Modern Acoustics and Signal Processing

Editor-in-ChiefWILLIAM M. HARTMANN

Michigan State University, East Lansing, Michigan

Editorial Board

YOICHI ANDO, Kobe University, Kobe, JapanWHITLOW W. L. AU, Hawaii Institute of Marine Biology, Kane’ohe, HawaiiARTHUR B. BAGGEROER, Massachusetts Institute of Technology, Cambridge,

MassachusettsNEVILLE H. FLETCHER, Australian National University, Canberra, AustraliaCHRISTOPHER R. FULLER, Virginia Polytechnic Institute and State University,

Blacksburg, VirginiaWILLIAM A. KUPERMAN, University of California San Diego, La Jolla, CaliforniaJOANNE L. MILLER, Northeastern University, Boston, MassachusettsMANFRED R. SCHROEDER, University of Gottingen, Gottingen, GermanyALEXANDRA I. TOLSTOY, A. Tolstoy Sciences, McLean, Virginia

For further volumes:http://www.springer.com/series/3754

Finn B. Jensen � William A. KupermanMichael B. Porter � Henrik Schmidt

Computational OceanAcoustics

ABC

Second Edition

Finn B. JensenNATO Undersea Research CentreViale San Bartolomeo 400La Spezia, 19126, ItalyJensen@nurc.nato.int

William A. KupermanMarine Physical Lab., Mail Code 0205Scripps Institution of OceanographyGilman Dr. 9500La Jolla, CA 92093-0238, USAwkuperman@ucsd.edu

Michael B. PorterHeat, Light, and Sound Research, Inc.3366 North Torrey Pines Court, Suite 310La Jolla, CA 92037, USA

Henrik SchmidtMassachusetts Institute of Technology (MIT)Massachusetts Avenue 77Cambridge, MA 02139, USA

ISBN 978-1-4419-8677-1 e-ISBN 978-1-4419-8678-8DOI 10.1007/978-1-4419-8678-8Springer New York Dordrecht Heidelberg London

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Series Preface

“...Soun is noght but air y-broke”

– Geoffrey Chauserend of the 14th century

Traditionally, acoustics has formed one of the fundamental branches of physics. Inthe twentieth century, the field has broadened considerably and become increas-ingly interdisciplinary. At the present time, specialists in modern acoustics can beencountered not only in Physics Departments, but also in Electrical and MechanicalEngineering Departments, as well as in Departments of Mathematics, Oceanogra-phy, and even Psychology. They work in areas spanning from musical instrumentsto architecture to problems related to speech perception. Today, six hundred yearsafter Chauser made his brilliant remark, we recognize that sound and acoustics isa discipline extremely broad in scope, literally covering waves and vibrations in allmedia at all frequencies and at all intensities.

The series of scientific literature, entitled Modern Acoustics and Signal Process-ing (MASP), covers all areas of today’s acoustics as an interdisciplinary field. Itoffers scientific monographs, graduate level textbooks, and reference materials insuch areas as architectural acoustics, structural sound and vibration, musical acous-tics, noise, bioacoustics, physiological and psychological acoustics, speech, oceanacoustics; underwater sound; and acoustical signal processing.

Acoustics is primarily a matter of communication. Whether it be speech or mu-sic, listening spaces or hearing, signaling in sonar or in ultrasonography, we seekto maximize our ability to convey information, and at the same time, to minimizethe effects of noise. Signaling has itself given birth to the field of signal process-ing, the analysis of all received acoustic information or, indeed, all information inany electronic form. With the extreme importance of acoustics for both modern sci-ence and industry in mind, AIP Press is initiating this series as a new and promisingpublishing venture. We hope that this venture will be beneficial to the entire interna-tional acoustical community, as represented by the Acoustical Society of America, afounding member of the American Institute of Physics, and other related societiesand professional interest groups.

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It is our hope that scientists and graduate students will find the books in thisseries useful in their research, teaching, and studies.

James Russell Lowell once wrote: “In creating, the only hard thing’s to begin.”This is such a beginning.

Robert T. BeyerSeries Editor-in-Chief

Preface to the Second Edition

This is the second edition of our book Computational Ocean Acoustics, revisedand supplemented, including much new material reflecting the progress in compu-tational acoustics and related signal processing issues over the past 17 years. Newmaterial appears throughout the book, but we should like to draw attention to thefollowing topics: the basic theory of waveguide invariants in Chap. 2, with a gen-eralization of the concept to realistic, range-dependent waveguides in Chap. 5. Thepresentation of ray methods in Chap. 3 has been significantly modified to provide amore intuitive development of the fundamental ray concepts. Some intricate issues(aliasing, etc.) related to discrete wavenumber integration in Chap. 4 have been ex-plained in detail, including several illustrative examples. Also, the extension of thewavenumber-integration technique to 3-D scattering and reverberation scenarios inhorizontally-stratified waveguides has been included in Chap. 4, together with sev-eral illustrative numerical examples. Within the framework of normal-mode theory(Chap. 5), there is new material on mode identification, as well as on normal modesin elastic media. In addition, a section on scattering from objects in a waveguidehas been added, together with a 3-D example of mode coupling around seamounts.Chap. 6 has been updated with recent developments in parabolic-equation modeling,notably improvements in dealing with elastic media, and 4-D code implementationsfor pulse propagation in general 3-D environments. In Chap. 7, we have added adescription of the virtual-source concept (VSC) for target scattering, and also ex-panded on the finite-element (FE) section to address the use of this technique fortarget scattering in ocean waveguides. Several numerical examples illustrate currentcapabilities in FE=VSC modeling. Chapter 8 contains new material on Doppler shiftin a waveguide, based both on wavenumber-integration and normal-mode theory.Chapter 9 has a new section dedicated to the extraction of time-domain Green’sfunctions from noise correlation functions. Finally, recent developments in sig-nal processing for sonar applications have been added to Chap. 10, specifically ontime-domain processing, vector-sensor beamforming, synthetic signal and sensorstimulation, and phase conjugation and time reversal. Finally, extensive use of colorillustrations throughout has improved the appearance of this book significantly.

The authors wish to thank the many colleagues who provided material forthe book or reviewed parts of the manuscript. These include Michael Ainslie,Michael Collins, Kevin Cockrell, Lee Culver, Gerald D’Spain, Stephanie Fried,

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Peter Gerstoft, Oleg Godin, Paul Hursky, Kevin LePage, Wenyu Luo, Ed McDonald,Peter Nielsen, Philippe Roux, Hee-Chun Song, Frederic Sturm, and Shane Walker.Special thanks go to Mario Zampolli for helping out with the section on finite-element applications. Three of the authors’ research reported here has been sup-ported by the U.S. Office of Naval Research, Ocean Acoustics Program, while oneauthor (FBJ) has been supported for 35 years by the NATO Undersea ResearchCentre. This support is gratefully acknowledged.

Since this book represents a major milestone in the authors’ research careers,we wish to acknowledge the unwavering support from our wives, Patrizia, Gaby,Laurel, and Satu, to whom this new edition is dedicated.

Finn B. JensenWilliam A. Kuperman

Michael B. PorterHenrik Schmidt

Preface to the First Edition

The importance of computers in scientific research today is universally recognized.Several new journals dedicated to the fields of Computer Science and Computa-tional Physics have appeared over the past decade, and it is evident that computershave already changed our approach to doing science, both experimentally andtheoretically.

A statement by Prof. Norman Zabusky in the October 1987 issue of PhysicsToday clearly points to the crux of the matter: “We are in the midst of a com-putational revolution that will change science and society as dramatically as theagricultural and industrial revolutions did. The discipline of computational scienceis already significantly affecting the way we do hard and soft science. Comput-ers with fast, interactive visualization peripherals have come of age and provide amode of working that is coequal with laboratory experiments and observations andwith theory and analysis. We can now grapple with nonlinear and complexly inter-coupled phenomena in a relatively short time and provide insight for quantitativeunderstanding and better prediction.”

The above comments on the impact of computers on research in virtually all dis-ciplines of science certainly also apply to the field of ocean acoustics, where wehave observed an explosive growth in the development and use of numerical mod-els since the mid-1970s. Numerical models have become standard research tools inacoustic laboratories, and computational acoustics is becoming an ever more impor-tant branch of the ocean acoustic science. Only the numerical approach allows usto include the full complexity of the acoustic problem, and moreover, a numericalexperiment is both faster and cheaper than an actual ocean experiment.

Since ocean acoustic modeling has now reached a mature state of development,we feel that the timing is right for a textbook on computational acoustics. Thebook is intended to present the state-of-the-art of numerical techniques as appliedto solving the wave equation in heterogeneous fluid–solid media. Emphasis is ondeveloping the theoretical foundation which directly leads to numerical implemen-tations for real ocean environments. Various computational schemes are discussedin detail, and, in contrast to standard textbooks where acoustic effects are demon-strated through analytical expressions, fundamental propagation features are hereillustrated graphically, often in color.

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It is anticipated that the text will be useful to both universities and governmentlaboratories. In universities, such material is covered in various departments includ-ing Applied Mathematics, Geology and Geophysics, and Ocean Engineering, or inspecial programs on Acoustics. The text would be appropriate either as a course di-rectly in computational ocean acoustics or as a principal reference in a more generalcourse on computational techniques for problems in wave propagation. The level ofthe text is suitable for either graduate students or undergraduates at the senior level.

The other important audience for this text is in government (especially Navy)and industry laboratories, which are involved in either the development or use ofpropagation models. For users, a principal problem is that of understanding both themathematical and numerical limitations of the various modeling techniques.

The authors all have many years of experience in the field. By involving fourauthors, each one being an expert on one or more of the topics listed in the table ofcontents, it was hoped to provide an in-depth treatment of all aspects of computa-tional ocean acoustics, ranging from the fundamentals of wave propagation theory,over particular solution techniques (rays, wavenumber integration, normal modes,parabolic equations), to the detection of signals in the presence of noise.

Much of the research which forms the basis for this book was carried out at theSACLANT Undersea Research Centre, La Spezia, Italy. This institution has hada consistent effort in acoustic modeling since the mid-1970s, and all four authorshave spent varying lengths of time at SACLANTCEN since then. It is the inspiringand fertile research environment at this Centre together with its excellent computingand graphics facilities which is the real reason for this book becoming a reality. Theauthors also wish to express their gratitude to the members of the SACLANTCENEnvironmental Modeling Group, Giancarlo Dreini, Carlo Ferla, Cinzia Isoppo andGiovanna Martinelli, for their assistance in the development and maintenance of thevarious acoustic models. We are also grateful to Alvaro Carrara for helping with theillustrations for the book.

Finally, we wish to thank the many colleagues who performed a critical reviewof parts of the manuscript at various stages of preparation. Among those are DaleEllis, Richard Evans, Joo-Thiam Goh, Frank Ingenito, Ed McDonald, John Perkins,Richard Pitre, George Rolt, and Alex Tolstoy. Special thanks go to Mike Collins forhelping us keep abreast of the most recent parabolic equation developments, and toKen Rolt for doing a thorough and detailed editing of the entire manuscript.

Finn B. JensenWilliam A. Kuperman

Michael B. PorterHenrik Schmidt

Contents

1 Fundamentals of Ocean Acoustics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 The Ocean-Acoustic Environment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.3 Some Acoustic Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.3.1 Sources and Receivers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101.3.2 Energy, Power and Intensity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.3.3 Relevant Units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131.3.4 Transmission Loss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

1.4 Sound Propagation in the Ocean . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151.4.1 Characteristic Propagation Paths . . . . . . . . . . . . . . . . . . . . . . . . . . 151.4.2 Deep Water . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171.4.3 Shallow Water . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281.4.4 Range-Dependent Environments .. . . . . . . . . . . . . . . . . . . . . . . . . 32

1.5 Volume Attenuation .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 351.5.1 Attenuation of Plane Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 351.5.2 Attenuation in Seawater . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

1.6 Bottom Loss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 381.6.1 Fluid–Fluid Interface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 401.6.2 Fluid–Solid Interface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 431.6.3 Layered Fluid Halfspace. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 451.6.4 Arbitrary Layering .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

1.7 Boundary and Volume Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 511.7.1 Surface Scattering. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 521.7.2 Bottom Scattering .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 541.7.3 Volume Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

1.8 Ambient Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 571.9 Sound Propagation Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

2 Wave Propagation Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 652.1 The Wave Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

2.1.1 The Nonlinear Wave Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 662.1.2 The Linear Wave Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

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2.2 The Helmholtz Equation.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 712.3 Homogeneous Media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

2.3.1 Coordinate Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 732.3.2 Source in Unbounded Medium .. . . . . . . . . . . . . . . . . . . . . . . . . . . 752.3.3 Source in Bounded Medium.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 772.3.4 Point Source in Fluid Halfspace. . . . . . . . . . . . . . . . . . . . . . . . . . . 802.3.5 Transmission Loss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

2.4 Layered Media and Waveguides . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 832.4.1 Integral Transform Techniques .. . . . . . . . . . . . . . . . . . . . . . . . . . . 842.4.2 Source in Fluid Halfspace . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 872.4.3 Reflection and Transmission . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 932.4.4 Ideal Fluid Waveguide .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1022.4.5 The Pekeris Waveguide .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1182.4.6 Waveguide Invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .133

2.5 Deep-Ocean Waveguides . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1392.5.1 Exact Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1402.5.2 WKB Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .143

Appendix 1: Principle of Reciprocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .147Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .149References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .152

3 Ray Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1553.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1553.2 Ray Bending . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1563.3 Mathematical Derivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .158

3.3.1 Solving the Eikonal Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1603.3.2 Solving the Transport Equation . . . . . . . . . . . . . . . . . . . . . . . . . . .1633.3.3 Ray Amplitudes and Jacobians . . . . . . . . . . . . . . . . . . . . . . . . . . . .1663.3.4 Initial Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1673.3.5 Intensity Calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .168

3.4 Ray Anomalies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1753.4.1 Caustics and Shadow Zones . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1753.4.2 Region of Validity of the Ray Solution . . . . . . . . . . . . . . . . . . .178

3.5 Gaussian Beams. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1803.5.1 Gaussian Beams in Free Space . . . . . . . . . . . . . . . . . . . . . . . . . . . .1803.5.2 Gaussian Beam Tracing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .181

3.6 Additional Mathematical Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1853.6.1 Alternate Forms of the Ray Equations .. . . . . . . . . . . . . . . . . . .1853.6.2 Treatment of Attenuation .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1873.6.3 Interfaces and Boundaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1883.6.4 Weak Interfaces and Ocean Sound-Speed

Discontinuities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1943.6.5 Fermat’s Principle. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1953.6.6 Simplifications for Stratified Media. . . . . . . . . . . . . . . . . . . . . . .196

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3.6.7 Snell’s Law .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1983.6.8 Reciprocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .199

3.7 Numerical Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2003.7.1 Direct Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2013.7.2 Cell Methods: n2 Linear . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2063.7.3 Cell Methods: c Linear . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2083.7.4 False Caustics and Profile Interpolation . . . . . . . . . . . . . . . . . .2113.7.5 Finding Eigenrays.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .213

3.8 Extensions and Related Techniques .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2163.8.1 The WKB Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2163.8.2 Ray Theory via the WKB Approximation .. . . . . . . . . . . . . . .2193.8.3 The Ray Invariant and the Waveguide Invariant .. . . . . . . . .2203.8.4 Hamiltonian Formulation of Ray Acoustics . . . . . . . . . . . . . .2233.8.5 Three-Dimensional Ray Tracing . . . . . . . . . . . . . . . . . . . . . . . . . .225

Appendix 1: Recipe for Simple Ray Code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .226Appendix 2: A Useful Property of the Jacobian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .227Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .228References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .230

4 Wavenumber Integration Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2334.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2334.2 Mathematical Derivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .235

4.2.1 Integral Transform Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2354.2.2 Homogeneous Fluid Layers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2374.2.3 n2-Linear Fluid Layers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2384.2.4 Homogeneous Elastic Layers . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2404.2.5 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2424.2.6 Attenuation .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .243

4.3 Numerical Solution of the Depth Equation . . . . . . . . . . . . . . . . . . . . . . . . . .2444.3.1 Direct Global Matrix Approach .. . . . . . . . . . . . . . . . . . . . . . . . . .2454.3.2 Propagator Matrix Approach .. . . . . . . . . . . . . . . . . . . . . . . . . . . . .2514.3.3 Invariant Embedding Approach . . . . . . . . . . . . . . . . . . . . . . . . . . .255

4.4 Reflection Coefficients. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2574.5 Wavenumber Integration.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .258

4.5.1 Fast Field Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2604.5.2 Truncation of Integration Interval . . . . . . . . . . . . . . . . . . . . . . . . .2614.5.3 Wavenumber Discretization: Aliasing . . . . . . . . . . . . . . . . . . . .2624.5.4 FFP: Fast Field Program.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2654.5.5 Complex Contour Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2684.5.6 Fast Hankel Transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2724.5.7 Trapezoidal Rule Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2764.5.8 Filon Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2784.5.9 Adaptive Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .279

4.6 Frequency Integration.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2804.7 Range-Dependent Propagation .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2814.8 3-D Wavenumber Integration.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .282

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4.9 Scattering and Reverberation in a Stratified Ocean . . . . . . . . . . . . . . . . .2844.9.1 Target Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2854.9.2 Rough Interface Reverberation .. . . . . . . . . . . . . . . . . . . . . . . . . . .2874.9.3 Scattering from Volume Inhomogeneities . . . . . . . . . . . . . . . .292

4.10 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2974.10.1 Waveguide with an Elastic Bottom . . . . . . . . . . . . . . . . . . . . . . .2974.10.2 The Bucker Waveguide . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3004.10.3 Beam Reflection and Transmission . . . . . . . . . . . . . . . . . . . . . . .3024.10.4 Arctic Propagation .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3054.10.5 Seabed Target Scattering and Reverberation . . . . . . . . . . . . .3094.10.6 Reverberation from Seabed Volume Inhomogeneities . . .314

Appendix 1: Recipe for Simple WI=FFP Code. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .320Appendix 2: Roughness Perturbation Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . .324Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .328References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .332

5 Normal Modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3375.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3375.2 Mathematical Derivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .338

5.2.1 Point Source in Cylindrical Geometry .. . . . . . . . . . . . . . . . . . .3385.2.2 Line Source in Plane Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . .341

5.3 Modal Expansion of the Green’s Function . . . . . . . . . . . . . . . . . . . . . . . . . .3435.4 The Isovelocity Problem.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3445.5 A Generalized Derivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3495.6 A Deep Water Problem: The Munk Profile . . . . . . . . . . . . . . . . . . . . . . . . . .3565.7 Numerical Approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .360

5.7.1 Finite-Difference Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3625.7.2 Layer Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3725.7.3 Shooting Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3735.7.4 Root Finders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3755.7.5 Choice of Numerical Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . .379

5.8 Prufer Transformations and Mode Counting . . . . . . . . . . . . . . . . . . . . . . . .3805.8.1 Top Halfspaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3825.8.2 Bottom Halfspaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .384

5.9 Modal Perturbation Theory .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3855.9.1 Modal Propagation Loss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3855.9.2 Modal Group Velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .390

5.10 Elastic Media .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3915.10.1 Governing Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3915.10.2 Numerical Discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3945.10.3 Shooting Methods and Compound Matrices . . . . . . . . . . . . .3955.10.4 Boundary and Interface Conditions .. . . . . . . . . . . . . . . . . . . . . .3975.10.5 Numerical Example .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .400

5.11 Normal Modes for Range-Dependent Environments.. . . . . . . . . . . . . . .4025.11.1 Coupled Modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4035.11.2 One-Way Coupled Modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .407

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5.11.3 The Adiabatic Approximation.. . . . . . . . . . . . . . . . . . . . . . . . . . . .4085.11.4 Example: A Warm-Core Eddy . . . . . . . . . . . . . . . . . . . . . . . . . . . .411

5.12 Scattering from Objects in a Waveguide .. . . . . . . . . . . . . . . . . . . . . . . . . . . .4135.12.1 Scattering Geometry .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4135.12.2 The Plane-Wave Scattering Function . . . . . . . . . . . . . . . . . . . . .4145.12.3 Scattering from Spherical Objects in a Waveguide.. . . . . .4175.12.4 Scattering from Non-Spherical Objects . . . . . . . . . . . . . . . . . . .421

5.13 Normal Modes for 3-D Varying Environments .. . . . . . . . . . . . . . . . . . . . .4235.13.1 Horizontal Refraction Equations .. . . . . . . . . . . . . . . . . . . . . . . . .4235.13.2 Global Propagation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4265.13.3 3-D Mode Coupling Around Seamounts . . . . . . . . . . . . . . . . .429

5.14 Waveguide Invariant and Dispersion for Realistic Environments . .4385.14.1 The Waveguide Invariant Is Variable! . . . . . . . . . . . . . . . . . . . . .4405.14.2 Range-Dependent Group Speed and

Adiabatic Mode Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4415.14.3 Waveguide Invariant for Range-Dependent

Environments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .443Appendix 1: Recipe for Simple Mode Code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .445Appendix 2: Evaluation of the Normalization Term . . . . . . . . . . . . . . . . . . . . . . . . .447Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .449References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .452

6 Parabolic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4576.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4576.2 Derivation of Parabolic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .458

6.2.1 Standard PE Derivation .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4586.2.2 Generalized PE Derivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4606.2.3 Expansion of the Square-Root Operator .. . . . . . . . . . . . . . . . .4616.2.4 Phase Errors and Angular Limitations . . . . . . . . . . . . . . . . . . . .466

6.3 The Elastic PE. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4706.4 Starting Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .472

6.4.1 Numerical Starters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4736.4.2 Analytical Starters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4766.4.3 Spectral Properties of Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . .482

6.5 Solutions by FFTs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4866.5.1 The Split-Step Fourier Algorithm .. . . . . . . . . . . . . . . . . . . . . . . .4876.5.2 Error Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4906.5.3 Numerical Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4936.5.4 Variable Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4956.5.5 Attenuation .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .496

6.6 Solutions by FDs and FEs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4976.6.1 Field Equations on Horizontal Interfaces . . . . . . . . . . . . . . . . .4986.6.2 IFD Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5006.6.3 Error Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5036.6.4 Numerical Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .505

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6.7 The Problem of Energy Conservation in PEs . . . . . . . . . . . . . . . . . . . . . . . .5066.8 Three-Dimensional PEs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .509

6.8.1 Expansion of the Square-Root Operator .. . . . . . . . . . . . . . . . .5096.9 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .512

6.9.1 Beam Propagation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5126.9.2 Propagation in a 2-D Wedge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5156.9.3 Propagation Over a Seamount . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5166.9.4 Propagation Over a Sloping Elastic Bottom .. . . . . . . . . . . . .5176.9.5 Propagation in a 3-D Wedge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .517

Appendix 1: Recipe for Simple PE Code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .523Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .526References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .527

7 Finite Differences and Finite Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5317.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5317.2 Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5327.3 Finite-Difference Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .534

7.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5347.3.2 Difference Approximations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5347.3.3 Convergence and Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5377.3.4 The Wave Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .538

7.4 Finite-Element Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5457.4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5457.4.2 Mathematical Derivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5477.4.3 The Acoustic Wave Equation.. . . . . . . . . . . . . . . . . . . . . . . . . . . . .5517.4.4 The Elastic Wave Equation .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5627.4.5 Coupled Fluid–Elastic Domains . . . . . . . . . . . . . . . . . . . . . . . . . .5657.4.6 Steady-State Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5667.4.7 Perfectly Matched Layers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5667.4.8 Time Recurrence .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .569

7.5 Boundary-Element Methods. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5707.5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5707.5.2 The Boundary-Integral Equation .. . . . . . . . . . . . . . . . . . . . . . . . .5717.5.3 Boundary-Element Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5737.5.4 Coupled Domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5767.5.5 Virtual Source Concept . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .580

7.6 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5847.6.1 Scattering by Arctic Ice Features . . . . . . . . . . . . . . . . . . . . . . . . .5847.6.2 Scattering from Objects Near Interfaces . . . . . . . . . . . . . . . . . .590

Appendix 1: Variational Formulation for Fluid–Elastic Interaction .. . . . . . . .598Appendix 2: Farfield Computations .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .604Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .605References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .607

Contents xvii

8 Broadband Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .6118.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .6118.2 Fourier Synthesis of Frequency-Domain Solutions . . . . . . . . . . . . . . . . .612

8.2.1 Evaluation by FFT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .6138.2.2 Complex Frequency Integration .. . . . . . . . . . . . . . . . . . . . . . . . . .616

8.3 Time-Domain Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .6178.3.1 Ray Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .6178.3.2 Spectral Integral Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .6198.3.3 Parabolic Equations .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .621

8.4 Doppler Shift in a Waveguide . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .6238.4.1 Wavenumber Integral Representation .. . . . . . . . . . . . . . . . . . . .6258.4.2 Normal Mode Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . .6288.4.3 Doppler Shift for Active Sonar . . . . . . . . . . . . . . . . . . . . . . . . . . . .630

8.5 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .6338.5.1 The Head-Wave Problem .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .6338.5.2 Mode Dispersion in a Waveguide . . . . . . . . . . . . . . . . . . . . . . . . .6368.5.3 3-D Wedge Propagation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .6408.5.4 Seismic Interface Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .6438.5.5 Deep-Water Propagation.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .6478.5.6 Surface-Duct Propagation with Leakage . . . . . . . . . . . . . . . . .6528.5.7 Acoustic Emission from Ice Fractures . . . . . . . . . . . . . . . . . . . .654

Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .657References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .659

9 Ambient Noise. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .6619.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .6619.2 Surface Noise in a Stratified Ocean . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .662

9.2.1 Mathematical Derivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .6639.2.2 Spatial Distribution of Noise Sources. . . . . . . . . . . . . . . . . . . . .6659.2.3 Wavenumber Integral Representation .. . . . . . . . . . . . . . . . . . . .6669.2.4 Normal Mode Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . .6679.2.5 Noise in a Homogeneous Halfspace . . . . . . . . . . . . . . . . . . . . . .6699.2.6 Noise in Stratified Media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .671

9.3 Extracting Time-Domain Green’s Functionsfrom Noise Correlation Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .6789.3.1 The Time-Domain Green’s Function . . . . . . . . . . . . . . . . . . . . .6809.3.2 Emergence of Coherent Wavefronts from Noise . . . . . . . . .6819.3.3 Data Examples of Extracting Wavefronts

from Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .6829.4 Surface Noise in a Three-Dimensional Ocean . . . . . . . . . . . . . . . . . . . . . . .686

9.4.1 Noise Modeling by Adiabatic Modes . . . . . . . . . . . . . . . . . . . . .6879.4.2 Simulated Noise Fields in 3-D Environments . . . . . . . . . . . .6919.4.3 Noise Modeling by PE. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .6949.4.4 Downslope and Deep-Ocean-Basin Noise Field . . . . . . . . .696

Appendix 1: Evaluation of the Cross-Spectral Density . . . . . . . . . . . . . . . . . . . . . .697Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .702References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .703

xviii Contents

10 Signals in Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .70510.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .70510.2 The Energetics of Signals in Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .706

10.2.1 Array Gain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .70610.2.2 Sonar Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .708

10.3 Plane-Wave Beamforming .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .71410.3.1 Linear Beamforming.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .71410.3.2 Adaptive Beamforming .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .71610.3.3 Multiple-Constraints Beamforming .. . . . . . . . . . . . . . . . . . . . . .72210.3.4 White-Noise Constraint Processor . . . . . . . . . . . . . . . . . . . . . . .723

10.4 Time-Domain Processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .72410.4.1 Isovelocity Time-Delay Beamforming.. . . . . . . . . . . . . . . . . . .72510.4.2 Non-Isovelocity Time-Delay Beamforming:

The Turning Point Filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .72510.4.3 Example: Passive Fathometer . . . . . . . . . . . . . . . . . . . . . . . . . . . . .726

10.5 Performance Prediction: Modeling, Beamformingand the Sonar Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .728

10.6 Matched-Field Processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .73010.7 Simulating Matched-Field Processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .733

10.7.1 Depth–Range Matched Field Processing . . . . . . . . . . . . . . . . .73410.7.2 Three-Dimensional Matched Field Processing .. . . . . . . . . .741

10.8 Vector-Sensor Beamforming . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .75010.9 Synthetic Signals and Sensor Stimulation . . . . . . . . . . . . . . . . . . . . . . . . . . .754

10.9.1 Stochastic Signal and Noise Model . . . . . . . . . . . . . . . . . . . . . . .75510.9.2 Snapshot Synthesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .75510.9.3 Signal Variability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .75610.9.4 Noise Realizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .757

10.10 Phase Conjugation and Time Reversal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .75810.10.1 Theory and Simulation for Phase

Conjugation=TRM in the Ocean . . . . . . . . . . . . . . . . . . . . . . . . . .75910.11 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .767Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .767References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .769

About the Authors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .773

Name Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .775

Subject Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .783