Modern Methods in Analytical Acoustics: Lecture Notes

746 Pages·1992·15.6 MB·English

by  D. G. Crighton

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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© Springer-Verlag Berlin Heidelberg 1992
Originally published by Springer-Verlag Berlin Heidelberg New York in 1992
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69/3830-543210 Printed on acid-free paper
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 .... xv
Acknowledgements xvii
Part I. The Classical Techniques of Wave Analysis 1
1 Complex Variable Theory
F.G. Leppington. . . . . 3
1.1 Complex Numbers ...... . 3
1.2 Exponential, Hyperbolic and Trigonometric
Functions . . . . . 7
1.3 Many-Valued Functions 10
1.4 Differentiation 16
1.5 Series Expansions 19
1.6 Integration. . . 23
1. 7 The Cauchy Integral 37
1.8 Analyticity. . . . 39
2 Generalized Functions
J.E. Ffowcs Williams. . 46
2.1 Introduction 46
2.2 The Two-Dimensional Delta Function 56
2.3 The Three-Dimensional Delta Function. 58
2.4 Convolution Algebra . . . . . 59
2.5 Development of Integral Equations . . 66
2.6 The Simple Harmonic Oscillator . . . 71
2.7 Green Functions for the Wave Equation 73
2.8 The Bending Wave Equation 75
2.9 The Reduced Wave Equation 76
2.10 Sound Waves with Damping 78
2.11 Internal Waves . . . . . 79
3 Fourier Transforms, Random Processes, Digital
Sampling and Wavelets
AP. Dowling . ........... . 80
3.1 Definitions and Formal Properties of Fourier
Transforms. . . . . . . . . . . . 80
viii
3.2 Transforms of Generalized Functions 88
3.3 Random Processes 94
3.4 Digital Sampling . 104
3.5 Wavelets. . . . 116
4 Asymptotic Evaluation of Integrals
F.G. Leppington. . . 124
4.1 Introduction 124
4.2 Watson's\Lemma . 128
4.3 Rapidly Oscillatory Integrals 130
4.4 Integrals with a Large Exponent 134
4.5 Diffraction Integrals 145
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 168
6.2 Heuristic Approach to Matching and its Pitfalls . 169
6.3 Formal Approach to Matching . . . . 173
6.4 Sound Generation by Forced Oscillations 179
6.5 Plane Wave Scattering. . . . . . . 184
6.6 Higher Approximations . . . . . . 187
6.7 Two-Dimensional Problems: Logarithmic Gauge
Functions . . . . . . . . . . . . . . 193
6.8 Purely Logarithmic Gauge Functions: Scattering
by Soft Bodies. . . 200
6.9 Composite Expansions 204
6.10 Conclusions 207
7 Multiple Scales
A.P. Dowling. . 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 311
11 Noise Source Mechanisms
J.E. Ffowcs Williams 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 346
11.14 A Semi-Infinite Plane . 348
11.15 Scattering by a Wedge 351
11.16 Scattering by a Rigid Disc 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 . . . . . 452
16.1 The Surface Pressure Spectrum on a Rigid Plane
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. . 525
18 Scattering and Diffraction
F.G. Leppington . . 527
18.1 Basic Equations 527
18.2 Exact Solutions 530
18.3 Approximate Solutions 538
18.4 Matched Asymptotic Expansions: Duct Problem 544
19 Inverse Scattering
F.G. Leppington . . 550
19.1 Introduction. . . . . . . 550
19.2 Method of Imbriale & Mittra . 553
19.3 Optimization Method 558
19.4 High-Frequency Limit 561
20 Resonators
M. Heckl .. 565
20.1 Introduction. . . . . . . . . 565
20.2 Single Degree of Freedom Systems 566
20.3 Multi Degree of Freedom Systems . 580
20.4 Continuous, Resonating Systems 586
21 Bubbles
D.G. Crighton 595
21.1 Introduction. . . . . . 595
21.2 Motion of a Single Bubble. 596
21.3 Sound Speed in Bubbly Liquid - Low Frequencies 600
21.4 Sound Speed in Bubbly Liquid - Dispersive Effects 602
21.5 Nonlinear Waves in Bubbly Liquid . . . . . 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