurement of its intensity
in the incident and refracted beams. If the sound intensity measured at points E and F (Fig.
38) is designated 11 and 12, respectively. the attenuation factor is
/1
~= 101og7;'
(12.1)
On the other hand, the attenuation factor according to the ray pattern is equal to
w= 10 log F,
where
F= CD
AB
(12.2)
52
MODELING OF SOUND PROPAGATION IN INHOMOGENEOUS MEDIA
[CH.3
TABLE 3
Exp.
No.
ei
1
11
ei
l. crn
B. dB
by (12.1)
50/0 salt solution and water( (t.c = 57 rn/sec)
2
3
19
27
34
22
30
1.3
1.5
1.0
I by (12.2)
4.6
3.0
2.0
6.4
3.4
2.8
10"!osalt solution and water (t.c = 114 rn/sec)
4
12
12
12
1.1
5.9
26
26
2.6
6.0
5.6
6.0
26
20"/0 salt solution and water (t, c = 228 rn/sec)
7
13
13
13
20
20
20
5
6
8
9
10
11
12
32
32
32
36
36
36
1.2
1.6
2.0
1.2
1.6
2.0
7.8
8.8
7.4
6.6
6.8
8.0
6.5
7.2
8.2
8.3
8.5
8.9
5.7
6.0
6.4
f----------
I
dl
r
I
o
I
Fig.38. Calculation of the sound attenuation in the
transition layer.
Assuming that the sound velocity gradient in the central part of the transition layer is
constant (Fig. 36 justifies this assumption), from the ray pattern we obtain
(12.3)
The latter equation assumes a particularly simple form when the range r is much larger
than the distance d from the transmitter to the layer. If ra » 1 and ~ C «cl' then [58]
(12.4)
The measured values of the attenuation of the sound intensity level are compared with the
values computed according to the ray pattern for certain experimental conditions (the acoustic
frequency is 500 kc) in Table 3.
Near-field model studies similar to those described above can also be carried out with the
transmitter above the transition layer, provided the velocity of sound therein decays with depth.
For the formation of such a transition layer an aqueous solution of ethyl alcohol is poured on
top of the water in the tank, where the concentration of the alcohol solution is lower than the
value for the maximum velocity of sound in the solution.
H3]
MEDIA WITH ASOUND VELOCITY GRADIENT
53
Let us consider the case when the transmitter
is placed in the homogeneous medium above a layer of positive discontinuity of the velocity of sound
(Fig.36). In this case the sound impinges on the discontinuity layer in the direction of increasing velocity
of sound.
It is evident in the ray pattern of the field of a
nondirectional transmitter that for a sufficient differential in the velocity of sound the sound intensity
decreases considerably in the discontinuity layer,
Fig. 39. Schematic ray pattern for
because underneath the layer the ray tube broadens
placement of the transmitter above
very rapidly. An analogous effect takes place in the
a positive discontinuity layer. c)
field of a directional transmitter. Moreover, in the
Central ray of the sound beam; 0)
latter case the illuminated zone can be depth limited.
lower outermost ray of the principal
This will happen when the outermost ray of the prinlobe of the transmitter directivity
cipal lobe of the transmitter directivity pattern sufpattern.
fers total internal reflection at the lower boundary of
the layer or above it (ray 0 in Fig. 39). Then the entire ZOne below that level turns out to be almost totally screened. The weak field observed in this zone is to be attributed primarily to the side
lobes of the transmitter directivity pattern (the corresponding rays are indicated in Fig. 39 by
dashed lines).
It is apparent from the ray pattern that the screened region is not formed in the field of
a nondirectional transmitter. As for the field of a directional transmitter, the screening effect can be eliminated by rotation of the transmitter axis downward through the requisite angle.
An analysis of the results of model measurements of the sound field affords a means for
assessing the influence of the positive discontinuity layer on the propagation of the sound beam
through it. The experimental pattern of the sound field as a whole is consistent with ray theory.
Horizontal sections of the field show that above the discontinuity layer the sound intensity
varies with increasing range according to a spherical power law, which gradually changes over
to a cylindricallaw. This variation of the field indicates the formation of asound channel in the
homogeneous surface layer.
Below the discontinuity layer the sound intensity at first decreases quite rapidly by an exponential law (zone of weak sonic illumination). but with increasing distance changes more slowly, by apower law.
The intensity reduction in the sound beam incident on the layer agrees with the calculated
value according to ray theory taking account of the fact that the sound intensity in the layer destreams, as weIl as to the proximity of the ice edge in the polar seas, there are zones with a
considerable horizontal gradient of the velocity of sound.
§ 13.
Modeling of Media with a Vertical-Horizontal
Sound Velocity Gradient
There are regions of the world ocean in which, due to the presence of warming or cooling
streams, as weIl as to the proximity of the ice edge in the polar seas, there are zones with a
considerable horizontal gradient of the velocity of sound.
In media of this type the stratification is upset as a result of the variation of the velocity
in the horizontal direction and the concomitant transformation of the law governing the depth
54
MODELING OF SOUND PROPAGATION IN INHOMOGENEOUS MEDIA
[eH. 3
O.-_____J~~~90~___________7r,~--~~--~~~10c,m/s~
"
5
5
Fig. 40. Vertical profile of the velocity of sound in a
medium with a vertical-horizontal velocity gradient at
various distances from the transmitter. 1) 39 cm; 2)
92 cm; 3) 112 cm; 4) 138 cm; 5) 175 cm.
c,m/sec
1510
1505
--------2
~-----J
7500
.-------4
50
100
150
200 r. cm
.......
' ----;--,-----
Furnace
Fig. 41. Horizontal profile of the velocity of
sound in a medium with a vertical-horizontal
velocity gradient at various depths. 1) 0.5 cm;
2) 1 cm; 3) 1.5 cm; 4) 4 cm.
variation of the velocity of sound along the path of propagation. This leads to a distortion of
the field pattern that would exist in the case of a plane-Iayered medium.
LUtie has been done toward the development of a procedure and methodology for the modeling of media characterized by a vertical-horizontal velocity gradient, although certainly model
experiments would be of great service in the investigation of the principles governing the sound
field in this theoretically intimidating case. We now give an example of how the modeling
method is applied to the problem.
In the Acoustic Department of Gorky State University we have developed an experimental
arrangement for the simulation of one type of the kind of medium in question. It is formed in
an experimental tank by nonuniform heating of the surface of the water along the tank. For
heating of the water an electric furnace of the construction described above (§ 9) is used. The
§l4]
SOUND PROPAGATION IN THE PRESENCE OF ROUGH FREE SURF ACE
55
furnace is suspended in the immediate proximity of the water surface, covering only a portion
of it. * This heater arrangement was such as to heat the upper layers of the water only directly
beneath the furnace and to create a specific distribution of the temperature and, hence, of the
velocity of sound in the tank both in the vertical and in the horizontal planes. The velocity distribution typically produced in the experiments after several hours from the start of heating is
illustrated in Figs. 40 and 41.
The depth dependence of the velocity of sound at several distances from the transmitter
is shown in Fig. 40.
The horizontal distribution of the velocity of sound at several depths is shown in Fig. 41.
The magnitude of the vertical velocity gradient was of the order a' = -0.1 m- 1, and for
the horizontal velocity gradient it was b' = 10- 3 m- 1• For a scaling factor M = 104 these values
would correspond to real ocean gradients of a = -10- 5 m- 1 and b = 10- 7 m- 1•
The velocity of sound at various points of the medium with a vertical-horizontal sound
velocity gradient and the sound field in it are measured by the customary procedure (§ § 6, 8).
§ 14.
Modeling of Sound Propagation in the Sea
in the Presence of a Rough Free Surface
Scattering of Sound Waves by a Rough Surface. The waveform surface
of the sea (wind waves, swells) scatters incident sound and elicits certain other secondary effects (the formation of a large quantity of air bubbles in the surface layer, ocean noise, etc.),
which in combination with scattering by the waves lead, on account of reverberation, to the attenuation and masking of acoustic signals transmitted through the water.
In the study of sound scattering by surface waves under model conditions tb.e secondary
ef.fects of the wave state are absent. Consequently. the diffraction of sound by the periodic
structure of the surface can be investigated in unadulterated form without the side effects of
the surface wave state. The scattering indicatrices can be found experimentally as a function
of the parameters of the surface, angle of incidence of the ray, and acoustic wavelength; the
sound field in the direction of specular reflection and in the direction of detection, etc., have
been investigated.
In the case of a surface with randomly distributed irregularities the statistical characteristics of the reflected wave can be determined and their correlation with the parameters of
the reflecting surface established. The study of these and other problems bearing on the scatte ring of sound by the uneven surface is instrumental, in particular, in the explication of the
important problem in underwater acoustics of the influence of the rough surface of the sea on
the principle governing the variation of the sound pressure with distance from the sound source.
Analogous problems also arise in connection with the propagation of radio waves over an
uneven surface and their diffraction by periodic structures. For this reason the acoustical
modeling method, with specific reservations due to the perpendicularity of the eIectrical and
magnetic fieIds, is equally applicable to the investigation of the scattering of electromagnetic
waves by ocean surface waves and irregular features of the local topography.
The method for investigating sound scattering by an uneven surface has been described
in a number of papers. The measurement of the amplitude and phase of the scattered waves
* The total length of the tank was 2 m, and the length of the furnace was 40 cm. The transmitter
was set up at the edge of the tank in the area not covered by the electric furnace.
56
MODELING OF SOUND PROPAGATION IN INHOMOGENEOUS MEDIA
M
Fig.42. Diagram of the setup for
recording the directional characteristic of sound scattering by a
model. S) Sound source; P) receiver; M) scattering model.
[CH.3
and their scattering directions begins with the selection
and design of the model of a surface with definite parameters. Then the sound source is oriented so that the
angle of incidence of the ray will have a desired value.
For a given setup of the model and wave source the
field is measured, if necessary, at several frequencies.
The same measurements are carried out atother angles
of incidence and for other models.
The distances at which it is required to place the
transmitter and receiver relative to the surface model
depend on the desired objective of the experiment.
The easiest to investigate is the scattering of sound by a model surface with irregularities
distributed according to a periodic law. In [35, 37, 96] the field scattered by surfaces having a
sinusoidal and sawtooth profile has been investigated.
In [40] the following condition is deduced on the diffraction resolution of the sound field
into individual beams in the case of periodic uneven surfaces of finite dimensions (the condition is readily verified in the model of [38]):
LA
r»-/.-,
(14.1)
where r is the distance from the receiver to the scattering surface, L is the length of the scattering area, A is the spatial period of the irregularities, and A. is the sound wavelength.
Condition (14.1) is fulfilled if L» A. and L» A and all parts of the scattering surface
are situated in the Fraunhofer zone relative to the source.
Distances r < LA/A. correspond to the geometrie domain. It is important to know the
characteristics of the scattered field and reflection coefficient in this domain with regard to
the investigation of sound scattering by the sea surface.
In order to record the directional scattering characteristics one must be able to move the
sound receiver around a circular arc with the model fixed at the center of the latter (Fig. 42).
If the model is floating in the water, the receiver must be moved about in a vertical plane
through the sound source, insofar as possible, between 0 and 180°, so as to ensure the reception
not only of forward-scattered signals, but back-scattered signals as welle
The directivity axis of the receiver must be aimed toward the center of revolution during
the measurements in order to obtain the maximum signal.
The sound pressure in the scattered field is measured by the customary procedure for
any acoustic signal. By the use of a circular-scan indicator one can reproduce the scattering
indicatrix directly on the display screen.
In [37] a signal indication procedure is described for the case of a model scattering surface immersed in water and oriented in a vertical plane. The receiver, attached to a metal
truss, was rotated by means of an electric motor with reduction gearing in the horizontal
plane along a circular arc centered at the site of the model. The electrical signal transmitted
from the receiver output was amplified, detected, and set in the form of a rectangular dc pulse
to the input of a pulse amplifier, to which was also sent a rectangular pulse (gate pulse) of
width 0.1 msec from a pulse generator. The gate pulse, which controlled a pulse stretcher,
acted to segregate the required portion of the received signal. From the pulse stretcher the
signal was delivered to a dc amplifier and then to the circular-scan indicator. The field was
§ 14]
SOUND PROPAGATION IN THE PRESENCE OF ROUGH FREE SURF ACE
57
recorded by photographing the display screen of the circular-scan indicator. The rotation of
the cathode ray in the indicator and the position of the receiver were synchronized by means
of a synchro transmission. In this way the experimental apparatus was capable of recording
the scattering indicatrices in apolar coordinate system.
For operation with highly directional transmitters and receivers it is not necessary to
deaden the inner walls of the tank, as in this case discrimination of the useful signals from the
pulses reflected by the walls of the tank is facilitated on the indicator.
The dimensions of the tank· are not too critical in the given problem. However, the small
dimensions of the tank not only require elevation of the acoustic frequency, but also the presence of a model with very small irregularities, which is not always easy to build, because these
irregularities may prove to be commensurate with the intrinsic roughness of the material.
For example, given a scaling factor M = 5000 asea height of 1 m will be simulated by irregularities only 0.2 mm in height. A further reduction in the height of the irregularities is
troublesome. The overall dimensions of the model arrangement can be increased if the scattered field, rather than being measured in a laboratory tank, is measured in an open reservoir
with the stationary model set up near the shore, as for example in [37-391.
The sound pressure in the signal scattered by the model deviates somewhat from the
true value obtained in the scattering of sound by a water-air interface of the same form. This
is because the conditions at the water-model interface, as stated (§ 5), are not entirely consistent with the conditions at the real \vater-air interface. It is necessary, therefore, to correct the measured values of the sound pressure by dividing them by the reflection coefficient
for sound incident at an appropriate angle ona plane reference model of the same dimensions
as the model of the uneven surface.
A model study of sound scattering by a surface with randomly distributed irregularities
is described in [97], in which multistep cork models of the type described above (§5) were used
for the scattering surface, their dimensions being large enough to cause the main part of the
sound beam to impinge completely on the surface. Inasmuch as the profile of the surface was
fixed in time, the model was moved through the water (in 5-cm steps) along the irradiated
area in order to obtain a statistical ensemble. The sound pressure in the reflection beam was
measured in the far zone for fixed angles of incidence (0, 30, 45, and 60°) within a certain frequency band. The pressure readings for various positions of the model (the model was moved
ten steps for every angle of incidence) were used to calculate the mean-square intensity of the
reflected signal. Thus, time averaging of the field was replaced by space averaging.
Using a model surface with irregularities distributed according to anormal (Gaussian)
law, the authors of [971 investigated the dependence of the sound intensity in the specularly
reflected wave on the acoustic frequency and slope and mean-square height of the irregularities,
thus permitting a comparison of the experimental results with the theory of sound scattering by
a surface [661.
It was also demonstrated in [97] that the intensity distribution in the scattered sound beam
could be analyzed to obtain a theoretical estimate of the correlation function of the irregularities
of the reflecting surface; it turned out that all the information conceming the surface is contained in the back-scattered field.
For the given investigation a surface model was used in which the profile and correlation function of the irregularities roughly conveyed the statistical properties of the wave state
of the ocean surface and those areas of the ocean bottom whose relief is to a certain extent
periodic. The analysis is applicable to asound whose wavelength is much greater than the rms
amplitude of the irregularities (in the experiments the acoustic frequency was about 80 kc).
58
MODELING OF SOUND PROPAGATION IN INHOMOGENEOUS MEDIA
[eH.3
However, this analysis is inapplicable for natural measurements on the sea, as ocean waves,
unlike the model rough sea surface moved through water, comprise a stationary random
process.
The time, space, and frequency correlations of the fluctuations of signals reflected at
various angles from a randomly rough surface covering a large number of Fresnel zones have
been investigated on a model whose profile and statistical properties were elose to the characteristics of the ocean surface [1]. This case is realized in actual situations involving the reflection of sound waves from the waveform surface of the sea.
In [84] experiments are described in which the scattering coefficient was measured in
water for sound at frequencies of 100 and 200 kc scattered by a model uneven surface whose
relief fit a normal distribution of irregularities, where the height of the latter was greater than
the radiated wavelength (the maximum vertical distance between the peak of the irregularity
and the trough was approximately equal to four wavelengths).
The foam plastic model represented a scaled-down copy of arelief map of the earth's
surface obtained from aeromagnetic recordings and, in the author's opinion, afforded a good
model of an uneven ocean bottom. It is shown in the paper that there is good agreement between the experimental data and the results of calculations based on the theory of Eckart [66] on
the assumption that the sound pressure in the material of the scattering model is equal to zero.
Experiments can also be set up in the laboratory tank to investigate the field scattered
not only by a stationary model, but also by waves on the surface of the water [76]. During the
experiment a narrow sound beam (of approximate width
is reflected from the uneven surface of the water. The signal received in the back-scattering direction is compared with the
incident signal on the surface. The experiment is depicted schematically in Block diagram 5,
in which the following nomenelature is used: 1) noise generator; 2) narrowband filter; 3) signal
generator; 4) transmitter; 5) receiver; 6) demodulator; 7) ac amplifier; 8) loudspeaker; 9)
wave height indicator; 10) dc amplifier; 11) multiband tape recorder.
3'
[J
I
[J
11
Block diagram 5.
The sound source transmits continuous sine waves (about 1.5 Mc) or signals modulated
with Gaussian noise in 'a selected frequency band (20 and 150 cps, respectively). The amplitudes
of the incident and reflected signals and height of the water waves near the scattering region
are recorded simultaneously on the tape recorder.
§l4]
SOUND PROPAGATION IN THE PRESENCE OF ROUGH FREE SURFACE
59
The form of the reflected signal envelope is analyzed concurrently with the statistical
properties of the wavy surface in order to calculate the probability density function of the amplitude in the scattered wave, to find the correlation between the form of the incident and reflected wave envelopes, and to determine certain other statistical characteristics of the scattered field.
Model measurements of sound scattering made it possible, on the one hand, to test the
theory of sound diffraction by an uneven surface [19,28,41,42,59, 66] within the limits of
its applicability, as is done in [35,37,96,97], and, on the other hand, to investigate experimentally the influence of an uneven surface on the sound field in cases that are difficult to
analyze theoretically (sound scattering by waves with a steep slope [72, 88] or with aperiod
commensurate with the acoustic wavelength; the field in the near zone, where the spectral
beams are unresol vable; etc.).
Influence of an Uneven Surface on Sound Propagation in the Sea.
It follows from straightforward geometric notions that an uneven surface, owing to the scatter-
ing of sound waves by it,. promotes the transfer of energy from the surface layer deeper into
the medium. This produces additional sound attenuation in the given layer.
The strongest scattering is experienced by rays impinging on the uneven surface at a
steep angle. With the stipulation that the sound wavelength be smaH in comparison with the
height of the irregularities (A), the limiting angle of incidence for specular reflection is determined by the expression ([57], p. 47)
At steeper incidence (e > ek) the wave is reflected diffusely. Consequently, the most
favorable conditions for sound scattering by the sea surface occur in a medium invested with a
n~gative sound velocity gradient near the surface, as in this case refraction causes the rays to
impinge on the surface at small grazing angles.
All other conditions being equal, the maximum scattering by the surface is experienced
by rays refracted in a surface sound channel.
Figure 43 illustrates the effect of a periodic uneven surface of trochoidal profile on the
fieId in a homogeneous medium. The sound amplitude was recorded by means of a camera
attachment to an oscilloscope (§6) and corresponds to different values of the parameters of a
foam plastic model of the free uneven surface. The last of the records illustrated was made for
a plane boundary. The distance from the transmitter is plotted on the horizontal axis in meters.
The acoustic frequency was equal to 530 kc, and the depth of the transmitter and receiver was
6 cm.
In the case of a medium with a negative sound velocity gradient, especially for a shallow
homogeneous surface layer (parameter s < 1), the wavy surface has little effect on the diffraction field in the shadow [12], but induces amplitude modulation with the spatial period of the
wave in the illuminated zone.
Spatial amplitude modulation, which is a consequence of the variation of the path difference
between the direct and reflected rays from the oscillating waveform surface, is easily observed
in the intensity level recordedon a logarithmic recorder.
Let us now examine in closer detail the sound field in a surface channel with an uneven
free surface.
The surface bounding the sound channel, by scattering sound, decreases the range of s ignal transmission and changes the angle of arrival of the rays at a point in the illuminated zone.
MODELING OF SOUND PROPAGATION IN INHOMOGENEOUS MEDIA
60
[CH.3
a
2
b
2
c
2
d
2
Fig. 43. Influence of an uneven surface on the field in a homogeneous medium.
a) A = 16 cm. A = 2 cm; b) A = 16 cm. A = 1 cm; c) A = 4 cm. A = 1 cm; d)
plane bound ary •
In the. investigation of the scattered sound it is recommended that a receiver with a sharp directivity pattern be used.
Unlike the sound receiver with a flat directivitypattern. which reacts to the total field,
the highly directional receiver enables one to measure the sound pressure separately in each
rayhitting a given point. The direction of the central axis of the receiver when oriented for
maximum reception is adopted as the direction of the ray in this case.
Consequently, the directional receiver can be used to determine the direction of arrival
of ~he rays at a point. This is valuable. in particular. for the separation of surface-scattered
sound from the direct sound.
Experiments performed by the author using the modeling method indicate that the average
level of the sound intensity in a channel bounded by an uneven surface decreases with distance
from the transmitter more rapidly than by a cylindrical law. The magnitude of the sound attenuation in the channel depends on the properties of the latter. Thus, an increase in the ver-
§l4]
SOUND PROPAGATION IN THE PRESENCE OF ROUGH FREE SURFACE
61
a
3
2
b
~
...
"
~
.
2
c
2
Fig. 44. Oscillograms of the field in a surface channel. a) Plane boundary between
water and air; b) model surface with randomly distributed irregularities (correlation
radius 1.53 mm, rms roughness height 0.26 mm); c) model periodic surface of trochoidal profile (A = 40 mm, A = 4 mm).
tical sound velocity gradient, resulting in strong positive sound refraction near the surface,
promotes the scattering of sound by surface irregularities.
The uneven surface not only changes the range of sound transmission in the channel, it
also alters the structure of the field, which be comes more diffuse. This is apparent, for example, in the oscillograms of Fig. 44, which show the sound pressure in a channel as a function of the distance in meters (on the horizontal axis) from the transmitter. In oscillogram a,
which corresponds to a plane water-air interface, the shadow and caustic zones are distinctly
marked. In oscillograms b and c, which represent the field in the presence of a model uneven
surface under the same experimental conditions, the caustics are not clearly pronounced. The
measurements were carried out at a frequency of 3 Mc, and the depths of the transmitter and .
receiver were 5 and 2.5 cm.
The results of the measurements at individual points of the channel using a highly directional receiver indicate that not only are the directions of the rays arriving at a point changed,
but new rays appear as well.
For this reason the scattered field partially illuminates the shadow zone.
Entropy Variati.on as a Measure of the Distortion of a Sound Signal. Using an underwater sound channel for the transmission of information, it is important
to know how and where the signal is distorted under the influence of noise. The capacity of the
channel depends on the degree of signal distortion. Entropy variation in the channel may serve
as an indicator of the degree of signal distortion.
The author has investigated this problem under model conditions. The sound channel
represents a linear system in which multiray propagation takes place. The parameters of the
multiray channel include the number of rays, the sound attenuation along the ray, and the delay
of one ray relative to another. In the case of slow variation of the channel parameters with time
it may be treated as a system with constant parameters [26]. Then the channel is represented
as a single-channel linear filter with noise that induces nonuniformity in its frequency characteristic. On this basis it is admissible to use statistical communication theory [64] to estimate
the channel capacity •
62
MODELING OF SOUND PROPAGATION IN INHOMOGENEOUS MEDIA
[eH. 3
In an underwater channel the refraction of sound and concomitant multiray character of
the channel introduce noise into the signal transmission due to the superposition of pulses
arriving at the reception point via different paths. Another source of noise is the scattering
of sound by volume and surface inhomogeneities of a localized character.
Under the conditions of the model experiments described here the random inhomogeneities
(density, temperature, etc.) are small and are disregarded. It is assumed that sound is scattered only by surface irregularities.
A measure of the variation of the capacity of the noisy channel, according to statistical
communication theory, is the channel information loss, which depends on the frequency characteristic of the channel. The average information acquired from a set of independent messages
is called the entropy of the probability distribution in the message.
According to [23], the entropy is defined in terms of the probabilities Pi of a set of transmitted messages as
n
ff n =
-
~ p;1ogp.,
1=1
and the entropy variation in the channel is expressed in terms of the frequency characteristic
of the system I(f) as follows:
,
1 ":
I:J.Hn=H n - H"=-WJlog21I(f)] dj,
(14.2)
o
where Hn and H~ are the entropies at the channel input and output, respectively, and w is the
frequency band of the signal.
The quantity .6.H n depends on the properties of the filter, but not on the transmitted signal ensemble. In the case of zero noise I(f) = 1, and Hri = Hn , i.e., the signal passes through
the filter without distortion. For an assessment of the quality of information transmission under model conditions the frequency characteristic is ascertained at several points of the channel for various surface states of the liquid.
The electroacoustical instrumentation for the model experiments comprises two sections:
a conventional pulse transmitting and receiving subsystem, which permits the received pulses
to be viewed on an oscilloscope screen, and a subsystem for observation of the frequency characteristic of the signal at any point of the medium.
The first subsystem is used to detect the characteristic zones of the channel (caustic,
shadow). The basic principle of the second subsystem for the measurement of the frequency
characteristics entails the following. A voltage whose frequency is linearly modulated (say,
at a repetition rate of 25 cps) within the limits of particular investigated portions of the characteristic is sent to the channel input. The output voltage from the channel is detected and
delivered to the vertical-deflection plates of the oscilloscope, to the horizontal-deflection
plates of which is sent a voltage of the same shape, phase, and amplitude with a modulation
voltage. The oscilloscope screen displays the envelope curve of the output signal, which in the
event of sufficiently slow frequency variation represents the frequency characteristic of the
investigated signal.
The original experimental data for determining the entropy variation in the channel are
afforded by photographie records of the frequency characteristics obtained at various points of
the channel. These curves are used in conjunction with Eq. (14.2) to calculate .6.H n , which is
U4]
a
SOUND PROPAGATION IN THE PRESENCE OF ROUGH FREE SURFACE
63
equal to the entropy difference in the frequency
band w at the channel input and its output at a
definite point. *
c
Next the relative entropy variation is
found:
MI=2010g ~~~ ,
b
d
where AH o is the entropy difference at a particular chosen point.
Fig. 45. Frequency characteristics recorded in a surface channel. a) Channel
bounded by plane water-air interface,
data recorded at a distance r = 40 cm
from the transmitter (near the first caustic); b) the same at r = 60 cm (at the first
caustic); c) channel bounded by a model
periodic surface (A = 20 mm, A = 2 mm).
r = 50 cm (near the first caustic); d) the
same at r = 100 cm (between caustics).
Sampie contours of several frequency
characteristics obtained by the author in a
channel at various distances from the transmitter are shown in Fig. 45.
The vertical distribution of the velocity
of sound in the given channel is characterized
by the following values of the relative velocity
gradient:
a = 0.1 m- 1 from the surface to a depth
z = 2.5 cm;
a = 1.1 m- 1 for 2.5 cm:s z :s 13 cm;
a = 0 for z > 13 cm.
The total depth of the water in the tank was 15 cm.
A noticeable feature is the relatively smooth form of the frequency characteristics at the
caustic and their erratic behavior at other points. This dependence of the frequency characteristic on the structure of the field in the channel is attributable to the fact that in the vicinity
of the caustic separate pulses arrive in phase. so that the signal distortion is slight. At other
points the multiray quality of the propagation strongly distorts the signal.
Numerical data obtained by the author. indicating the entropy variation (in decibels) at
different points of the given channel relative to the entropy value at the first caustic for a
plane liquid surface, are presented in Table 4.
The measurements were performed at a frequency of 900 kc and a pulsewidth of 20 Il-sec.
Following is a summary of the results of an analysis of the model experiments:
1) The minimum entropy variation is observed at the caustics, the first caustic being
the most propitious in this respect.
2) As a result of sound scattering by surface irregularities, beginning with a certain
distance from the transmitter, the entropy variation increases on the average with distance
from the transmitter; this indicates a violation of the field structure typical of channels (caustics, shadow zone).
*In plotting the curve for log2 II(f) I it is essential to observe the normalization condition, which
UI
dictates that the integral
S/(J)df •
o
which is proportional to the "total power of the process"
in the given frequency band, remain constant at all points of reception.
64
MODELING OF SOUND PROPAGATION IN INHOMOGENEOUS MEDIA
[eH. 3
TABLE 4
C':>
'0
@
C"J
Type of surface
t::
CI)
oe
<ll
1'0
'"
()
...
Plane
P eriodic with trochoidal
profile
A=20 rnrn
A=2 rnrn
o
5.8
1.4
2.5
7.8
2.8
'"
()
~
;:I
I 5.1 I 2.0
7.5
5.1
6.2
3.3
7.5
8.0
These conclusions make it possible to ascertain the zones in which the signal is transmitted with minimum distortion if the structure of the field is known. On the other hand, from
the entropy variation measured experimentally at various points one can assess the structure
of the sound field along its path of propagation.
§ 15.
Modeling of Media with Random Inhomogeneities
During the propagation of sound in the ocean fluctuations are observed in the amplitude
and phase of the sound signals. These fluctuations are elicited predominantly by random spacetime temperature variations in the water, which correspond to random variations of the acoustic refractive index in the medium.
Random temperature inhomogeneities formed in the medium (temperature microstructure) scatter sound transmitted through the medium.
An analogous effect is observed during the propagation of sound and electromagnetic
waves in an inhomogeneous atmosphere.
The majority of inhomogeneities have a linear scale greater than or. at best, of the
same order of magnitude as the radiation wavelength. At short distances the inhomogeneities
induce geometrical effects, such as focusing and defocusing of the waves. At sufficient distances from the wave source rays traversing different paths in the inhomogeneous medium interfere with one another. The phase and amplitude of the signal vary in time and space according to a probabilistic law. which correlates with the space-time variation of the acoustic
refractive index in the medium.
Laboratory investigations of the principles governing wave propagation in a medium with
random inhomogeneities are best conducted on the basis of the considerations set forth in the
beginning of the book. In particular. the theory is tested more expediently and at lower material
cost in model experiments than in field investigations of sound propagation in the ocean or radio
waves in the earth's atmosphere.
An underwater sound tank for model studies of the sound field in a given medium must
have, over and above the equipment described previously, an apparatus for the creation of random inhomogeneities in the medium.
The problem can be solved by two conceptually dissimilar techniques: 1) the generation
of turbulence containing local inhomogeneities in the water, where the refractive index fluctuates in space and time [78, 89, 100]; 2) the use of a statiopary model of a medium containing
randomly ·distributed volume inhomogeneities, plus the creation of statistical ensembles with
the movement of asound ray in the medium [87].
In the first modeling technique temperature fluctuations are formed in the medium either
by the introduction of water into the tank through the bottom at a temperature higher than that
§l5]
MEDIA WITH RANDOM INHOMOGENEITlES
65
in the tank or by heating of the water with an electric furnace in the lower part of the tank.
In both cases vertical thermal convection is set up in the medium.
In [78] a tank is described in which hot water at a temperature of about 45°C is injected
through a grating with several openings built into the bottom of the tank. The temperature of
the rest of the water in the tank is 19°C prior to the beginning of the experiment.
The electric furnace used in other arrangements comprises a heating grid of insulated
copper wire. In [89, 100] a heater is described which consists of a wire 600 m in length and a
power rating of 18 kW. The h~ating grid is suspended at a distance of 30 cm over the bottom of
the tank. The local temperature gradient formed between the wires of the grid also promote the
formation of microinhomogeneities in the medium. The temperature of the heated water is
regulated by the current through the heater.
For measurement of the water temperature at different points thermocouples or thermistor thermometers are used , being hooked up as one arm of a \Vheatstone bridge. The
average vertical drift rate of the hot water in the tank is determined from temperature graph8
recorded simultaneously by thermocouples arranged vertically. These graphs are also used
to calculate the spatial correlation of the temperature in the vertical direction. Thermocouples
oriented in the horizontal plane are used to acquire data on the spatial correlation in the horizontal direction.
The measurements of the temperature microstructure and acoustical measurements were
carried out one or two hours after the heater was turned on. In this period, as tests showed,
the turbulence terias to become approximately uniform, remaining that way for several hours,
and the average temperature in the water increases slowly and uniformly. After a certain
amount of time the water temperature equalizes throughout the entire volume.
The uniformity of the turbulence is tested by measuring the mean-square temperature
deviation at various points of the medium from the mean temperature at the same points.
The region in which the mean-square temperature deviation is constant determines
the scale of the uniform turbulence. The mean-square deviation and radius of correlation of
the temperature prove roughly constant in a horizontal layer of adequate thickness for complete
confinement of asound beam. With increasing height over the heater these variables slowly
change, making it possible to measure the fluctuations of the sound field in media with different microstructural parameters.
The acoustic refractive index in the turbulent medium varies between the limits from 0.995
to 1.005.
The electroacoustical instrumentation used for the investigation of the sound field in the
indicated medium has the same operating principle as that described before.
A sound signal transmitted through the medium is picked up by nondirectional sound receiver, transformed into an electrical signal, amplified , and sent through a peak detector and
on to an oscilloscope , where it is photographed from the screen. An image of the signal is
obtained on the film. Pulses are transmitted by the sound source at a proper frequency for
ensuring the transmission of each subsequent pulse through the somewhat changed medium (for
example, one pulse after 1 or 2 sec). This forms a statistical ensemble of sound pulse peaks I
which is used for processing of the results of the sound field measurements.
The procedure described above for the modeling of media with randoI? inhomogeneities
has been used in [89, 100] to test the dependence of the coefficient of variation of the pressure
in the sound pulses on the acoustic frequency and separation of the source and receiver. The
coefficient of variation V, defined by the relation V 2 =
1p I~ 1P 1_ 2, depends on the cor-
(/Pf2 -
66
MODELING OF SOUND PROPAGATION IN INHOMOGENEOUS MEDIA
[CH.3
relation function of the acoustic refractive index in the
medium, and for a Gaussian correlation function, as shown
by the theory, varies as the square root of the distance
0
00
[92]. This result has been corroborated experimentaUy.
The sound field in a medium has been investigated [100]
Fig. 46. Diagram of the modelat a distance of 90-270 cm from the transmitter. The
ing of a medium with random inlower distance limit was dictated by the fact that the
homogeneities (viewed upward
measurements were to be performed in the far zone, while
through the tank). S) Sound
the upper limit was set by the dimensions of the heating
source; P) receiver; 0) axis of
surface, because all the measurements were made over
rotation.
the central area of the heater. The sound field was determined by means of five receivers suspended at a fixed
distance from the transmitter. The outputs of aU the receivers were connected to oscilloscopes.
The signal transit time along the path of propagation was much shorter than the period of time
in which the properties of the medium could vary significantly. Consequently, the coefficient of
variation for each of the five distances could be ascertained from data taken at equal time intervals.
000000000
000
0
0 0
0 0 000
0
S *-u-Ov ~ -<l r:J>-o-oo-O-<>-*-P
o 0 0
0 0
0
0000
o
00
0000
0
In estimating the experimental error, it is important to realize that the departure of conditions in the medium from statisticaI uniformity introduces error into the measurement results, because the parameters of the temperature microstructure are determined at one point,
while the field fluctuations are produced by scatterers along the entire path from the sound
source to the receiver. Moreover, the results of measurements not too far from the heater
(closer than 30 cm) are affected by refraction of the sound beam due to local thermal gradients.
In [78] a similar procedure has been used to measure the width of the transmission band
of a medium containing random inhomogeneities. Suppose that two continuous sound waves* are
passed simultaneously through the medium, one wave having a constant frequency and the other
varying above or below that frequency. The sound signals of different frequencies picked up by
the receiver are recorded separatelyon two channels of a recording device. At the reception
point the correlation coefficient of the signals received at the two frequencies is determined.
If the frequencies cOincided, the correlation coefficient would be equal to one. With a disparity
between the frequencies it assurnes a value of 0.5. The given frequency difference is characterized by the transmission bandwidth of the medium. In [78] the constant frequency was 500 kc,
and the bandwidth of the medium turned out to be 250 kc in the frequency range below 500 kc,
and at higher frequencies it was 750 kc.
In another technique [87] for the propagation of waves through a medium containing
random inhomogeneities the latter were created by means of thin-walled elastic (neoprene with
latex) spherical balloons 10 cm in diameter, filled with a mixture of glycerin and water and
immersed in pure water (Fig. 46). The velocity of sound in the mixture was 0.1 to1% higher
than in'the water, depending on the glycerin content.
A total of 100 baUoons were prepared, lots of ten being filied with a mixture of identical
composition. Since the balloons were heavier than water, they could be suspended inside the
tank; they were placed so as to form a two-dimensional random configuration. In [87] a procedure is described for the measurement of the amplitude and phase of plane waves in the
inhomogeneous medium described above. In different experiments the acoustic frequency was
varied over the range from 124 to 240 kc; the separation of the transmitter and receiver was
varied from 90 to 360 cm. For a fixed separation both transducers were rotated simultaneously
*The tank was deadened with a mat of treated horsehair.
§l6]
MODEL EXPERIMENTS WITH INTERNAL WAVES
67
through 360 in the horizontal plane about the hypothetical vertical axis through the middle of
the line joining the two transducers. This procedure made it possible to obtain different statistical ensembles for the variation of the refractive index along the sound ray.
0
For each separation of the transmitter and receiver the amplitude of the received signal
was recorded. The amplitude records made for a large number of ensembles at given frequencies and transducer separations were used to compute the ensemble average of the pressure amplitude, effective radius of sound scattering by the inhomogeneities, and certain other
statistical variables.
The method of modeling by means of stationary inhomogeneities affords greater reliability with respect to knowledge of the size and shape of the inhomogeneities than is afforded by
the simulation of thermal turbulence in the medium. Fixed inhomogeneities have a great deal
in common with the reallarge-scale inhomogeneities encountered in the ocean and atmosphere,
even though they are not exactly consistent with the standard postulates of the theory of wave
scattering by inhomogeneities (the medium in the tank is not strictly isotropie and homogeneous;
the Gaussian autocorrelation function postulated in theory does entirely correspond to either
the model or the modeled medium; the waves cannot be regarded as perfectly plane; etc.).
Nevertheless, the results of the model experiment exhibit good agreement with the theory of
wave propagation in a medium containing random inhomogeneities distributed according to a
normal law [92], thus indicating the broad domain of its applicability.
Consequently, our experience to date in the modeling of media containing random inhomogeneities permits us to conclude that acoustical modeling has very definite advantages both
in the investigation of the statistics of the medium itself and in the investigation of the statistics
of the scattered signal. In particular, acoustical modeling should prove useful for verification
of the theory of scattering primarily of radio waves in the troposphere. It is necessary in this
case, however, to recognize that the results of acoustical modeling can only be applied in a
qualitative sense to the propagation of radio waves in the atmosphere, because the variation of
the acoustic refractive index in the turbulent model medium turns out to be much greater than
the fluctuations of the refractive index for radio waves in the atmosphere.
§ 16.
Model Experiments with Internal Waves
Internal waves refer to gravity waves having a larger amplitude in the depths of the water
than on its surface [30]. They give rise to distortion of the isotherms and,accordingly, of the.
surfaces of constant sound velocity. The periodic deformations of the constant-velocity surface,
in turn, cause a variation of the field geometry (variation of the shadow boundaries, focusing
and defocusing of the rays, etc.).
A statistical analysis of the temperature fluctuations under model conditions* in a zone of
abrupt negative temperature change (~tOI ~z = -4 deg/cm) indicates the onset of internal waves
under these conditions as welle The situation is illustrated in Fig. 47, which shows the autocorrelation coefficient of the temperature fluctuations observed at a point in the vicinity of the
temperature discontinuity (the rms deviation of the temperature in the given case was 0.04").
The autocorrelation dependence of the temperature fluctuations is adequately described by a
function of the form
R ('t) = e- (':'0)2 cos a't ,
where a and TO are constants characterizing the process.
* The investigations were carried out in the laboratory of the Acoustics Department of Gorky
state University with the assistance of D. A. Selivanovskii and Yu. N. Cherkashin.
MODELING OF SOUND PROPAGATION IN INHOMOGENEOUS MEDIA
68
10
JOt, sec
20
F ig. 47. Generation of internal waves
in a temperature discontinuity layer.
~. -=~::;-;~---;~~:j~o;5~=~~=~~~-;
:;:- --
.-- - - - - - -- 2::--- _-_-1.6~ -~----
---
--------
.---- ~ - - -- -- -':::2""=
--- ------- ---
=----: _ -......-_--:_--------=
~~
~-_:----
Fig.48. Diagram of an experiment
for the investigation of sound scattering by an internal wave. 1) Tank;
2) water; 3) turpentine; 4) piston
vibrator for the generation of an internal wave; 5, 6) identical acoustic
transmitter and receiver with a ±10°
principal directivity lobe at 500 kc.
[eH. 3
Besides the internal waves produced in a temperature discontinuity layer by a natural uncontrolled
process, internal density waves can be simulated
artificially at an interface between two immiscible
liquids.
A schematic diagram of an experiment set up
to investigate the scattering of sound by an internal
wave [13] formed at a water -turpentine interface is
shown in Fig. 48 (the difference between the acoustic refractive indices at the interface is ~n = 0.15).
The experiment was carried out in a bottomless
tank set up near the surface of the water in an underwater acoustical pond •
Suppose that the transmitter sends a narrow
grazing beam of sound pulses along the interval
wave. Each pulse as it propagates successively
covers longer segments of the internal wave from
the transmitter and is scattered by those segments.
Therefore, the scattered signal picked up by the
sound receiver in the direction of detection has a
duration twice the transit time of the pulse along
the entire internal wave, the remaining segments
of the received signal corresponding to the field
scattered by different zones of the internal wave,
the near zone and farther away.
For the measurement of the signal scattered by the internal wave the same procedure is
used as for the investigation of the signal scattered by an uneven surface (§ 14).
Another method can be used to study the signal scattering. It is based on the fact that
the interna! wave modulates the amplitude of a pulse reflected from it, so that the spectrum
of the scattered field contains, in addition to the fundamental. two additional satellite frequencies.* If the internal wave has a random character. the amplitude of the scattered field
will be modulated by a random function. For observation of the effect of sound modulation by
the interna! wave a standard signal generator, mixing device, and selective intermediate amplifier are connected into the receiving section of the circuit. The receiving device also contains a gate pulse stage, which is used to display any desired portion of the received signal on
the oscilloscope.
For measurement of the field scattered by a particular segment of the interna! wave, distance marks are projected on the oscilloscope screen to ascertain the position of the portion
of the scattered signa! corresponding to that segment. The magnitude of the scattering dp is
estimated from the mean level of the received portion of the signal. The scattering magnitude
is characterized by the ratio of dp to the pressure Po of the wave incident on the scattering
area of the surface. For the determination of Po the receiver is placed in the position of the
scattering surface area.
* This effect is analogous to the Raman scattering of light.
§ 17]
SOUND SCATTERING BY BODIES IN WAT ER
69
In another method for measuring the scattered field the gated portion of the received signal is added to the signal from the standard generator in the mixer; the resultant differencefrequency signal is transmitted through an if filter and amplified.
The ensuing signal frequency characteristic is calibrated. This is done with two standard signal generators, one of which is used to generate the fundamental frequency in the signal, the other for modulation corresponding to the influence of an internal wave. Now in the
mixer the signal from the receiver is replaced by the modulated voltage from the two standard
signal generators.
By the proper choice of voltages on the generators it is possible to obtain the same width
on the part of the frequency characteristic as in signal transmission from the sound receiver.
By measurement of the voltage at both standard signal generators the magnitude of the sound
scattering by the internal wave is estimated.
With a certain change of the experimental conditions, using the same arrangement, one
can study the variation of the width of the sound beam during its transmission through the internal wave [14], In this oase a transmitter with a narrow directivity pattern (l/Jo< 10") is
placed in the turpentine, where its transmits sound pulses to the water boundary. The sound
field is measured with a receiving transducer identical to the transmitter, the receiver being
moved across the beam in the horizontal plane.
The width of the sound beam in the water after refraction by the internal wave is compared with the width of the beam refracted by a plane stationary interface between the media.
During the experiment the sound intensity exhibits a ripple, which is attributed to the alternating focusing and defocusing of sound by the internal wave as it intersects the beam. The oscillations of the sound intensity level in the beam due to the internal wave can thus be measured.
§ 17.
Sound Scattering by Bodies in Water
The scattering of plane waves by bodies of simple configuration (sphere, long cylinder)
in a homogeneous infinite medium has been studied on many occasions. Rayleigh gave a theoretical solution of the problem for the limiting case of sound scattering by a sphere of small
diameter relative to the sound wavelength ([59], p. 334). Later the problem was solved without
the stipulation of small radius on the part of the sphere or cylinder [48]. There is no general solution to the problem of the field scattered by bodies of arbitrary configuration. However, the investigation of wave scattering by bodies of various configurations has significant
bearing on many branches of science and technology. In particular, underwater acoustics is
concerned both with signals reflected from solid objects of various configurations and with the
field scattered by liquid spheroidal scatterers, i.e., by so-called local density and sound velocity microinhomogeneities distributed throughout the oceanic medium according to some
stochastic law. In acoustics, as in the theory of electromagnetic wave propagation, scatterers
whose dimensions are commensurate with or greater than the wavelength play the most crucial role.
Given a proper choice of modeling scale, the scattered field can be investigated under
model conditions. The acoustical modeling of wave scattering by bodies is also used for the
investigation of the radar detection of objects of diverse configuration [91].
Laboratory experiments on the scattering of sound have been conducted by and large on
spherical and cylindrical bodies. The main problem of all the investigations was to test the
existing theories and, concomitantly, to explain certain properties of the scattered field unaccounted for in the theoretical solution of the problem. As a rule, in all the experiments polar
70
MODELING OF SOUND PROPAGATION IN INHOMOGENEOUS MEDIA
[eH. 3
diagrams (indicatrices) of the scattered field have been recorded and the back-scattering of
sound (echo signal) measured.
These problems are dealt with in [44, 70, 73, 80, 81, 83], in which experiments are described on the scattering of sound waves by cylindrical and spherical bodies.
The scattered field is measured in underwater acoustical ponds and tanks. The principal
mechanical equipment in the tank consists of a fixed holder for the source, a holder for the
reflecting object, and a holder for the hydrophone. The latter is suspended on the end of a
movable arm, which can be rotated about the vertical axis through the point of suspension of
the fixed object. With rotation of the arm the hydrophone measures the sound field scattered
by the object at the same distance in different directions.
In [70, 81] this procedure was used to measure the azimuthal characteristics of the field
scattered by spherical and cylindrical bodies. The distance from the transmitter and hydrophone to the scattering object was 3 m, which meant that the measurements were performed
well inside the far zone of the field at the experimental frequencies (50 to 150 kc). The angular
width of the sound beam from the transmitter was between 5.7 and 16°, depending on the fre,quency. The dimensions of the scattering object were chosen with a view toward the possibility
of studying the scattering of sound by the bodies over a wide range of variation of the dimensionless variable ka (k = 27r/>"; a is the radius of the sphere or circular cylinder).
This quantity was varied from 4.1 to 57 for the sphere and from 0.33 to 24.4 for the cylinder, i.e., the scattering was investigated both on bodies that were both smaU and large relative to the sound wavelength. The bodies were suspended on one or two fine nylon filaments
(about 1 mm thick), which were acoustically transmissive. For the investigation of the field
scattered by a cylinder the latter was suspended by its longitudinal axis or perpendicularly to
the axis. The field was determined in the plane perpendicular to the cylinder axis or in the
horizontal plane through the axis in various orientations relative to the direction of the incident beam.
The arm on which the hydrophone was mounted was revolved by an electric motor about
the scattering body at an angular velocity of one revolution every ten minutes.
The depth of the transmitter, receiver, and center of the obstac1e was 3.6 m; the distance
of the measurement plane from the bottom of the tank was 12 m. As a result of this arrangement of the objects and the relatively narrow directivity pattern of the transmitter it was not
necessary to deaden the tank.
The scattered field was displayed on the screen of an oscilloscope and recorded on an
automatie level recorder with a circular scan, which produced a direct polar diagram of the
scattered field.
In the experiments described above the scattering was investigated as a function of the
frequency, scattering angle, and material of the scattering object for pulses of various widths.
It was demonstrated as a result of the model investigations, for example, that metal scattering bodies (aluminum, brass) cannot be regarded as perfectly solid, because they transmit
an appreciable amount of energy, complicating the structure of the scattered signal by virtue
of wave interference.
The intensity of an echo signal from a sphere for large values of ka is independent of the
acoustic frequency and is a Httle less than the value calculated on the assumption of a perfectly
solid medium.
In the case of long pulses (pulsewidth greater than the acoustic length of the sphere diameter) the echo intensity fluctuates strongly with variation of the frequency, indicating inter-
§l7]
SOUND SCATTERING BY BODIES IN W ATER
71
ference between the sound reflected from the outer and inner
surfaces of the sphere. This effect is not observed in the transmission of short pulses.
Fig. 49. Diagram of an experiment on the scattering
of sound by an obstacle with
acoustical properties similar to those of the surrounding medium (viewed from
above). 1) Transmitter;
2, 3) main and auxiliary hydrophones; 4) scattering
sphere.
The back-scattered field of long pulses (within the angular
range of 180' is highly erratic. The back-scattered field of
short pulses, on the other hand, is fairly uniform within the
same angular limits.
The experiments on the scattering of sound by a cylinder
revealed that the field distribution in the plane perpendicular
to the long axis at the half-way point is independent of the length
of the cylinder; this indicates that the longitudinal vibrational
modes of the cylinder do not affect the field scattered perpendicularly to the axis. On the other hand, an analysis of light
pulses reflected from a cylinder stipulate the existence of surface waves propagating around the outside surface of the cylinder.
The polar diagrams depend on the length of the cylinder, length of the pulse, and the quantity ka. The scattered field of long pulses also depends on whether the cylinder is solid or
hollow. The strongest echo comes from points where there is a sharp variation in the slope
of the reflecting surface and a discontinuity (end of a hOllOW open cylinder).
In similar experiments the signal phase and, hence, the geometry of the wave front in
the presence of a scattering obstacle are determined [73].
The experimental arrangement for investigating scattered fields has been used to model
the scattering of sound in the ocean by fish tissues and certain elastic and plastic materials [83].
An interesting object of investigation is the scattering of sound by obstacles whose acoustical properties are similar to the surrounding medium. This has bearing on the model investigation of the scattering attributes of turbulence and temperature microinhomogeneities of the
oceanic medium.
Investigations along these lines are described in [82, 86, 87]. The experimental setup is
illustratedschematically in Fig. 49. A transmitter and two identical cylindrical nondirectional
hydrophones, i.e., one main hydrophone and one auxiliary, were placed in a tank containing pure
water (previously treated by chlorination and the sedimentation of suspended particles). The
stationary hydrophones were equidistant relative to the transmitter. A liquid sphere served as
the scatterer, being moved in a horizontal plane about the main hydrophone, which was at the
center. The distance from the transmitter to each of the hydrophones was 1.35 m. The distance from the main hydrophone to the sphere was 25 cm (this being the radius of revolution
of the sphere). The transmitter, hydrophones, and circular trajectory of the sphere were all
in one plane, which was parallel to the surface of the water. In this way the experimental
arrangement was capable of measuring the scattered field in the equatorial plane of the s[here.
The scattering sphere represented a thin-skinned neoprene balloon (6.3 or 12.5 cm in
diameter) filled with liquid. The sphere, filled with water from the tank, did not scatter sound,
indicating the total acoustic transmissivity of the balloon material (at a frequency of 30 kc).
Two liquids similar to water in their acoustical properties were used to fill the balloons:
1) a mixture of ethyl alcohol and 4.7% (by volume) acetyl tetrabromide (the density and
velocity of sound are close to 1 and 0.8 relative to water);
2) . a mixture of 50% (bY volume) glycerin and distilled water (the density and velocity of
sound are equal to 1.15 and 1.2, respectively).
72
MODELING OF SOUND PROPAGATION IN INHOMOGENEOUS MEDIA
Fig. 50. Schematic arrangement
for the analog modeling of wave
fields in inhomogeneous media.
1) Two-dimensional waveguide;
2) horn-type waveguide; 3) tank.
[CH.3
The auxiliary hydrophone was used to isolate the
scattered field by itself from the total field. A cylindrical transmitter with a uniform directivity pattern generated an approximately equal sound field at the sites of
the two hydrophones in the absence of the scattering
sphere. A slight disparity in the readings of the hydrophones could be attributed to the difference in the signals
arriving at the hydrophones after reflection from the walls
and surface of the water.
After the insertion of the scattering sphere the
sound pressure was read at both hydrophones, where in
order to obtain the amplitude and phase of the scattered
signal the readings of the auxiliary hydrophone were
subtracted from the readings of the main hydrophone, and the initial pressure difference at the
hydrophones (in the absence of the scatter) was taken into account.
The experimentally determined polar (azimuthal) characteristics of the scattering from
a liquid sphere concur with the theory[68], attesting tothe applicability of the modeling method.
As the material of the present section indicates, the modeling method has been used for
the solution of a very limited range of problems of soundscattering by obstacles. It is recommended that the applications of the modeling method be broadened, extending it to problems
whose theoretical solution is impracticable. Typical of such problems is the investigation of
sound scattering by bodies of complex configuration in inhomogeneous media (specifically, in
layered-inhomogeneous media with a vertical sound velocity distribution), in a liquid with diverse forms on the part of the free surface and bottom, etc.
§ 18.
Analog Modeling of Wave Fields in
Inhomogeneous Media
In addition to the modeling of inhomogeneous media by the method of space-time simula....
tion, which is realized by means of equipment arrangements that more or less preserve the
features of the media under investigation, there is an alternative approach to ths problem. In
[31-34] an experimental arrangement is described for the investigation of the field in inhomogeneous media by the method of analog modeling in the underwater acoustical tank of the
Acoustics Department at Moscow State Univers ity. The method is based on the analogy between the propagation of waves in regularly inhomogeneous media and in variable cross-section
waveguides. Corresponding to this is the analogy between the propagation of a cylindrical wave
in free space and the propagation of an individual normal mode in a planB waveguide •
The heart of the analog model is a two-dimensional waveguide (Fig. 50) formed by two
aeoustieally high-reflecting surfaces, namely the surfaee of the water and a rectangular platform eoated with a sound-absorbing material and suspended below the water level. A suitable
eoating invested with the attributes of an acoustieally compliant medium is hard sponge rubber
(porous ebonite) 5 mm thiek. For the exclusion of reflection from the edges of the waveguide
the latter has attaehed to its four sides waveguides of horn-type eonfiguration with generatrices
in the form of ellipsoidal ares. The inner surfaees of the horns also have sponge rubber cemented to them. Sound is transmitted through the horns inside the tank, where it is totally
scattered as a result of multiple refleetion from the walls and bottom of the tank, surface of the
water, and undersurfaee of the suspended platform.
The total working area of the platform in the experiments deseribed here was 250 x 400
em 2• The tank had a length of 10 m and an equal width and depth of 4 m.
§l9]
73
ON THE APPLICABILITY OF RAY REPRESENT ATIONS
a
b
z,cm
.Fig. 51. Three vertical profiles of the velocity of sound
(a) and diagram of a diffracted ray in the shadow zone for
antiwaveguide sound propagation corresponding to the vertical profile 2 (b).
The acoustical measurements in the waveguide (measurement of the amplitude and phase
of the sound pressure) were carried out by conventional techniques using a coordinate positioning device.
In waveguides with a constant or variable cross section representing a shallow-ocean
model one or more normal modes can be excited by the transmitter. By varying the form of
the lower boundary of the layer and its angle of inclination relative to the horizontal one can
simulate a variety of inhomogeneous media. In [31], for example, a wedge-shaped waveguide
was used to model a plane-Iayered medium in which the acoustical refractive index depends
only on one coordinate. In [33] the results obtained by Kuznetsov from a detailed investigation of the amplitude and phase structure of the sound field in a liquid wedge are presented.
The application of raster techniques made it possible to assess the efficacy of the given method
of modeling inholnogeneous media.
§ 19.
On the Applicability of Ray Representations
in Underwater Acoustics
Ray representations, which afford easy visualization, serve as the foundation for the
simplest methods of calculating sound fields.
One of the principal limitations on ray acoustics is its unsuitability at grazing angles
that fit the inequality ([20], p. 109)
a"A.
sinB« ( z;-
)1 13,
(19.1)
where a is the relative gradient of the velocity of sound.
This condition is formulated for aplane wave incident on a reflecting layer at a grazing
angle O. In the event the wave is spherical or has a certain directivity pattern in the wave
zone, the above inequality may prove to be unfulfilled for a portion of the sound beam. Then it
is still possible to use the geometrical method for calculation of the field, but only for that portions of the beam.
An analysis of the results of model experiments and a comparison with theory indicates
that the geometrical method in its elementary form (law of divergence of ray tubes) or modified
74
MODELlNG OF SOUND PROPAGATION IN INHOMOGENEOUS MEDIA
[eH. 3
form (method of steepest des cents , quasi-geometrical method - [20], p. 220) is applicable in underwater acoustics for calculation of the field in a weakly inhomogeneous medium (aA« 1);
for calculation of the attenuation of asound beam in transition through asound veloeity discontinuity layer (excluding extreme grazing rays); for calculation of the near fieldof a transmitter in any layered-inhomogeneous media and the Held in the case of asound source and receiver situated in a homogeneous surface layer of thickness Ho at distances r« sHVA when the
medium below the surface layer has a negative sound velocity gradient; and for calculation of
the Held at or near caustics and in certain other cases.
In addition, the admissibility of physical interpretation of the Held in a shadow zone by
means of ray representations can also be tested experimentally under modeling conditions [16].
Such an interpretation has been given in a number of theoretical papers [45, 46, 98]. Besides
the "direct rays" indigenous to the illuminated zone, "diffracted rays" and their wave fronts in
the shadow zone are investigated. The notion is eonsistent with the geometric-acoustical treatment as an approximation of the wave treatment.
The existence of diffracted rays in a shadow zone can be verified by measuring the time
and direction of arrival of the diffraction signal at the point of observation.
The arrival time of the diffraction signal in the shadow zone is ealculated from the notion
of energy transfer from the illuminated to the shadow zone.
According to [94] the transit time of a pulse from asound source S (Fig. 51) to a point Q
in the shadow zone in a medium with a negative sound velocity gradient is calculated aeeording
to the equation*
(19.2)
where r' = PQ is the distance from the point of observation (Q) to the shadow boundary (P),
tSA and tBQ are the transit times of the pulse along the respective paths SA and BQ, and Co is
the velocity of sound at the liquid surface.
Following this scheme, we can also use (19.2) to calculate the transit time of a pulse in
the first shadow zone in asound channel with a bilinear sound velocity profile (Fig. 52). In
this case the transit time of the pulse along the ray segments in the illuminated zone is calculated by the usual ray-theory relations (HO).
In cases where the trajectories of the diffracted rays are not known, the calculation of
the signal arrival time in the shadow zone proves difficult and can be carried out only approximately near the boundary of the illuminated zone.
For a schematic representation of the given method, from a point inside the shadow zone
we drop a perpendicular to the ray separating the shadow from the illuminated zone and assume
that the arrival time of the diffraction pulse at the given point is equal to its arrival time at the
point on the boundary ray at the base of the perpendicular (point D in Fig. 52b). t
The above method is based on the notion of the "cross diffusion" of sound energy into the
shadow zone along the wave front and is only applicable to the shadow zone near the boundary
of the illuminated zone, because deeper into the shadow the energy flux transferred through
the side walls of the ray tubes is rapidly depleted.
* The given method is tantamount to adetermination of the transit time of the principal part of
the signal by the stationary phase method [22].
tIn Fig. 52 the line QD is not perpendieular to the boundary ray because different seales have
been chosen for the coordinate axes.
H9]
ON THE APPLICABILITY OF RAY REPRESENTATIONS
TABLE 5
Range r, cm
te
I
I
lJSec
40 (shadow boundary)
48
50
60
35(shadow boundary)
36
43
45
47
51
54
57
0
11
I
245
280
300
360
213
220
260
270
280
320
340
360
(
10
f1sec
Jßec
243
280
305
366
208
214
258
270
282
307
326
344
243
275
290
Curve 1
The same
208
210
225
245
Curve 2
The same
"
"
-
"
"
"
"
"
"
-
-
/,x
/x
1600
7500
a
Vertical sound
velocity
profile in Fig. 52
12
X"""
1700c,mjsec
~ •
x~• •
x..............
,
X~
20
•
x,
,
X'x
x
1
z,cm
b
2
Fig. 52. Calculation of the arrival time of a signal
at a point inside the shadow zone. a) Vertical sound
velocity profile; b) ray pattern.
75
MODELING OF SOUND PROPAGATION IN INHOMOGENEOUS MEDIA
76
Or-________ors~O~--------~'O~or.cm
10
15
18
21
23
10
[CH.3
In light of the foregoing the pulse transit time
from the source to the point Q is equal to
(19.3)
The pulse transit time into the shadow has been
calculated according to Eqs. (19.2) and (19.3) in media
having different vertical velocity profiles and is compared in the ensuing tables with the corresponding experimentally measured values of the transit time.
1. Underwater Sound Channel with
Bi li n e a r Ve I 0 c i tyP ro f i I e (Fig. 52). The
20
experimental conditions were as follows: width of
z.cm
sound pulses, 50 J,lsec; carrier frequency, 1000 kc;
Fig. 53. Determination of the direcpulse repetition rate, 50 cps; transmitter in the form
tion of a diffracted ray in the shadow
of a hollow semicylinder of barium titanate ceramic
zone of an underwater sound channel.
with a broad directivity pattern in the vertical plane
(±80° at the principallobe); cylindrical receiver with
a uniform directivity pattern in the vertical plane; error of time measurement on the oscilloscope screen, ±5 Ilsec. The transmitter and receiver were placed at an equal depth above the
channel axis (z = 3 cm).
The arrival times t 1 and t 2 of a pulse in the first shadow zone, calculated according to
Eqs. (19.2) and (19.3), respectively, and the corresponding experimental data (te) are summarized in Table 5. It is apparent from the table that the time calculated according to (19.2)
satisfactorily agrees with the experimental value within the experimental error limits; the time
according to (19.3), as expected, agrees with the experimental only for points near the shadow
boundary.
2. Antichannel with a Gradual Decrease of the Sound Velocity with
Depth (Fig.51). The experimental conditions were as folIows: width of the sound pulses,
40 J,lsec; carrier density, 500 kc; width of the principal directivity lobe in the vertical plane,
±6.5°; all other experimental conditions the same as in the preceding case. The measurements were carried out with the transmitter and receiver at equal depth (z). Inasmuch as
the transmitter had a sharp directivity pattern, in order to eliminate the effect of the side
lobes of the transmitter directivity pattern on the diffraction field in the shadow zone, the
transmitter was rotated upward through an angle sufficient to bring the central ray of the sound
beam in contact with the surface of the water.
The arrival times of a pulse at a point inside the shadow zone, calculated according to
Eq. (19.2), and the corresponding experimental data (te), which are consistent with thetheoretical, are shown in Table 6.
The use of highly directional receivers for measurement of the field in the shadow zone
indicates that the diffraction field has a directivity similar to the field in the illuminated zone.
At each point of the shadow it is possible to determine a certain optimum placement of the receiver corresponding to the peak reception of sound. By analogy with the illuminated zone,
with this placement the direction of the diffracted ray at the given point of the shadow is ascertained. Carrying out the measurements at different points, one determines a family of diffracted rays in the shadow. This is illustrated in Fig. 53, which shows one such ray in the
shadow zone formed in a medium with an underwater sound channel. Notice particularly the
similarity of the diffracted ray (2) and the ray bounding the shadow zone (1). The measurements
were carried out at a frequency of 1 Mc. The transmitter was submerged to the depth of the
channel axis (z = 6 cm).
§19]
ON THE APPLICABILITY OF RAY REPRESENT ATIONS
77
TABLE 6
Distanee r. ein
le
Il see
74
84
54
64
74
459
516
300
370
430
11
I
jlSee
z'em
462
529
306
369
433
5
5
3
3
3
Vertieal sound
velocity
profile in Fig. 51
Curve 1
The same
Curve 2
The same
..
TABLE 7
h, cm
z. cm
" cm
<pO
I
<f>:
',li
13°10'
13 10
13 10
13 40
13 40
15
15
15
19
19
21 30
21 30
64
64
64
54
54
43
43
43
47
47
52
52
I
5
5
5
3
3
3
3
3
3
3
3
3
3
3
3
3
3
2
2
2
3
3
5
5
64
74
84
54
64
44
54
64
54
64
54
64
14
16
14
15
14
18
16
18
22
20
24
24
cm
Vertical sound
velocity
profile in Fig. 51
Curve 1
The same
" 2
Curve
The same
Curve 3
The same
..
..
"
"
"
The angle cP of inclination of the diffracted ray relative to the horizontal is the same,
within the experimental error limits (±1"), at different points situated at the same level and
coincides (except in the near-surface layer) with the angle CPo of inclination of the boundary
ray as calculated from the ray pattern. This follows from the data of Table 7, which gives,
in addition to the distance r from the reception point and distance ro from the shadow boundary.
the horizontal level z of the measurements and depth hof the transmitter. The measurements
were carried out at a frequency of 500 kc.
Thus, it is reasonable to conclude on the basis of the results of model experiments
that ray representations can be used to analyze the sound field in a shadow zone.
In all of the model experiments described the geometric-acoustical condition was satisfied (with the exception of grazing angles less than 2 to 5") [see (19.1)]:
B = aJ. ~ 2'lt sin 3 fJ ,
since the dimensionless number B had an order of magnitude no larger than 10- 2 or 10- 3•
We now consider the possibiliW of modeling a medium in which the propagation of sound
does not obey the laws of geometrical acoustics.
In this case the following conditions must be met:
B ): 2'lt sin 3fJ •
If, for example, e :::; 10°, then B ;::: 0.03. This condition requires that an acoustic frequency no higher than 1 00 kc be used in model experiments for asound velocity gradient a =
2.5 m- 1•
Model investigations conducted in a tank at low frequencies are frought with several
engineering difficulties (§6). Consequently, the improvement of modeling techniques for the
78
MODELING OF SOUND PROPAGATION IN INHOMOGENEOUS MEDIA
[CH.3
purpose of studying wave effects in an experimental tank requires a search for modeling materials by which it will be possible to produce layered-inhomogeneous media invested with significantly larger velocity gradients. This would mean the avoidance of a drastic reduction in
frequency.
Another approach calls for the use of large-scale modeling, which can be accomplished
in large ponds or acoustical field test stations at sufficiently low frequencies.
Summarizing the overall results of Chapter 3, we conclude that the techniques for modeling inhomogeneous media are weIl suited to the investigation of many phenomena observed in
connection with the propagation of sound in the sea.
APPENDIX
ACOUSTICAL MODELING
OF RADIO WAVE PROPAGATION
IN THE EARTH'S ATMOSPHERE
The earth's atmosphere may be treated roughly as a layered-inhomogeneous medium
to the extent that its physical properties vary far more rapidly in the vertical direction than
in the surfaces perpendicular to it. Consequently, if the influence of irregular inhomogeneities
is eliminated, acoustical modeling of the field in layered-inhomogeneous media can be applied
to the investigation of the propagation of radio waves in the atmosphere. The applicability of
the method is limited in connection with the impossibility of modeling such specific characteristics of electromagnetic waves as the transverse character of the field, the influence of the
earth's magnetic field, the absorption of electromagnetic waves, etc. Despite these limitations, acoustical modeling makes it possible to carry out at least a qualitative study of the
principles governing radio wave propagation in the troposphere, as weIl as in the ionosphere.
This uncovers additional opportunities for gaining insight into the specific characteristics of
radio wave propagation in the atmosphere. As the reader is aware, model experiments have
the virtues of comparative simplicity, low cost, and high accuracy. The application of modeling
techniques permits the rapid acquisition of essential information on the field, an asset that is
particularly valuable in that it affords a physically transparent picture of what is happening
during the propagation of radio waves in the atmosphere.
For the acoustical modeling of radio waves in the atmosphere the following similarity
relation must be satisfied:
(1)
where rand z are the horizontal range and height, respectively, in the atmosphere, ro and Zo
are the corresponding variables in the acoustical model, A is the electromagnetic wavelength,
A0 is the sound wavelength in the model, and M is the scaling factor.
It follows from (1) that
(2)
where fand f 0 are the electromagnetic and sound wave frequencies, respectively, and c and
Co are the respective velocities of electromagnetic waves in the atmosphere and sound waves
in the model. If the model medium is water, m is approximately equal to 2 105•
0
79
APPENDIX
80
In the modeling experiment the water-air interface corresponds to the boundary between
the atmosphere and the earth's surface. The absorption of electromagnetic waves in the atmosphere cannot be simulated in an acoustical model and special corrections must be made
for it. For this purpose the absorption of sound in the model medium is calculated,* and corrections are introduced into th€ experimental values of the sound pressure. Once the resulting values, expressed in conditional units and regarded as the values of the electric (or
magnetic) field strength, have been referred to definite distances in the atmosphere by means
of the similari ty relation, new corrections are introduced into those values to account for the
attenuation of the electromagnetic waves.
Propagation of Radio Waves in the Troposphere
The laws of radio wave propagation in the troposphere depend on the wavelength.
A singular feature of long waves (2 to 20 km) is the waveguide character of their propagation between the earth and the ionosphere, which function as the conducting walls of a waveguide, guiding the flow of waves and stimulating their further propagation.
The calculation of the longwave field in general requires a rigorous solution of the waveguide problem for a spherical earth surrounded by an inhomogeneous ionosphere. However,
inasmuch as the wavelength and height of the ionosphere are much smaller than the earth's
radius, the effect of the earth's curvature is significant only at large distances from the wave
source. The stratified structure of the earth's crustt and ionosphere mayaiso be disregarded
and both media replaced by homogeneous media having a reflectivity equivalent to that of the
corresponding inhomogeneous media.
Consequently, the propagation of long radio waves in the atmosphere is similar to the
propagation of sound in a shallow sea and can be modeled analogously by means of sound waves
in a thin layer of water with two perfectly reflecting boundaries (the metal bottom of the tank
below, corresponding to the upper boundary of wave propagation, and the air interface above,
corresponding to the lower boundary of the troposphere, i.e., the earth).
The sound field associated with waveguide propagation is represented by the spectrum of
normal modes.
In cases when the width of the waveguide is large in comparison with the wavelength the
number of normal modes forming the field turns out to be very I arge , even at considerable distances. Therefore, the waveguide method of analyzing the field presents a difficult problem
and must be replaced by the geometrical method.
The foregoing case is realized in the troposphere during the propagation of ultrashort
radio waves and decimeter waves.
The dielectric constant and, accordingly, the refractive index in the troposphere may
gene rally be regarded as real and mainly decreasing with altitude. This guarantees the kind
of wave refraction in the medium that intensifies the transport of the waves past the horizon.
The occasional violation of monotonicity in the variation of the dielectric constant can in the
case of ultrashort waves produce height-limited local zones of a waveguide character.
Since only the relative curvature of the rays and earth's surface and not their absolute
values are significant with regard to wave propagation, it is permissible in the calculations to
* Tabulated data are used for the coefficient of sound attenuation.
t This assumption is justified by the fact that radio waves penetrate only into the upper surface
layer of the earth.
ACOUSTICAL MODELING OF RADIOWAVE PROPAGATION IN THE ATMOSPHERE
81
invoke the concept of the equivalent radius R' of the earth's surface such that the atmosphere
constitutes a homogeneous sphere. According to [61]
where R o is the average geometric radius of the earth and no is the refractive index at the earth's
surfaee (z = 0), no -1 = 36 . 10- 5 m-i.
For the incidence of rays ·on the earth at small angles
1
--Ro
p'
where p is the radius of eurvature of the ray.
For the so-called "standard atmosphere" [20]
R' ~ : Ro ~ 8500 m.
The field pattern is unchanged, but the analysis of the problem is simplified if the earth
is regarded as plane and the rays as curved so as to leave unaltered the previous relative eurvature of the rays and the earth. This ease eorresponds to the following reduced refractive
index for radio waves [52]:
n' (z) = n (z)
+ ~o .
(3)
Let us examine the possibilities of the acoustieal modeling method for the investigation
of the propagation of ultrashort waves in the troposphere for eertain models of the reduced
refraetive index.
1. The value of n(z) deereases linearly with height, where Idn/dz I < I/R o. Here, according to (3), the redueed refraetive index decreases linearly with height. In this case beyond
the horizon a geometrie shadow zone is formed, whieh is bounded by the ray tangent to the
earth's surface 175].
The acoustieal model of this situation entails the propagation of sound waves in water
with a negative depth-constant gradient of the velocity of sound. The field is investigated
near the plane air interface, which in the case of radio waves in the troposphere eorresponds
to the interface with the earth.
The attenuation of the field intensity level in the geometrie shadow zone per unit length is
equal, aceording to (11.1), to ß1 = I:l.ß / I:l.r = 6.66q, where q = (9a 2k) 1/3 and k = 21T / A.
Considering waves of length A = 5 m (f = 60 Me) in a standard atmosphere (dn/dz =
-3.9 • 10- 8 m- i , a = dn' /dz = 1l.8 • 10- 8 m- i), we find q = 6.55 . 10- 5 m- i and ß i = 43.5 • 10- 5
dB/m.
The distanee from the source, situated at a height h = 100 m above the earth, to the boundary of the shadow zone in this case (for a radius of the earth Ro = 6400 km) is r o = 36 km.
Inasmuch as the attenuation of radio waves in the shadow zone is smaH, the only way this
ease ean be realized in the underwater acoustieal tank is to have the latter of suffieient length
and to use high frequencies. Suppose that the sealing factor is M = 3600. Then the distanee
from the source to the shadow boundary is r 0 = 10m; the sound wavelength is about
1.5 mm (frequeney f 0 = 1 Me); the relative sound velocity gradient ao = -4.2 • 10-4 m-i. A
82
APPENDIX
z
a
_~n,
d '-------"----n
----/:
'z
b
.-----..,.-------C(Z)
z
Fig. 54. Acoustical modeling of an earth surface waveguide. a) Vertical profile of the reduced refractive index and behavior of a radio ray for a plane earth; b) vertical profile of the velocity of sound in the underwater
acoustical model and corresponding ray pattern.
medium in which the reduced refractive index is constant to a certain height and then increases
linearly is simulated analogously in the underwater acoustical tank.
2. A waveguide is formed near the surface of the earth (Fig. 54a). According to the experimental data [52] the gradient of the refractive index in the surface layer has an approximate
value dn'/dz = -13.2 • 10- 8 rn-i. and the maximum wavelength (in cm) sustained by a waveguide
of height d (in meters) can be computed from the relation i\max = 0.085d3/ 2• from which it follows that for the standard height of the waveguide (several tens of meters) the waveguide propagation effect is primarily associated with superhigh frequencies.
The distribution of the velocity of sound with depth in the medium used to model the surface atmosphere waveguide is illustrated in Fig. 54b.
Suppose that waves of length i\ = 50 c;m propagate in a surface waveguide of height d =
100 m. Let us adopt a scaling factor M = 2000. We then obtain the following parameters for
the acoustical model: d o = 5 cm; i\ 0 = 0.25 mm (f 0 = 6 Mc); sound velocity gradient. ao =
2.6 • 10-4 rn-i. A tank of length lo = 10 m is equivalent to a range l = 20 km.
Both of the cases analyzed above indicate that the acoustical modeling of the propagation
of ultrashort waves in the layer next to the earth's surface is possible. but requires the use of
long tanks (greater than 10 m), which are not always realizable in experimental laboratory practice. Consequently. the acoustical modeling of ultrashortwave propagation in the troposphere
is best pursued under semimodel conditions in naturallakes or bays. in which a surface sound
channel is formed during the cold season under the influence of various meteorological factors.
The dimensions of the natural bodies of water are such as to permit the modeling of longrange radio wave propagation. The acoustical measurements can be conducted at lower ultrasonic frequencies thar.. in a tank. thus diminishing the attenuation of sound in the water.
It follows from the foregoing that natural underwater acoustical ponds can be used not only
for the large-scale modeling of the sound field in the ocean, but also for the investigation of
radio wave propagation in the atmosphere.
ACOUSTICAL MODELING OF RADIOWAVE PROPAGATION IN THE ATMOSPHERE
Pro p aga t ion
0 fRa d i 0
83
W ave s i n t hel 0 nos p her e
The principal experimental method for the investigation of the ionosphere involves the
transmission of radio signals into it and the reception of the radio echo from different strata
of the ionosphere. This method is also used for the detection of local reflecting objects in the
ionosphere and for the determination of the laws of shortwave radio communications.
A relatively weak ionization of the atmosphere is observed at fairly low altitudes, from
60 to 90 km; this zone (the D layer) is important only with regard to the propagation of long
waves [2], The electron concentration in the ionosphere grows appreciably at a height of about
100 km above the earth, where as a result of the dependence of the electron concentration only
on the height z (disregarding sporadic concentration fluctuations) the ionosphere may be thought
of as a layered-inhomogeneous medium with adefinite law of variation of the dielectric constant with height. This law will differ in different layers of the ionosphere. Moreover, it varies
with time under the influence of factors beyond the realm of our problem. To a first approximation a linear or parabolic variation of the dielectric constant as a function of the height is
adopted as a mathematical model of the structure of the ionosphere.
An analogous distribution of the square of sound velocity with depth is created in the
underwater acoustical tank by astutely chosen liquid solutions (§9). This affords the possibility
of acoustical modeling of the propagation of short radio waves in the ionosphere.
In model experiments not only can the amplitude of a signal reflected from the ionosphere
and received on the earth be estimated, it is also possible to study in detail the structure of the
field inside the ionosphere itself.
We now examine the way in which modeling of a parabolic layer can be implemented in an
underwater acoustical tank.
In this layer the dielectric constant varies according to the law [85]
E
= n 2 (z) = 1 _ e2N (z) = 1 _ (R)2
m1tf2
f
(E.. _~)
Zm
z~'
(4)
where e and mare the charge and mass of the electron, f is the frequency, z is the height
measured from the lower boundary of the ionosphere (assuming that the height of that boundary
above the earth is h = 100 km), Zm is the height of the layer of maximum electron concentrati on (Nm) in the parabolic layer, and fk is the frequency at which a vertical ray is reflected
from the given layer (critical frequency).
Aeeording to [2], at a height of 120 to 130 km a region of elevated electron concentration
ean exist inside the parabolic layer (at medium latitudes about 1.5 . 105 electrons/cm3 by day,
5 ' 103 to 104 electrons/cm3 by night). The thiekness of this region is 15 to 20 km (E layer).
However, the maximum electron eoneentration Nm in the parabolic layer is found at a height of
about 300 km (F layer); it reaehes a value of 2 • 106 by day and up to 2 • 105 by night.
At the point of refleetion of a vertieal ray n = O. Henee, aecording to (4) the critical frequency (in Me) is expressed by the formula
(5)
The critieal frequency for the level of maximum electron eoncentration z m is equal to
about 13 Me by day and 4 Me by night.
Consequently. according to (4) the refractive index varies in the parabolic layer (daytime)
by the law
84
APPENDIX
z
a
b
-ho
0
Zro
Zm
Zmo
Zr
o
-h L -_ _ _ _ _---:!--_ _
n
Zo
Fig.55. Acoustical modeling of a parabolic layer in the
ionosphere. a) Vertical profile of the radio wave refractive index; b) vertical profile of the velocity of sound
in the corresponding acoustical model.
)2
13 (0 .Olz - 2.5 . 1O-Sz2).
n2 (z) = 1 - ( T
(6)
The E layer exerts an influence on the propagation of waves at a frequency of 0.75 to 1
Me, the F layer on higher-frequency waves, on the order of 3 to 30 Me. For waves with a
frequency below 1.5 to 3 Me the anisotropicity of the ionosphere must be taken into account;
thus, the acoustical modeling of wave propagation in the ionosphere at such wavelengths is
highly impracticable.
The level at which waves of the given length are reflected depends on the grazing angle
81 of the ray as it enters the ionosphere.
The horizontal distance between the points of entry of the ray into the ionosphere and its
exit is expressed by the formula [85]
The horizontal range of ray reception at the earth's surfaee is equal to
where h is the height of the lower boundary of the ionosphere.
The parabolic layer (Fig. 55a) can be modeled in a tank by means of a stratified medium
such as one comprising an aqueous solution of ethyl (or methyl) alcohol with a eoneentration
that increases upward from the bottom, onto which is poured pure water, to the surface, where
the solution is pure alcohol, and the velocity of sound becomes a maximum at the depth corresponding to a eoncentration of about 35% alcohol (Fig. 55b). The acoustic refractive index
has a minimum at this depth, which corresponds in the ionosphere, according to (4), to the level
at which the electron concentration is maximal.
The modeling of the reflection of radio waves from the stratified ionosphere at a frequency f < fk is facilitated by the fact that these waves are reflected below the layer of maximum electron concentration. Consequently, there is no need for the formation of asound veloeity maximum (refractive index minimum) in the tank; instead, it suffices to simulate the
ionospherie layer up to the height Zr (dashed line in Fig. 55a) at which are reflected the rays
hitting the ionosphere at steepest incidence.
ACOUSTICAL MODELING OF RADIOWAVE PROPAGATION IN THE ATMOSPHERE
85
In conclusion we consider an example of the modeling of radio wave propagation in the
ionosphere. As revealed by calculations according to Eq. (6), waves of length 50 mare completely reflected in a layer of height 22 km above the lower boundary of the ionosphere. We
assume a sealing faetor M = 105 • Then the indieated ionospheric layer corresponds to a transition layer with a thickness of 22 cm, and a height h = 100 km eorresponds to a depth of 1 m.
The acoustic frequency in the model experiment is equal to 3 Mc.
The refractive index gradient, which is numerically equal to the relative wave velocity
gradient, is expressed as follows according to (6):
I
dn ~"2
1 ( f13 )2(0.01 - 5 . 1O-5z ).
a = I,Tz
Forz «zm, as occurs inthe givencase,
a
Setting f
= ( ~fO.OO5 km -1.
= 6 Mc, we obtain a ~ 2 • 10-5 m- 1•
This gradient is simulated in the tank by asound velocity gradient ao ~ 2 m- 1• However,
the largest value of ao that can be obtained in a transition layer of thickness llzm = 22 cm between the water and alcohol is equal to 1.6 m- 1• This follows trom ~he relation aollz m ~ 0.35,
which ensues in turn from the graph of the velocity of sound in an aqueous solution of ethyl
alcohol as a function of the concentration (§8). Therefore, the thickness of the transition layer
between the pure water and alcohol must be decreased to 17.5 cm, which corresponds to an
ionospheric layer 17.5 km thick. According to the geometrical theory, in this layer all waves
incident on the lower boundary of the ionosphere at angles 8 :s 8 0 will be reflected, where
80 = cos- 1 n and n is the refractive index for 50-m waves at the upper boundary of the layer
(i.e., at z = 17.5 km). The value of n is found from Eq. (6). In the present case the angle 8 0 is
roughly equal to 75°.
If the ionosphere is modeled by an aqueous solution of sodium chloride, then with regard
for the relation aollz m ~ 0.15, which applies to this case, we obtain llzm = 7.5 cm in the solution and, accordingly, 7.5 km in the ionosphere.
The maximum grazing angle of a ray reflected in this layer turns out to be about 40°.
For a scaling factor M = 105 a tank of length 10 m eorresponds to a range of 1000 km.
Thus, it is possible to investigate the multiple reflection of radio waves from the ionosphere
and earth in model studies.
A practical difficuHy is the eonsiderable attenuation of sound at a frequency of 3.0 Me in
the solution due to absorption. The latter can be abated by lowering the acoustic frequency, but
then the maximum range of propagation that can be simulated in a tank of the given length is
diminished accordingly.
The foregoing examples foster the conclusion that the application of the method of acoustical modeling to the investigation of radio wave propagation in the earth's atmosphere is entirely feasible under laboratory experimental conditions.
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