Unique in addressing two different problems sound visualization and manipulation in a unified way Advances in signal processing technology are enabling ever more accurate visualization of existing sound fields and precisely defined sound field production. The idea of explaining both the problem of sound visualization and the problem of the manipulation of sound within one book supports this inter-related area of study. With rapid development of array technologies, it is possible to do much in terms of visualization and manipulation, among other technologies involved with the spatial distribution of sound. This book aims to explore various basic functions for the visualization and manipulation and demonstrate to the reader how these properties determine the quality of visualization and manipulation. The first half of the book introduces some basic and general concepts and theories and the second part of the book explains a number of techniques in sound visualization and manipulation. It offers a unified presentation to two very different topics - sound field visualization techniques based on microphone arrays, and techniques for generation of controlled sound fields using loudspeaker arrays. The authors emphasize the similarities between these two physical problems and between the mathematical methods used for solving them. With extensive examples throughout the book, chapters include: Acoustic Wave Equation and its Basic Physical Measures, Acoustic Wave Equation and its Basic Physical Measures, Basic Theory of Sound Visualization, Acoustic Holography, Beamforming, Basic Theory of Sound Manipulation, Sound Focusing, and Sound Field Reproduction. The first book to combine both the visualization and manipulation of sound technologies in one comprehensive volume Presents the basic concepts using simple one dimensional cases and then extends the concept to three dimensional cases, enabling easier understanding of the fundamental concepts through the use of minimum mathematics Provides a solid understanding of associated physics as well as mathematical concepts for understanding the technologies, addressing diffraction problems in an integrated format by using Kirchhoff-Helmholtz integral equation Uses extensive examples demonstrating the benefits and drawbacks of various applications, including beamforming and acoustic holography A valuable resource for post /graduate students, acoustic engineers, audio and noise control system developers

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 Piston 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 Fresnel 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

6.5.3 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 ∈ ) 340

6.7.3 Source Outside, Listener Outside (V0 ⊂ , r ∈ ) 341

6.7.4 Source Outside, Listener Inside (V0 ⊂ , 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 B Description of Sound Field 385**

B.1 Three-Dimensional Acoustic Wave Equation 385

B.1.1 Conservation of Mass 385

B.1.2 Conservation of Momentum 385

B.1.3 Equation of State 388

B.1.4 Velocity Potential Function 390

B.1.5 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