Acoustics of Layered Media II: Point Sources and Bounded Beams

530 Pages·1999·11.448 MB·English

by  Professor Leonid M. Brekhovskikh, Dr. Oleg A. Godin (auth.)

Springer Series on
Waye Phenomena 10
Edited by L.B. Felsen
Springer-Verlag Berlin Heidelberg GmbH
Springer Series on
Waye Phenomena
Editors: L. M. Brekhovskikh L. B. Felsen H.A. Haus
Managing Editor: H. K.V Lotsch
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Mechanics of Continua Resonance Acoustic Spectroscopy
and Wave Dynamics By N. D. Veksler
2nd Edition
By L. M. Brekhovskikh, Y. Goncharov Volume 12
Scalar Wave Theory
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Plane and Quasi-Plane Waves
Caustics, Catastrophes, and Wave Fields
By L. M. Brekhovskikh, O. A. Godin
By Yu. A. Kravtsov, Yu. I. OrJov
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of Inhomogeneous Media Electromagnetic Pulse Propagation
By Yu. A. Kravtsov, Yu. I. Orlov in Causal Dielectrics
By K. E. Oughstun, G. C. Sherman
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Acoustics of Layered Media II Surface Acoustic Waves
2nd Edition in Inhomogeneous Media
Point Sources and Bounded Beams By S. Y. Biryukov, Y. Y. Gulyaev, Y. Y. Krylov
By L. M. Brekhovskikh, O. A. Godin and Y. P. Plessky
L.M. Brekhovskikh O.A. Godin
Acoustics
of Layered Media IT
Point Sources and Bounded Beams
Second, Updated and Enlarged Edition
With 45 Figures
Springer
Professor Leonid M. Brekhovskikh, Academician
P.P. Shirshov Institute of Oceanology, Russian Academy of Sciences, Nakhimovsky pr. 36,
117218 Moscow, Russia
Dr. Oleg A. Godin
School of Earth and Ocean Sciences, University of Victoria, P.O. Box 1700,
Victoria, B.C. V8W 2Y2, Canada and
P.P. Shirshov Institute of Oceanology, Russian Academy of Sciences, Nakhimovsky pr. 36,
117218 Moscow, Russia
Series Editors:
Professor Leonid M. Brekhovskikh, Academician
P. P. Shirshov Institute of Oceanology, Russian Academy of Sciences, Nakhimovsky pr. 36,
117218 Moscow, Russia
Professor Leopold B. Felsen, Ph. D.
Department of Electrical Engineering, Polytechnic University, Six Metrotech Center
Brooklyn, NY 11201, USA
Professor Hermann A. Haus, Ph.D.
Department of Electrical Engineering & Computer Sciences, MIT, Cambridge, MA 02139, USA
Managing Editor: Dr.-Ing. Helmut K.V. Lotsch
Springer-Verlag, Tiergartenstrasse 17, D-69121 Heidelberg, Germany
ISSN 0931-7252
ISBN 978-3-642-08489-8
Library of Congress Cataloging-in-Publication Data applied for.
Die Deutsche Bibliothek-CIP-Einheitsaufnahme
Brekhovskikh, Leonid M.: Acoustics of layered media / L. M. Brekhovskikh; O. A. Godin.
2. Point sources and bounded beams. -2., corc. ed. - 1999
(Springer series on wave phenomena ; VoI. 10)
ISBN 978-3-642-08489-8 ISBN 978-3-662-03889-5 (eBook)
DOI 10.1007/978-3-662-03889-5
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Preface
In this second, revised and expanded edition of Acoustics of Layered Media.
II: Point Sources and Bounded Beams, a number of modifications have been
made to reflect recent progress in research on the subject and on the basis
of experience with the first edition. A more detailed discussion is presented
of the wave propagation in two- and three-dimensionally inhomogeneous en
vironments as well as of acoustic energy conservation and the reciprocity
properties of sound fields. Major additions include a new chapter on waves in
arbitrarily inhomogeneous moving media (Chap. 8) and two new appendices
devoted to a detailed discussion of sound propagation in a range-dependent
waveguide in terms of modes (Append. B) and within the parabolic approxi
mation (Append. C). Furthermore, more than 300 references on recent pub
lications have been added throughout the text to aim the reader towards
"growth points" in the field.
The goal of the book remains to provide an understanding of physical phe
nomena in acoustic wave propagation in inhomogeneous media and to develop
efficient mathematical methods to adequately describe these phenomena. In
addition to an insight into the underlying physics and mathematical methods
employed, accurate quantitative modeling of wave fields requires, of course,
computer simulations. Computer freeware implementing some of the theo
retical approaches discussed in this book can be found on the World Wide
Web, for instance, in the Ocean Acoustics Library (http:j /oalib.njit.edu). The
computer-oriented reader will find this frequently updated Web site a valu
able complement to our presentation. Nowadays, it is particularly suitable to
use the Web in following the software progress because numerical techniques
in wave propagation develop even more rapidly than the theory evolves, and
hardly any printed book can keep pace with this development.
Based on Acoustics of Layered Media. I: Plane and Quasi-Plane Waves
and Acoustics of Layered Media. II: Point Sources and Bounded Beams, senior
undergraduate courses on wave propagation in inhomogeneous media and on
ocean acoustics have been taught at the Moscow IllI:,titute of Physics and
Technology for a number of years. The authors are indebted to their students
and to many colleagues for numerous comments which have helped to correct
typographical errors and to improve the text of the first edition.
VI Preface
Most of the preparation of the second edition has been done by one of the
authors (OC); the other (LB) is grateful to him for that.
Moscow, Russia L. M. Brekhovskikh
Victoria, B.C., Canada O. A. Godin
January 1999
Preface to the First Edition
This is the sequel to our book Acoustics of Layered Media I: Plane and Quasi
Plane Waves (Springer Ser. Wave Phenom., Vol. 5). Taken together, these two
monographs present a systematic exposition of the theory of sound propaga
tion in inhomogeneous media, which starts from first principles and includes
recent results. More advanced topics are considered in this second volume.
Although the theory of wave beams and fields of localized sources is more
sophisticated than the theory of quasi-plane waves, it embraces a much wider
range of interesting problems that are also important for applications.
We exploit the results of Acoustics of Layered Media I, as long as it is expe
dient to consider sound fields as a superposition of plane or quasi-plane waves.
However, the knowledgeable reader will view this book as self-contained.
Similar topics have been treated in the book by L.M. Brekhovskikh, Waves
in Layered Media, the English version of the second edition of which was
published by Academic Press in 1980. Since Waves in Layered Media became
very popular, we have tried here to retain its spirit. However, the majority
of this text is devoted to new material which reflects the significant progress
of the theory during recent years. In particular, acoustic fields in a moving
fluid are considered and much attention is paid to sound propagation in range
dependent environments, which is currently on the leading edge of research
activities. Old topics are treated from new points of view afforded by recently
devised theoretical methods.
Although the book is devoted to acoustical waves in fluids, most of the
developed approaches are equally useful in studying elastic waves in solids
and also electromagnetic waves.
The authors are grateful to S.V. Burenkov, V.V. Goncharov, V.M.
Kurtepov and A.G. Voronovich for discussions of many issues treated in
the monograph, and to T.!. Tzyplakova for great help in preparing the
manuscript.
Moscow L.M. Brekhovskikh
February, 1992 GA. Godin
Contents
1. Reflection and Refraction of Spherical Waves ........... 1
1.1 Integral Representation of the Sound Field .............. 1
1.2 Reflected Wave ...................................... 5
1.3 Refracted Wave ...................................... 16
1.4 Very Large or Very Small Ratio of Media Densities.
Reflection from an Impedance Boundary ............... . 21
1.5 Weak Boundaries ................................... . 26
1.6 Reflection from a Moving Medium 33
2. Reflection of Bounded Wave Beams 41
2.1 Displacement of a Reflected Beam ..................... . 42
2.1.1 Classical Expression for Displacement ........... . 42
2.1.2 Examples of Beam Displacement ............... . 44
2.2 Incidence Angle Close to Angle of Total Reflection ...... . 47
2.2.1 Displacement of the Maximum
of the Beam Envelope ......................... . 47
2.2.2 The Role of Absorption ....................... . 52
2.2.3 Displacement of the "Centroid" of a Beam ...... . 53
2.3 Approach to Beam Displacement
Using Energy Considerations ......................... . 58
2.4 Incidence Angle Close to 7r /2 ......................... . 63
2.5 Reflection from a Boundary with Refraction Index
Close to Unity ...................................... . 65
2.6 The Goos-Hanchen Effect ............................ . 71
2.7 "Nonspecular Effects"
Accompanying Beam Reflection ........................ 72
2.7.1 Longitudinal Displacement of a Beam ............ 73
2.7.2 Deviation of the Beam Reflection Angle
from the Angle of Incidence .................... 76
2.8 Some Remarks About Beam Reflection
at a Fluid-Solid Interface ............................. 79
2.9 Concluding Remarks 79
3. The Lateral Wave ....................................... 81
3.1 Physical Interpretation and Significance ................. 81
X Contents
3.2 The Ray Approach ................................... 85
3.2.1 Ray Displacement upon Reflection ............... 85
3.2.2 Caustics of Usual and Diffracted Rays ........... 86
3.2.3 Lateral Rays in a Moving Medium ............... 89
3.3 Region of Observation of a Lateral Wave ................ 91
3.3.1 Two Lossy Homogeneous Halfspaces in Contact ... 91
3.3.2 Physical Interpretation ......................... 93
3.3.3 The General Case ............................. 95
3.4 Lateral Waves in Layered Media ....................... 95
3.4.1 Very Large Horizontal Source-Receiver
Separations ................................... 95
3.4.2 Review of Other Problems ...................... 99
3.5 Lateral Wave Generation by a Directional Source ........ 102
3.5.1 Lateral Waves in Sound Beam Reflection ......... 102
3.5.2 Distributed Sound Source 107
3.6 Weakly Uneven Boundaries ............................ 108
3.6.1 The Mean Field ............................... 109
3.6.2 Random Lateral Wave
from a Plane Incident Wave 112
3.6.3 Random Lateral Wave from a Point Source 114
4. Exact Theory of the Sound Field
in Inhomogeneous Moving Media ....................... 121
4.1 Wave Equation
for Nonstationary (Nonsteady-State) Moving Media ...... 121
4.1.1 Linearization of Hydrodynamics Equations ....... 121
4.1.2 Exact Wave Equations ......................... 124
4.1.3 Sound Wave Equation
for a Medium with Slow Currents ............... 126
4.2 Reciprocity Relations ................................. 130
4.2.1 Reciprocity Principle for a Medium at Rest ....... 130
4.2.2 Layered Moving Media. Flow Reversal Theorem 131
4.2.3 Flow Reversal Theorem and the Reciprocity
Principle for Homogeneous Media and
Homogeneous Flow ............................ 133
4.3 Exact Solutions of the Wave Equations
for a Point Source .................................... 135
4.3.1 The Point Source in Homogeneous Moving Media 135
4.3.2 Integral Representation of the Field
in a Layered Medium .......................... 142
4.3.3 Sound Field in a Medium Where Sound Velocity
Is a Linear Function of z ....................... 144
4.3.4 Sound Field in a Medium Where the Squared
Refraction Index Is a Linear
or Quadratic Function of Coordinates ............ 146
Contents XI
4.4 Discrete Spectrum of a Field. Normal Modes ............ 150
4.4.1 Discrete Spectrum in a Medium at Rest .......... 151
4.4.2 A Linear Source in a Waveguide ................. 152
4.4.3 Discrete Spectrum of a Field of a Point Source
in a Moving Medium .......................... 155
4.4.4 More About the Structure of the Discrete
Spectrum of a Point Source
in a Moving Medium .......................... 160
4.4.5 Formulas of More Convenience .................. 166
4.5 Phase and Group Velocities of Modes ................... 169
4.5.1 Generalized Orthogonality of Modes ............. 170
4.5.2 Mode Phase and Group Velocities
in a Medium at Rest ........................... 174
4.5.3 Phase and Group Velocities in a Moving Medium .. 177
4.6 The Epstein Waveguide ............................... 182
4.6.1 Waveguide with a Free Boundary ................ 182
4.6.2 Waveguide with an Absolutely Rigid Boundary 186
4.6.3 Comparison with Results Obtained
in the WKB Approximation .................... 188
5. High Frequency Sound Fields ........................... 193
5.1 Geometrical Acoustics Approximation
for a Localized Source ................................ 193
5.1.1 Ray Series. Eikonal Function ................... 193
5.1.2 Ray Equations. Ray Tube. Power Density Flow 194
5.1.3 A Three-Dimensionally Inhomogeneous Moving
Medium ...................................... 196
5.1.4 Layered Media and Horizontal Flow ............. 200
5.2 Ray Acoustics as a Limiting Case of Wave Theory ....... 202
5.2.1 The Case of a Moving Medium. . . . . . . . . . . . . . . . .. 202
5.2.2 Waveguide Sound Field in the Ray Approximation 205
6. The Field at and near a Caustic 209
6.1 Simple Caustics ...................................... 210
6.1.1 Definition .................................... 210
6.1.2 Caustics in Waveguides. Qualitative Results ...... 211
6.1.3 The Sound Field near an Ordinary Point
of a Caustic .................................. 211
6.1.4 Field near a Caustic in Terms of Ray Quantities 215
6.1.5 Limits of Validity ............................. 216
6.2 Reference Functions Method ........................... 217
6.2.1 Caustics in Media at Rest ...................... 217
6.2.2 The Reference Functions Method
for Solving One-Dimensional Wave Equations ..... 221