Sound Visualization and Manipulation

Professor Yang-Hann Kim and Jung-Woo Choi; both of the Center for Noise and Vibration Control (NOVIC), Department of Mechanical Engineering, KAIST (Korea Advanced Institute of Science and Engineering, Korea.Professor Kim gained his B.S. in Naval Architecture and Marine Engineering from Seoul National University, Korea, in 1977, and his Ph.D in Acoustics and Vibration in M.E. (O.E. Program), M.I.T., USA, in 1985. He has been working in the field of sound visualization and manipulation for more than 20 years, and has taught acoustics to undergraduate and graduate students. His research interests include Sound Visualization, Active Noise/Vibration Control, Sound Focusing, Structural Acoustics, and Duct Acoustics. He has written two books and contributed to numerous journals and conference papers. Professor Kim was awarded the Excellence Award in Teaching from Mechanical Engineering, KAIST (Dynamics, 2010).Professor Choi gained his Ph.D in Acoustics and Vibration in Mechanical Engineering from KAIST in 2005. He was Visiting Post-Doctoral Researcher Institute of Sound and Vibration Research (ISVR) at the University of Southampton, UK, in 2006, and worked as a Senior Engineer in the Acoustic and Sound Technology Lab., Samsung Electronics in 2011 before taking up his current post. Professor Choi's research interests include Sound Focusing, Sound Field Reproduction, Sound Visualization, and Active Noise/Vibration Control. He has written extensively on the topic in numerous journals and conference proceedings.

• About the Author xi Preface xiiiAcknowledgments xviiPart I ESSENCE OF ACOUSTICS1 Acoustic Wave Equation and Its Basic Physical Measures 31.1 Introduction 31.2 One-Dimensional Acoustic Wave Equation 31.2.1 Impedance 91.3 Three-Dimensional Wave Equation 101.4 Acoustic Intensity and Energy 111.4.1 Complex-Valued Pressure and Intensity 161.5 The Units of Sound 181.6 Analysis Methods of Linear Acoustic Wave Equation 271.6.1 Acoustic Wave Equation and Boundary Condition 281.6.2 Eigenfunctions and Modal Expansion Theory 311.6.3 Integral Approach Using Green’s Function 351.7 Solutions of the Wave Equation 391.7.1 Plane Wave 401.7.2 Spherical Wave 411.8 Chapter Summary 46References 462 Radiation, Scattering, and Diffraction 492.1 Introduction/Study Objectives 492.2 Radiation of a Breathing Sphere and a Trembling Sphere 502.3 Radiation from a Baffled Piston 582.4 Radiation from a Finite Vibrating Plate 652.5 Diffraction and Scattering 702.6 Chapter Summary 792.7 Essentials of Radiation, Scattering, and Diffraction 802.7.1 Radiated Sound Field from an Infinitely Baffled Circular Piston 802.7.2 Sound Field at an Arbitrary Position Radiated by an Infinitely Baffled Circular Piston 812.7.3 Understanding Radiation, Scattering, and Diffraction Using the Kirchhoff–Helmholtz Integral Equation 822.7.4 Scattered Sound Field Using the Rayleigh Integral Equation 96References 97Part II SOUND VISUALIZATION3 Acoustic Holography 1033.1 Introduction 1033.2 The Methodology of Acoustic Source Identification 1033.3 Acoustic Holography: Measurement, Prediction, and Analysis 1063.3.1 Introduction and Problem Definitions 1063.3.2 Prediction Process 1073.3.3 Mathematical Derivations of Three Acoustic Holography Methods and Their Discrete Forms 1133.3.4 Measurement 1193.3.5 Analysis of Acoustic Holography 1243.4 Summary 129References 1304 Beamforming 1374.1 Introduction 1374.2 Problem Statement 1384.3 Model-Based Beamforming 1404.3.1 Plane and Spherical Wave Beamforming 1404.3.2 The Array Configuration 1424.4 Signal-Based Beamforming 1454.4.1 Construction of Correlation Matrix in Time Domain 1464.4.2 Construction of Correlation Matrix in Frequency Domain 1514.4.3 Correlation Matrix of Multiple Sound Sources 1524.5 Correlation-Based Scan Vector Design 1604.5.1 Minimum Variance Beamformer 1604.5.2 Linear Prediction 1644.6 Subspace-Based Approaches 1704.6.1 Basic Principles 1704.6.2 MUSIC Beamformer 1734.6.3 ESPRIT 1804.7 Wideband Processing Technique 1824.7.1 Frequency-Domain Approach: Mapping to the Beam Space 1824.7.2 Coherent Subspace Method (CSM) 1844.7.3 Partial Field Decomposition in Beam Space 1854.7.4 Time-Domain Technique 1904.7.5 Moving-Source Localization 1984.8 Post-Processing Techniques 2044.8.1 Deconvolution and Beamforming 2044.8.2 Nonnegativity Constraint 2074.8.3 Nonnegative Least-Squares Algorithm 2094.8.4 DAMAS 210References 212Part III SOUND MANIPULATION5 Sound Focusing 2195.1 Introduction 2195.2 Descriptions of the Problem of Sound Focusing 2215.2.1 Free-Field Radiation from Loudspeaker Arrays 2215.2.2 Descriptions of a Sound Field Depending on the Distance from the Array 2215.2.3 Fresnel Approximation 2235.2.4 Farfield Description of the Rayleigh Integral (Fraunhofer Approximation) 2255.2.5 Descriptors of Directivity 2275.3 Summing Operator (+) 2305.3.1 Delay-and-Sum Technique 2305.3.2 Beam Shaping and Steering 2315.3.3 Wavenumber Cone and Diffraction Limit 2335.3.4 Frequency Invariant Radiation Pattern 2365.3.5 Discrete Array and Grating Lobes 2375.4 Product Theorem (×) 2405.4.1 Convolution and Multiplication of Sound Beams 2405.4.2 On-Axis Pressure Response 2435.5 Differential Operator and Super-Directivity (−) 2455.5.1 Endfire Differential Patterns 2455.5.2 Combination of Delay-and-Sum and Endfire Differential Patterns 2525.5.3 Broadside Differential Pattern 2525.5.4 Combination of the Delay-and-Sum and Broadside Differential Patterns 2585.6 Optimization with Energy Ratios (÷) 2595.6.1 Problem Statement 2595.6.2 Capon’s Minimum Variance Estimator (Minimum Variance Beamformer) 2615.6.3 Acoustic Brightness and Contrast Control 2625.6.4 Further Analysis of Acoustic Brightness and Contrast Control 2735.6.5 Application Examples 276References 2806 Sound Field Reproduction 2836.1 Introduction 2836.2 Problem Statement 2846.2.1 Concept of Sound Field Reproduction 2846.2.2 Objective of Sound Field Reproduction 2846.3 Reproduction of One-Dimensional Sound Field 2866.3.1 Field-Matching Approach 2866.3.2 Mode-Matching Approach 2886.3.3 Integral Approach 2896.3.4 Single-Layer Potential 2956.4 Reproduction of a 3D Sound Field 2966.4.1 Problem Statement and Associated Variables 2966.5 Field-Matching Approach 2986.5.1 Inverse Problem 2986.5.2 Regularization of an Inverse Problem 3056.5.3 Selection of the Regularization Parameter 3096.6 Mode-Matching Approach 3116.6.1 Encoding and Decoding of Sound Field 3116.6.2 Mode-Matching with Plane Waves 3136.6.3 Mode-Matching with Spherical Harmonics 3206.7 Surface Integral Equations 3376.7.1 Source Inside, Listener Inside (V0 ⊂ V , r ∈ V ) 3376.7.2 Source Inside, Listener Outside (V0 ⊂ V , r ∈ ) 3406.7.3 Source Outside, Listener Outside (V0 ⊂ , r ∈ ) 3416.7.4 Source Outside, Listener Inside (V0 ⊂ , r ∈ V ) 3426.7.5 Listener on the Control Surface 3426.7.6 Summary of Integral Equations 3446.7.7 Nonradiating Sound Field and Nonuniqueness Problem 3446.8 Single-layer Formula 3466.8.1 Single-layer Formula for Exterior Virtual Source 3466.8.2 Integral Formulas for Interior Virtual Source 355References 369Appendix A Useful Formulas 371A.1 Fourier Transform 371A.1.1 Fourier Transform Table 371A.2 Dirac Delta Function 374A.3 Derivative of Matrices 374A.3.1 Derivative of Real-Valued Matrix 374A.3.2 Derivative of Complex-Valued Function 375A.3.3 Derivative of Complex Matrix 376A.4 Inverse Problem 376A.4.1 Overdetermined Linear Equations and Least Squares (LS) Solution 377A.4.2 Underdetermined Linear Equations and Minimum-Norm Problem 378A.4.3 Method of Lagrange Multiplier 379A.4.4 Regularized Least Squares 380A.4.5 Singular Value Decomposition 380A.4.6 Total Least Squares (TLS) 382Appendix B Description of Sound Field 385B.1 Three-Dimensional Acoustic Wave Equation 385B.1.1 Conservation of Mass 385B.1.2 Conservation of Momentum 385B.1.3 Equation of State 388B.1.4 Velocity Potential Function 390B.1.5 Complex Intensity 391B.1.6 Singular Sources 392B.2 Wavenumber Domain Representation of the Rayleigh Integral 398B.2.1 Fourier Transform of Free-Field Green’s Function (Weyl’s Identity) 398B.2.2 High Frequency Approximation (Stationary Phase Approximation) 399B.3 Separation of Variables in Spherical Coordinates 400B.3.1 Angle Functions: Associated Legendre Functions 400B.3.2 Angle Functions: Spherical Harmonics 402B.3.3 Radial Functions 404B.3.4 Radial Functions: Spherical Bessel and Hankel Functions 404B.3.5 Description of Sound Fields by Spherical Basis Function 408B.3.6 Representation of the Green’s Function 409References 411Index 413

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