MODERN METHODS IN ANALYTICAL ACOUSTICS Lecture Notes

D.G. Crighton, A.P. Dowling, J.E. Ffowcs Williams, M. Heckl and F.G. Leppington

MODERN METHODS IN ANALYTICAL ACOUSTICS Lecture Notes

With 175 Figures

Springer-Verlag Berlin Heidelberg GmbH

Cover illustration: Ch. 10, Fig. 8. Sound rays in the deep ocean with a Munk-profile for a source at a depth of 1000 m. (After Porter & Bucker, 1987.)

ISBN 978-3-540-19737-9

British Library Cataloguing-in-Publication Data Modern methods in analytical acoustics: lecture notes.

I. Crighton, D. G., 1942-620.2

ISBN 978-3-540-19737-9

Library of Congress Cataloging-in-Publication Data Modern methods in analytical acoustics: lecture notes / D.G. Crighton

. . . [et al.]. p. cm.

Includes index. ISBN 978-3-540-19737-9 ISBN 978-1-4471-0399-8 (eBook) DOI 10.1007/978-1-4471-0399-8 1. Sound-waves. 2. Vibrations. I. Crighton, D.G., 1942-

QC243.M63 1992 91-41977 620.2 - dc20 CIP

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D.G. Crighton Fellow of St John's College, Cambridge, and The Professor of Applied Mathematics and Head of the Department of Applied Mathematics and Theoretical Physics University of Cambridge

A.P. Dowling Fellow of Sidney Sussex College, Cambridge, and Reader in Acoustics, Department of Engineering University of Cambridge

J.E. Ffowcs Williams Fellow of Emmanuel College, Cambridge, and The Rank Professor of Engineering (Acoustics), Department of Engineering University of Cambridge

M. Heckl Professor of the Institut fur Technische Akustik, Technische Universitat Berlin

F.G. Leppington Professor of Applied Mathematics, and Head of the Mathematics Department, The Imperial College of Science and Technology London.

CONTENTS

Preface .... Acknowledgements

Part I. The Classical Techniques of Wave Analysis

1 Complex Variable Theory F.G. Leppington. . . . .

1.1 Complex Numbers ...... . 1.2 Exponential, Hyperbolic and Trigonometric

Functions . . . . . 1.3 Many-Valued Functions 1.4 Differentiation 1.5 Series Expansions 1.6 Integration. . . 1. 7 The Cauchy Integral 1.8 Analyticity. . . .

2 Generalized Functions J.E. Ffowcs Williams. .

2.1 Introduction 2.2 The Two-Dimensional Delta Function 2.3 The Three-Dimensional Delta Function. 2.4 Convolution Algebra . . . . . 2.5 Development of Integral Equations . . 2.6 The Simple Harmonic Oscillator . . . 2.7 Green Functions for the Wave Equation 2.8 The Bending Wave Equation 2.9 The Reduced Wave Equation 2.10 Sound Waves with Damping 2.11 Internal Waves . . . . .

3 Fourier Transforms, Random Processes, Digital Sampling and Wavelets AP. Dowling . ........... .

3.1 Definitions and Formal Properties of Fourier Transforms. . . . . . . . . . . .

xv xvii

1

3

3

7 10 16 19 23 37 39

46

46 56 58 59 66 71 73 75 76 78 79

80

80

viii

3.2 Transforms of Generalized Functions 3.3 Random Processes 3.4 Digital Sampling . 3.5 Wavelets. . . .

88 94

104 116

4 Asymptotic Evaluation of Integrals F.G. Leppington. . . 124

124 128 130 134 145

4.1 Introduction 4.2 Watson's\Lemma . 4.3 Rapidly Oscillatory Integrals 4.4 Integrals with a Large Exponent 4.5 Diffraction Integrals

5 Wiener-Hopf Technique F.G. Leppington. . . . . 148

5.1 Introduction . . . . . . . . . . . 148 5.2 Wiener-Hopf Procedure for Vibrating String

Problem . . . . . . . . . . . . . 151 5.3 Wiener-Hopf Procedure for Integral Equation 155 5.4 Properties of Wiener-Hopf Decompositions. 157 5.5 Diffraction by Semi-infinite Rigid Plate:

Sommerfeld Problem . . . . . . . . 163

6 Matched Asymptotic Expansions Applied to Acoustics Dp. Crighton . . . . . . . . . . . . .. 168

6.1 Introduction 6.2 Heuristic Approach to Matching and its Pitfalls . 6.3 Formal Approach to Matching . . . . 6.4 Sound Generation by Forced Oscillations 6.5 Plane Wave Scattering. . . . . . . 6.6 Higher Approximations . . . . . . 6.7 Two-Dimensional Problems: Logarithmic Gauge

Functions . . . . . . . . . . . . . . 6.8 Purely Logarithmic Gauge Functions: Scattering

by Soft Bodies. . . 6.9 Composite Expansions 6.10 Conclusions

7 Multiple Scales A.P. Dowling. .

168 169 173 179 184 187

193

200 204 207

209

7.1 The Damped Harmonic Oscillator. . . . .. 209 7.2 The Effect of a Gradual Sound-Speed Variation on

Plane Waves . . . . . . . . 214 7.3 Comparison with the WKB Method 219 7.4 Ray Theory. . . . . . . . . 222

ix

8 Statistical Energy Analysis AI.lleckl ....... . 233

8.1 Introduction . . . . . . . . . 233 8.2 Power Flow Between Two Resonators 234 8.3 Power Flow in Multi-Modal Systems . 236 8.4 Thermodynamic Analogy . 253 8.5 Applications . . . . 255 8.6 Limits of Applicability. . 258

9 Mean Energy and Momentum Effects in Waves AI. lleckl . . . . . . . . . . . . . .. 260

9.1 Introduction . . . . . . . . . . .. 260 9.2 Hamilton's Principle and the Wave Equation. 261 9.3 Sound Intensity in Gases and Liquids 266 9.4 Intensity of Waves in Solids. . . 276 9.5 Mean Momentum and Wave Drag. 278

10 Numerical Methods AI.lleckl . .... 283

10.1 Introduction. . . . . . . . . . .. 283 10.2 Finite-Element Methods (FEM) in Acoustics 283 10.3 Boundary Element Methods (BEM) 287 10.4 Source Substitution Methods. . . 291 10.5 Applications of Fourier Transforms 297 10.6 Ray Tracing. . . . 302 10.7 Concluding Remarks . . . . . 309

Part II. The Generation of Unsteady Fields

11 Noise Source Mechanisms J.E. Ffowcs Williams

311

313

11.1 The Equations of Fluid Motion . . . . 313 11.2 Wave Equation for Compressible Fluids. 318 11.3 Inhomogeneous Wave Equation. 319 11.4 Monopole and Multipole Sources 319 11.5 Dipole Sources. . . . . 320 11.6 Quadrupole Sources . . . 321 11.7 The Flow Noise Equations 322 11.8 Sound and Pseudo-Sound . 323, 11.9 Sound Induced by Convected Turbulence 325 11.10 Boundary Effects on Flow Noise . . . 334 11.11 Surface Effects as a Problem in Diffraction 342 11.12 The Infinite Plane . . . . . . . .. 343

x

11.13 The Rigid Sphere 11.14 A Semi-Infinite Plane . 11.15 Scattering by a Wedge 11.16 Scattering by a Rigid Disc

346 348 351 352

12 Vortex Sound AP. Dowling . 355

12.1 Sound Radiation by a Compact Turbulent Eddy in an Unbounded Space . . . . . . .. 355

12.2 Howe's Acoustic Analogy . . . . . . .. 360 12.3 A Line Vortex Near a Semi-Infinite Rigid Plane. 367 12.4. Vortex Sound Near a Cylinder . . 370 12.5 Vortex Sound Near Compact Bodies . . 373

13 Thermoacoustic Sources and Instabilities AP. Dowling . . . . . . . . . . . . 378

13.1 Introduction to Thermoacoustic Sources. 378 13.2 Combustion Noise . . . . . . . . 381 13.3 The Diffusion of Mass and Heat. . . . 389 13.4 Acceleration of Density Inhomogeneities. 390 13.5 Turbulent Two-Phase Flow 391 13.6 Thermoacoustic Instabilities 399

Appendix. . . . . . . 404

14 Effects of Motion on Acoustic Sources A.P. Dowling . . . . . . . . . . . 406

14.1 Effects of Motion on Elementary Sources 406 14.2 Taylor's Transformation . . . . . . 415 14.3 Howe's Method. . . . . . . . . . 419 14.4 The Method of Matched Asymptotic Expansions. 422 14.5 The Lighthill Theory . . . . . . . 425

15 Propeller and Helicopter Noise J.E. Ffowcs Williams 428

15.1 Introduction. . . . . 428 15.2 Spectral Decomposition . 430 15.3 Time Domain Methods. 436 15.4 Broad-Band Propeller Noise 437 15.5 The Sound of Point Sources in Circular Motion 439

16 Flow Noise on Surfaces A.P. Dowling . . . . .

16.1 The Surface Pressure Spectrum on a Rigid Plane

452

Wall . . . . . . . ......... 452

xi

16.2 Surface Curvature 466 16.3 Flexible Surfaces . 473 16.4 Mean Flow Effects 485 16.5 Scattering from the Convective Peak to Low

Wavenumbers . . . . . . . 492 16.6 The Role of Surface Shear Stress 501 16.7 Summary . . . . . . . . 504

17 Fluid-Loading Interaction with Vibrating Surfaces D.G. Crighton . . . . . . . . . . 510

17.1 Introduction. . . . . . . . . . 510 17.2 The Different Roles of Fluid Loading . 511 17.3 Intrinsic Fluid-Loading Parameter 520 17.4 The Free Waves on a Fluid-Loaded Plate 521

Part III. Wave Modification. .

18 Scattering and Diffraction F.G. Leppington . .

18.1 Basic Equations 18.2 Exact Solutions 18.3 Approximate Solutions 18.4 Matched Asymptotic Expansions: Duct Problem

19 Inverse Scattering F.G. Leppington . .

19.1 Introduction. . . . . . . 19.2 Method of Imbriale & Mittra . 19.3 Optimization Method 19.4 High-Frequency Limit

20 Resonators M. Heckl ..

20.1 Introduction. . . . . . . . . 20.2 Single Degree of Freedom Systems 20.3 Multi Degree of Freedom Systems . 20.4 Continuous, Resonating Systems

21 Bubbles D.G. Crighton

21.1 Introduction. . . . . . 21.2 Motion of a Single Bubble. 21.3 Sound Speed in Bubbly Liquid - Low Frequencies 21.4 Sound Speed in Bubbly Liquid - Dispersive Effects 21.5 Nonlinear Waves in Bubbly Liquid . . . . .

525

527

527 530 538 544

550

550 553 558 561

565

565 566 580 586

595

595 596 600 602 606

xii

22 Reverberation M.Heckl . .. 610

22.1 Introduction. .. .......... 610 22.2 Decay of Resonances in Systems with Few Modes 610 22.3 Reverberation in Systems with Many Modes (e.g.

Large Rooms) . . . . . . 612 22.4 Examples of Sound Absorbers . . . .. 620 22.5 Sound Fields in Large Rooms . . . .. 625 22.6 Reverberation Caused by Many Scatterers . 628

23 Solitons D.G. Crighton 631

23.1 Introduction. . . . . . . 631 23.2 Water Waves; Linear Theory 634 23.3 Fourier Transform Solution of Initial-Value

Problem . . . . . . . . . 635 23.4 Weakly Nonlinear Theory. . . 637 23.5 The Korteweg-de Vries Equation 639 23.6 Inverse Spectral Transform . . 640 23.7 Direct Spectral or Scattering Problem 641 23.8 Time Evolution of the Spectral Data 642 23.9 Inverse Spectral Problem . . . . . 643

24 Nonlinear Acoustics D.G. Crighton . . . 648

24.1 Introduction. . . . . . . . . . . .. 648 24.2 Linear Local Behaviour, Nonlinear Cumulative

Behaviour . . . . . . . . . . . 649 24.3 Characteristic Solution for Simple Waves. 654 24.4 The Fubini Solution. . . . . . . .. 657 24.5 Multi-Valued Waveforms and Shock Waves 658 24.6 Thermoviscous Diffusion - The Burgers Equation 661 24.7 Shock-Wave Structure . 663 24.8 The Fay Solution. . . 665 24.9 Effects of Area Change . 668

25 Chaotic Dynamics and Applications in Acoustics D.G. Crighton . . . . . . . . . 671

25.1 Introduction. . . . . . . . . 671 25.2 Sensitivity to Initial Conditions 675 25.3 The Period-Doubling Route to Chaos 677 25.4 Other Routes to Chaos . . 686 25.5 Characterization of Chaos . 692 25.6 Summary . . . . . . 702

xiii

26 Anti-Sound J.E. Ffowcs Williams 705

26.1 Introduction. . 705 26.2 One-Dimensional Waves 708 26.3 One-Dimensional Waves Between Two Coherent

Sources . . . . . . . . . . . . .. 711 26.4 One-Dimensional Anti-Sound . . . . .. 713 26.5 The Optimal Physically Realizable Controller 719

Subject Index . . . . . . . . . . . . . .. 727

PREFACE

This book has grown out of lecture notes for a course specially constructed for the Admiralty some 25 years ago. The then Head of Sonar at Portland, Mr S. D. Mason, knew the importance of keeping abreast of fundamental advances in the subject and recognized the value of understanding the i9.eas and mastering the techniques used to develop new knowledge. Thirty scientists in his charge attended an eight hour session once a month in which four lecturers presented the analytical techniques of selected topics relevant to modern SONAR, illustrating their material with reference to particular recently published scientific papers. The course evolved over twelve months to cover noise and unsteady flow and was immediately repeated to a general extra-mural audience at Imperial College; that five day course was oversubs­cribed with professionals coming from several centres, mainly from the aeronautical and naval industries of Europe and the U.S.A. Since then the course has been given many times in Europe and the U.S.A., the scope has been expanded and the lecturing team has grown, but the primary aim of teaching the analytical techniques alongside specific areas of wave motion and unsteady fluid mechanics has remained. Sadly Charles Ellen, a founder member of the group, is no longer with us and is much missed. Each time the course was given the notes developed, evolving out of their lecture note style towards something different. But they remain essentially notes on which lectures are based that we now wish to put on a more permanent record. Some topics are more mature than others and some did not appear in particular courses.

The lecturers enjoyed the courses very much, speaking always on topics they felt important to the fundamental understanding of noise, vibration and fluid mechanisms. They felt the audience enjoyed them too. We hope the readers of this collection of notes will find them both instructive and fun to read.

There are many who have helped with the production of these notes to whom we owe a great debt. In particular, Nigel Peake made valuable suggestions on a number of chapters; and Naomi Coyle, who has typed the notes in camera ready form, deserves our special thanks for her calm efficiency.

November 1991 JEFfW

ACKNOWLEDGEMENTS

We are grateful for permission to reproduce various figures from the authors named in each caption, and from publishers as follows:

Academic Press (Figure 16.3) - Journal of Sound and Vibration (Figure 15.4); American Institute of Aeronautics & Astronautics (Figure 15.3); American Institute of Physics - Journal of the Acoustical Society of America (Figures 10.8,25.15, 25.16, 25.17, 25.18); Annual Reviews Inc. (Figures 21.1, 21.3); Cambridge University Press - Journal of Fluid Mechanics (Figures 12.5, 25.10, 25.11, 25.12, 25.13); Editions de Physique (Figure 3.5); Elsevier (Figure 9.2); Institute of Electrical and Electronics Engineers, Inc. (Figure 19.3); McGraw-Hill (Figure 25.1); "Nature" (Figure 25.14); North-Holland (Figures 25.4, 25.5, 25.6, 25.7, 25.8, 25.9); von Karman Institute for Fluid Dynamics (Figure 15.5).

Chapter 6 derives from lecture notes first given in a course at David Taylor Research Center, Md. in 1976. DGC gratefully acknowledges permission of the Center for reproduction of these notes here.

Chapter 16 has been prepared with the support of Ferranti­Thomson Sonar Systems UK Ltd. and the Ministry of Defence, which is gratefully acknowledged.

We are also indebted to the United States Office of Naval Research for support in the preparation of some parts of the material presented in these lecture notes.