About the Author ............................................... xi
Preface ...................................................... xiii
Acknowledgments .............................................. xvii
Part I ESSENCE OF ACOUSTICS
1 Acoustic Wave Equation and Its Basic Physical Measures ....... 3
1.1 Introduction ............................................ 3
1.2 One-Dimensional Acoustic Wave Equation .................. 3
1.2.1 Impedance ........................................ 9
1.3 Three-Dimensional Wave Equation ........................ 10
1.4 Acoustic Intensity and Energy .......................... 11
1.4.1 Complex-Valued Pressure and Intensity ........... 16
1.5 The Units of Sound ..................................... 18
1.6 Analysis Methods of Linear Acoustic Wave Equation ...... 27
1.6.1 Acoustic Wave Equation and Boundary Condition ... 28
1.6.2 Eigenfunctions and Modal Expansion Theory ....... 31
1.6.3 Integral Approach Using Green's Function ........ 35
1.7 Solutions of the Wave Equation ......................... 39
1.7.1 Plane Wave ...................................... 40
1.7.2 Spherical Wave .................................. 41
1.8 Chapter Summary ........................................ 46
References .................................................. 46
2 Radiation, Scattering, and Diffraction ...................... 49
2.1 Introduction/Study Objectives .......................... 49
2.2 Radiation of a Breathing Sphere and a Trembling
Sphere ................................................. 50
2.3 Radiation from a Baffled Piston4 ....................... 58
2.4 Radiation from a Finite Vibrating\Plate ................ 65
2.5 Diffraction and Scattering ............................. 70
2.6 Chapter Summary ........................................ 79
2.7 Essentials of Radiation, Scattering, and Diffraction ... 80
2.7.1 Radiated Sound Field from an Infinitely
Baffled Circular Piston ......................... 80
2.7.2 Sound Field at an Arbitrary Position Radiated
by an Infinitely Baffled Circular Piston ........ 81
2.7.3 Understanding Radiation, Scattering, and
Diffraction Using the Kirchhoff-Helmholtz
Integral Equation ............................... 82
2.7.4 Scattered Sound Field Using the Rayleigh
Integral Equation ............................... 96
References .................................................. 97
Part II SOUND VISUALIZATION
3 Acoustic Holography ........................................ 103
3.1 Introduction .......................................... 103
3.2 The Methodology of Acoustic Source Identification ..... 103
3.3 Acoustic Holography: Measurement, Prediction, and
Analysis .............................................. 106
3.3.1 Introduction and Problem Definitions ........... 106
3.3.2 Prediction Process ............................. 107
3.3.3 Mathematical Derivations of Three Acoustic
Holography Methods and Their Discrete Forms .... 113
3.3.4 Measurement .................................... 119
3.3.5 Analysis of Acoustic Holography ................ 124
3.4 Summary ............................................... 129
References ................................................. 130
4 Beamforming ................................................ 137
4.1 Introduction .......................................... 137
4.2 Problem Statement ..................................... 138
4.3 Model-Based Beamforming ............................... 140
4.3.1 Plane and Spherical Wave Beamforming ........... 140
4.3.2 The Array Configuration ........................ 142
4.4 Signal-Based Beamforming .............................. 145
4.4.1 Construction of Correlation Matrix in Time
Domain ......................................... 146
4.4.2 Construction of Correlation Matrix in
Frequency Domain ............................... 151
4.4.3 Correlation Matrix of Multiple Sound Sources ... 152
4.5 Correlation-Based Scan Vector Design .................. 160
4.5.1 Minimum Variance Beamformer .................... 160
4.5.2 Linear Prediction .............................. 164
4.6 Subspace-Based Approaches ............................. 170
4.6.1 Basic Principles ............................... 170
4.6.2 MUSIC Beamformer ............................... 173
4.6.3 ESPRIT ......................................... 180
4.7 Wideband Processing Technique ......................... 182
4.7.1 Frequency-Domain Approach: Mapping to the
Beam Space ..................................... 182
4.7.2 Coherent Subspace Method (CSM) ................. 184
4.7.3 Partial Field Decomposition in Beam Space ...... 185
4.7.4 Time-Domain Technique .......................... 190
4.7.5 Moving-Source Localization ..................... 198
4.8 Post-Processing Techniques ............................ 204
4.8.1 Deconvolution and Beamforming .................. 204
4.8.2 Nonnegativity Constraint ....................... 207
4.8.3 Nonnegative Least-Squares Algorithm ............ 209
4.8.4 DAMAS .......................................... 210
References ................................................. 212
Part III SOUND MANIPULATION
5 Sound Focusing ............................................. 219
5.1 Introduction .......................................... 219
5.2 Descriptions of the Problem of Sound Focusing ......... 221
5.2.1 Free-Field Radiation from Loudspeaker Arrays ... 221
5.2.2 Descriptions of a Sound Field Depending on
the Distance from the Array .................... 221
5.2.3 FresneI Approximation .......................... 223
5.2.4 Farfield Description of the Rayleigh Integral
(Fraunhofer Approximation) ..................... 225
5.2.5 Descriptors of Directivity ..................... 227
5.3 Summing Operator (+) .................................. 230
5.3.1 Delay-and-Sum Technique ........................ 230
5.3.2 Beam Shaping and Steering ...................... 231
5.3.3 Wavenumber Cone and Diffraction Limit .......... 233
5.3.4 Frequency Invariant Radiation Pattern .......... 236
5.3.5 Discrete Array and Grating Lobes ............... 237
5.4 Product Theorem (×) ................................... 240
5.4.1 Convolution and Multiplication of Sound Beams .. 240
5.4.2 On-Axis Pressure Response ...................... 243
5.5 Differential Operator and Super-Directivity (-) ....... 245
5.5.1 Endfire Differential Patterns .................. 245
5.5.2 Combination of Delay-and-Sum and Endfire
Differential Patterns .......................... 252
5.5.3 Broadside Differential Pattern ................. 252
5.5.4 Combination of the Delay-and-Sum and
Broadside Differential Patterns ................ 258
5.6 Optimization with Energy Ratios (÷) ................... 259
5.6.1 Problem Statement .............................. 259
5.6.2 Capon's Minimum Variance Estimator (Minimum
Variance Beamformer) ........................... 261
5.6.3 Acoustic Brightness and Contrast Control ....... 262
5.6.4 Further Analysis of Acoustic Brightness and
Contrast Control ............................... 273
5.6.5 Application Examples ........................... 276
References ................................................. 280
6 Sound Field Reproduction ................................... 283
6.1 Introduction .......................................... 283
6.2 Problem Statement ..................................... 284
6.2.1 Concept of Sound Field Reproduction ............ 284
6.2.2 Objective of Sound Field Reproduction .......... 284
6.3 Reproduction of One-Dimensional Sound Field ........... 286
6.3.1 Field-Matching Approach ........................ 286
6.3.2 Mode-Matching Approach ......................... 288
6.3.3 Integral Approach .............................. 289
6.3.4 Single-Layer Potential ......................... 295
6.4 Reproduction of a 3D Sound Field ...................... 296
6.4.1 Problem Statement and Associated Variables ..... 296
6.5 Field-Matching Approach ............................... 298
6.5.1 Inverse Problem ................................ 298
6.5.2 Regularization of an Inverse Problem ........... 305
6.5.5 Selection of the Regularization Parameter ...... 309
6.6 Mode-Matching Approach ................................ 311
6.6.1 Encoding and Decoding of Sound Field ........... 311
6.6.2 Mode-Matching with Plane Waves ................. 313
6.6.3 Mode-Matching with Spherical Harmonics ......... 320
6.7 Surface Integral Equations ............................ 337
6.7.1 Source Inside, Listener Inside (V0  V,
r  V) ......................................... 337
6.7.2 Source Inside, Listener Outside (V0 V,
r Λ) ......................................... 340
6.7.3 Source Outside, Listener Outside (V0 Λ,
r Λ) ......................................... 341
6.7.4 Source Outside, Listener Inside (V0 Λ,
r V) ......................................... 342
6.7.5 Listener on the Control Surface ................ 342
6.7.6 Summary of Integral Equations .................. 344
6.7.7 Nonradiating Sound Field and Nonuniqueness
Problem ........................................ 344
6.8 Single-layer Formula .................................. 346
6.8.1 Single-layer Formula for Exterior Virtual
Source ......................................... 346
6.8.2 Integral Formulas for Interior Virtual Source .. 355
References ................................................. 369
Appendix A Useful Formulas ................................... 371
A.1 Fourier Transform ..................................... 371
A.1.1 Fourier Transform Table ........................ 371
A.2 Dirac Delta Function .................................. 374
A.3 Derivative of Matrices ................................ 374
A.3.1 Derivative of Real-Valued Matrix ............... 374
A.3.2 Derivative of Complex-Valued Function .......... 375
A.3.3 Derivative of Complex Matrix ................... 376
A.4 Inverse Problem ....................................... 376
A.4.1 Overdetermined Linear Equations and Least
Squares (LS) Solution .......................... 377
A.4.2 Underdetermined Linear Equations and
Minimum-Norm Problem ........................... 378
A.4.3 Method of Lagrange Multiplier .................. 379
A.4.4 Regularized Least Squares ...................... 380
A.4.5 Singular Value Decomposition ................... 380
A.4.6 Total Least Squares (TLS) ...................... 382
Appendix В Description of Sound Field ........................ 385
B.l Three-Dimensional Acoustic Wave Equation .............. 385
B.l.l Conservation of Mass ........................... 385
B.1.2 Conservation of Momentum ....................... 385
B.1.3 Equation of State .............................. 388
B.l.4 Velocity Potential Function .................... 390
B.l.5 Complex Intensity .............................. 391
B.1.6 Singular Sources ............................... 392
B.2 Wavenumber Domain Representation of the Rayleigh
Integral .............................................. 398
B.2.1 Fourier Transform of Free-Field Green's
Function (Weyl's Identity) ..................... 398
B.2.2 High Frequency Approximation (Stationary
Phase Approximation) ........................... 399
B.3 Separation of Variables in Spherical Coordinates ...... 400
B.3.1 Angle Functions: Associated Legendre
Functions ...................................... 400
B.3.2 Angle Functions: Spherical Harmonics ........... 402
B.3.3 Radial Functions ............................... 404
B.3.4 Radial Functions: Spherical Bessel and Hankel
Functions ...................................... 404
B.3.5 Description of Sound Fields by Spherical
Basis Function ................................. 408
B.3.6 Representation of the Green's Function ......... 409
References ................................................. 411
Index ......................................................... 413
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