Wave Propagation and Underwater Acoustics

294 Pages·1977·4.76 MB·English

by  Joseph B. Keller (auth.)

Lecture Notes
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Edited yb .J Ehlers, ,nehcn3(M .K Hepp, ,hciriLZ
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Managing Editor: W. BeiglbSck, Heidelberg
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Wave noitagaporP
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Edited yb
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Springer-Verlag
Berlin Heidelberg New York 1977
Editors
Joseph B. Keller
Courant Institute
of Mathematical Sciences
New York University
New York, New York/USA
John S. Papadakis
New London Laboratory
Naval Underwater Systems Center
New London, Connecticut/USA
ISBN 3-540-08527-0 Springer-Verlag Berlin Heidelberg New York
ISBN 0-387-08527-0 Springer-Verlag New York Heidelberg Berlin
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WAVE PROPAGATION AND UNDERWAT~ ACOUSTICS
Joseph .B Keller* and John S. Papadakis~* Editors
Courant Institute of Mathematical Sciences, New York University
Department of Mathematics, University of Rhode Island and
Naval Underwater Systems Center, New London Laboratory
Preface
A "Workshop on Wave Propagation and Underwater Acoustics" was held from
November 19 to November 21, 1974 in Mystic, Connecticut. It was sponsored by the
Acoustics Branch of the Office of Naval Research under the aegis of Hugo Bezdek.
The workshop was conceived at the New London Laboratory of the Naval Underwater
Systems Center and organized by the following committee of members of that
laboratory:
Chairman: John .S Papadakis, Department of Mathematics,
University of Rhode Island (Consultant)
L. T. Einstein
R. .H Mellen
Henry Weinberg
Among the twenty-one lectures at the workshop was a set of six surveys of
various aspects of the field. Those surveys were presented by five members and one
former visiting member of the Courant Institute of Mathematical Sciences, New York
University. They were prepared with the intention that they would be expanded,
combined and published together as a general survey of the mathematical theory of
underwater sound propagation. These notes are the result. They would not have
appeared without the untiring effort of Professor John .S Papadakis, who guided
them through the editorial process. I wish to thank him particularly for this. I
also thank the entire committee for having asked me to present a set of survey
lectures, and for then agreeing to let me share the presentation with my colleagues.
Joseph .B Keller
WAVE PROPAGATION AND UNDERWATER ACOUSTICS
ELBAT FO STNETNOC
I. SURVEY OF WAVE PROPAGATION AND UNDERWATER ACOUSTICS
(~eph B. Keller)
i. Introduction ......................................................... i
2. Wave propagation in a deterministic medium ........................... 2
3. Wave propagation in a stochastic medium .............................. 8
References ........................................................... 12
II. EXACT AND ASYMPTOTIC REPRESENTATIONS OF THE SOUND FIELD IN A STRATIFIED OCEAN
(DalJit S. Ahluwalia and Jo~B. ~ )
O. Introduction ......................................................... 14
i. Formulation and fundamental equations ................................ 16
2. Time harmonic waves .................................................. 19
3. The homogeneous ocean of constant depth
3.1 Introduction ................................................... 21
3.2 Normal mode representation ..................................... 23
3.3 Hankel transform representation ................................ 25
3.3A Appendix ..................... ~ ................................. 26
3.4 Ray representation ............................................. 27
3.5 Connections between the representations ............. ........... 30
3.5A Appendix ....................................................... 34
4. The inhomogeneous stratified ocean of constant depth
4.1 Introduction ................................................... 36
4.2 Normal mode representation ..................................... 38
4.3 Hankel transform representation ................................ 39
4.4 Multiple scattering representation ............................. h0
4.5 Connections between the representations ........................ h3
4.5A Appendix ....................................................... 45
IV
5. Asymptotic representations for the inhomogeneous stratified
ocean of constant depth
5.1 Introduction ........................................ ~ .......... 46
5.2 Asymptotic form of the modal representation .................... 46
5.2A Appendix ....................................................... 54
5.2B Appendix ....................................................... 61
5.2C Appendix ....................................................... 63
5.3 Asymptotic form of the Hankel transform representation ......... 64
5.4 Asymptotic form of the multiple scattering representation ...... 67
5.4.1 Explicit asymptotic form of the multiple scattering
representation ................................................. 68
5.5 Connections between the asymptotic forms of the
representations ................................................ 71
6. The ray representation
6.1 Introduction ................................................... 76
6.2 Geometrical construction of the ray representation ............. 77
6.3 Analytic derivation of the ray representation .................. 78
6,4 The ray representation for the stratified ocean of
constant depth ................................................. 82
References ........................................................... 85
III. HORIZONTAL RAYS AND VERTICAL MODES
(Robert Burridge and Henry Weinberg)
I. Introduction ......................................................... 86
2. Acoustic propagation in an almost stratified medium .................. 88
.3 Uniform asymptotic expansions in regions containing caustics ......... 97
3.I The field near a smooth caustic ................................ 97
3.2 A point source in an almost stratified medium .................. 105
4. Space-time rays for more general time dependence ..................... 109
4.1 The ray theory ................................................. 109
4.2 The excitation due to a point source ........................... i14
4.3 The Airy phase ................................................. 118
4.4 The precursor and other phenomena requiring special
treatment ...................................................... 12 3
VII
.5 Two theoretical examples .............................................. 125
5.1 Homogeneous medium, one free horizontal boundary, one
rigid boundary with small constant slope ......................... 125
5.2 Propagation in deep water for which the sound speed
increases with depth ............................................. 130
6. Long range acoustic propagation in a deep ocean. ...................... 13h
6.1 Environmental parameters ......................................... 13h
6.2 The computer program ............................................. 139
6.3 Comparison of computed amplitudes with observational data ........ lh3
References ............................................................ 150
IV. WAVE PROPAGATION IN A RANDOMLY INHOMOGENEOUS OCEAN
(Werner Kohler and George C. Papanicolaou)
0. Introduction ......................................................... 153
i. The physical problem ................................................. 155
2. Asymptotic analysis of stochastic equations .......................... 163
.3 Application of asymptotic methods to coupled mode equations .......... 170
4. Coupled power equations .............................................. 177
.5 Quasi-static and slowly-varying coupled power equations .............. 180
Appendix. First and second order perturbation theory for
Boltzmann-llke equations .................................... 187
6. Coupled fluctuation equations .......................................... 192
7. Depth-dependent quantities ............................................. 196
8. High frequency approximation to coupled power equations ................ 197
Appendix A. Numerical study .......................................... 213
Appendix B. Diffusion approximation for coupled power equations
with radiation loss ............................. ~ ........ 217
References ............................................................. 222
V. THE PARABOLIC APPROXIMATION METHOD
(Fred D. Tappert)
i. Basic concepts ......................................................... 224
2. Derivations of parabolic equations ..................................... 238
.3 Asymptotic analysis .................................................... 266
h. Summary ....................................... . ........................ 280
IIIV
5. Acknowledgment ................................................... 281
Appendix A. Historical survey of parabolic wave
equation applications .............................. 282
References ....................................................... 582
CHAPTER I
SURVEY OF WAVE PROPAGATION AND ~ERWA~ ACOUSTICS
Joseph B. Keller
Courant Institute of Mathematical Sciences
New York University
251 Mercer Street
New York, NY 10012
.i Introduction
Underwater acoustics, the science of sound propagation in the ocean, has been
developed extensively during the last forty years in response to practical needs.
By now the theory .si so well developed that it provides a general understanding
and a detailed description of how sound travels in the ocean, and of the
mechanisms affecting it. The theory can also he used to make quantitative
Calculations of the sound field produced by a given source. However, there are
difficulties which limit the accuracy of such calculations. The first is the lack
of adequate information about the sound velocity in the ocean as a function of
position and time. The second is the analytical and computational difficulty of
Calculating the sound field in terms of the properties of the ocean. The
mathematical methods which have been devised to overcome this latter difficulty
Sme the subject of these notes.
The analysis of underwater sound propagation is based upon the physical
Principles of theoretical acoustics. These principles lead to a wave equation for
the acoustic pressure, together with suitable boundary conditions at the ocean
Surface and bottom, and initial conditions. The properties of the ocean which
enter into this formulation are the sound speed c(x,y,z,t) , the bottom depth h(x,y),
the surface elevation n(x,y,t) and the ambient water velocity R(x,y,z,t).
Absorption, which results from viscous dissipation, heat conduction, chemical
reaction, scattering by particulate matter, etc. is usually accounted for by an
absorption coefficient which depends upon position and frequency. In the analysis
of time harmonic fields it is combined with the sound speed to yield a complex
refractive index. Absorption by the bottom is usually accounted for by a bottom
impedance, or sometimes by a bottom reflection coefficient.
Most of the theoretical analyses ignore the surface elevation, the ambient
water velocity, and the absorption in the fluid and in the bottom. Some of these
effects are taken into account afterwards in an ad hoc manner. For the most part we
shall follow the common procedure of ignoring them.
Initially, the theory concerned the deterministic problem of propagation in an
ocean of prescribed constant or gradually varying properties. However, as experi-
mental technique improved, it was found that the observed sound field undergoes
extensive and rapid fluctuations. These fluctuations are caused by fluctuations in
the properties of the ocean. To analyze them the ocean is represented as a random
medium, and the problem of sound propagation in a random medium is considered. The
theory of this kind of propagation is not as well developed as that of propagation
in a deterministic medium, as we shall see. We shall first describe the theory of
the deterministic case and then describe that of the random case.
.2 Wave pro~a~ation in a deterministic medium
Let us consider first the simplest case, that of a time harmonic point source in
an unbounded homogeneous ocean. The resulting sound field is a spherical wave.
Secondly, suppose the ocean is bounded above by a horizontal plane free surface on
which the acoustic pressure p vanishes. Then p is the sum of two spherical ~aves,
one from the source and another from the image of the source in the plane surface,
multiplied by the reflection coefficient R = -1 . The interference between these
two waves leads to an oscillation in the magnitude of p which is sometimes
referred to as the Lloyd mirror effect. Thirdly, let the ocean be bounded above by
a horizontal plane free surface on which p = 0 and below by a horizontal plane
bottom on which the normal derivative 8p/Sn = O. Then p is the sum of an
S
infinite number of spherical waves from the source and from an infinite set of images
of it in the two planes.
The image method of constructing p , which leads to the above results, does not
generalize to the case of an inhomogeneous ocean nor to the case of non-planar
bolmdaries. Furthermore, at horizontal distances from the source which are large
compared with the depth, many of the spherical waves have nearly the same phase, or
arrival time. This makes it difficult to calculate p because the successive waves
nearly cancel one another.
These disadvantages of the image method can be overcome, in part, by the method
of normal modes, which was introduced and developed by C.L. Pekeris i. That method
applies to any horizontally stratified ocean of constant depth. It leads to a
representation of p as the sum of an infinite number of normal modes. Only a
finite number of them are propagating and the rest are evanescent. Thus, at large
distances from the source only the propagating modes are important, so there p is
represented by a finite sum.
The method of normal modes is restricted to horizontally stratified oceans of
constant depth. Furthermore, at distances from the source which are not large
compared with the depth, the evanescent waves are not negligible, so many of them
must be taken into account in calculating p.
The latter difficulty, but not the former, can be overcome by the Hankel
transform method, which was utilized by L. Brekovskikh 2 and others. This yields a
representation of p as an integral involving Bessel functions and solutions of the
normal mode equation. Although this integral is convenient for evaluation at short
ranges, it is not so convenient at long rs/iges, where the normal mode representation
is more useful.
A third representation of p in a horizontally stratified ocean of constant
depth is given by the method of multiple scattering. This method is a generalization
of the image method from the case of a homogeneous ocean to that of a horizontally
stratified one. In it p is represented as a stun of waves: one wave emerging
directly from the source, another wave which represents scattering of the direct
Wave by the medium above the source, a third wave which results from scattering of
the direct wave by the medlumbelow the source, and successive multiply scattered