Fourier Acoustics : Sound Radiation and Nearfield Acoustical Holography

Intended a both a textbook and a reference, Fourier Acoustics develops the theory of sound radiation uniquely from the viewpoint of Fourier Analysis. This powerful perspective of sound radiation provides the reader with a comprehensive and practical understanding which will enable him or her to diagnose and solve sound and vibration problems in the 21st Century. As a result of this perspective, Fourier Acoustics is able to present thoroughly and simply, for the first time in book form, the theory of nearfield acoustical holography, an important technique which has revolutionised the measurement of sound. Relying little on material outside the book, Fourier Acoustics will be invaluable as a graduate level text as well as a reference for researchers in academia and industry. Key Features * The physics of wave propogation and sound vibration in homogeneous media *Acoustics, such as radiation of sound, and radiation from vibrating surfaces *Inverse problems, such as the theory of nearfield acoustical holography *Mathematics of specialized functions, such as spherical harmonics Front Cover......Page 1 Fourier Acoustics: Sound Radiation and Nearfield Acoustical Holography......Page 4 Copyright Page......Page 5 Contents......Page 6 Preface......Page 12 1.2 The Fourier Transform......Page 16 1.3 Fourier Series......Page 19 1.4 Fourier–Bessel (Hankel) Transforms......Page 20 1.5 The Dirac Delta Function......Page 21 1.6 The Rectangle Function......Page 22 1.8 Continuous Fourier Transform and the DFT......Page 23 Problems......Page 28 2.2 The Wave Equation and Euler's Equation......Page 30 2.3 Instantaneous Acoustic Intensity......Page 32 2.4 Steady State......Page 33 2.5 Time Averaged Acoustic Intensity......Page 34 2.6 Plane Wave Expansion......Page 35 2.7 Infinite Plate Vibrating in a Normal Mode......Page 41 2.8 Wavenumber Space: k-space......Page 42 2.9 The Angular Spectrum: Fourier Acoustics......Page 46 2.10 Derivation of Rayleigh's Integrals......Page 49 2.11 Farfield Radiation: Planar Sources......Page 53 2.12 Radiated Power......Page 67 2.13 Vibration & Radiation: Infinite Point-driven Plate......Page 71 2.14 Vibration & Radiation: Finite, Simply Supported Plate......Page 77 2.15 Supersonic Intensity......Page 92 Problems......Page 98 3.1 Introduction......Page 104 3.2 Overview of the Theory......Page 105 3.3 Presentation of Theory for a One-Dimensional Radiator......Page 106 3.4 Ill Conditioning Due to Measurement Noise......Page 108 3.5 The k-space Filter......Page 109 3.6 Modification of the Filter Shape......Page 112 3.7 Measurement Noise and the Standoff Distance......Page 113 3.8 Determination of the k-space Filter......Page 115 3.9 Finite Measurement Aperture Effects......Page 118 3.10 Discretization and Aliasing......Page 120 3.11 Use of the DFT to Solve the Holography Equation......Page 122 3.12 Reconstruction of Other Quantities......Page 127 Problems......Page 128 4.2 The Wave Equation......Page 130 4.3 General Solution......Page 136 4.4 The Helical Wave Spectrum: Fourier Acoustics......Page 140 4.5 The Rayleigh-like Integrals......Page 148 4.6 Farfield Radiation - Cylindrical Sources......Page 152 4.7 Radiated Power......Page 162 Problems......Page 163 5.2 Overview of the Inverse Problem......Page 164 5.3 Computer Implementation of NAH......Page 169 5.4 Experimental Results......Page 175 Problems......Page 196 6.2 The Wave Equation......Page 198 6.3 The Angle Functions......Page 201 6.4 Radial Functions......Page 208 6.5 Multipoles......Page 212 6.6 Spherical Harmonic Directivity Patterns......Page 219 6.7 General Solution for Exterior Problems......Page 221 6.8 General Solution for Interior Problems......Page 232 6.9 Transient Radiation - Exterior Problems......Page 236 6.10 Scattering from Spheres......Page 239 Problems......Page 247 7.1 Introduction......Page 250 7.2 Formulation of the Inverse Problem- Exterior Domain......Page 251 7.3 Interior NAH......Page 253 7.4 Scattering Nearfield Holography......Page 260 Problems......Page 264 8.2 Green's Theorem......Page 266 8.3 The Interior Helmholtz Integral Equation......Page 267 8.4 HIE for Radiation Problems (Exterior Domain)......Page 275 8.5 HIE for Scattering Problems......Page 277 8.6 Green Functions & the Inhomogeneous Wave Equation......Page 279 8.7 Simple Source Formulation......Page 282 8.8 The Dirichlet and Neumann Green Functions......Page 287 8.9 Construction by Eigenfunction Expansion......Page 292 8.10 Evanescent Neumann & Dirichlet Green Functions......Page 296 8.11 Arbitrarily Shaped Bodies......Page 303 8.12 Conformal NAH for Arbitrary Geometry......Page 306 Problems......Page 308 Index......Page 311 Front Cover 1 Fourier Acoustics: Sound Radiation and Nearfield Acoustical Holography 4 Copyright Page 5 Contents 6 Preface 12 Chapter 1. Fourier Transforms & Special Functions 16 1.1 Introduction 16 1.2 The Fourier Transform 16 1.3 Fourier Series 19 1.4 Fourier鈥揃essel (Hankel) Transforms 20 1.5 The Dirac Delta Function 21 1.6 The Rectangle Function 22 1.7 The Comb Function 23 1.8 Continuous Fourier Transform and the DFT 23 Problems 28 Chapter 2. Plane Waves 30 2.1 Introduction 30 2.2 The Wave Equation and Euler's Equation 30 2.3 Instantaneous Acoustic Intensity 32 2.4 Steady State 33 2.5 Time Averaged Acoustic Intensity 34 2.6 Plane Wave Expansion 35 2.7 Infinite Plate Vibrating in a Normal Mode 41 2.8 Wavenumber Space: k-space 42 2.9 The Angular Spectrum: Fourier Acoustics 46 2.10 Derivation of Rayleigh's Integrals 49 2.11 Farfield Radiation: Planar Sources 53 2.12 Radiated Power 67 2.13 Vibration & Radiation: Infinite Point-driven Plate 71 2.14 Vibration & Radiation: Finite, Simply Supported Plate 77 2.15 Supersonic Intensity 92 Problems 98 Chapter 3. The Inverse Problem: Planar NAH 104 3.1 Introduction 104 3.2 Overview of the Theory 105 3.3 Presentation of Theory for a One-Dimensional Radiator 106 3.4 Ill Conditioning Due to Measurement Noise 108 3.5 The k-space Filter 109 3.6 Modification of the Filter Shape 112 3.7 Measurement Noise and the Standoff Distance 113 3.8 Determination of the k-space Filter 115 3.9 Finite Measurement Aperture Effects 118 3.10 Discretization and Aliasing 120 3.11 Use of the DFT to Solve the Holography Equation 122 3.12 Reconstruction of Other Quantities 127 Problems 128 Chapter 4. Cylindrical Waves 130 4.1 Introduction 130 4.2 The Wave Equation 130 4.3 General Solution 136 4.4 The Helical Wave Spectrum: Fourier Acoustics 140 4.5 The Rayleigh-like Integrals 148 4.6 Farfield Radiation - Cylindrical Sources 152 4.7 Radiated Power 162 Problems 163 Chapter 5. The Inverse Problem: Cylindrical NAH 164 5.1 Introduction 164 5.2 Overview of the Inverse Problem 164 5.3 Computer Implementation of NAH 169 5.4 Experimental Results 175 Problems 196 Chapter 6. Spherical Waves 198 6.1 Introduction 198 6.2 The Wave Equation 198 6.3 The Angle Functions 201 6.4 Radial Functions 208 6.5 Multipoles 212 6.6 Spherical Harmonic Directivity Patterns 219 6.7 General Solution for Exterior Problems 221 6.8 General Solution for Interior Problems 232 6.9 Transient Radiation - Exterior Problems 236 6.10 Scattering from Spheres 239 Problems 247 Chapter 7. Spherical NAH 250 7.1 Introduction 250 7.2 Formulation of the Inverse Problem- Exterior Domain 251 7.3 Interior NAH 253 7.4 Scattering Nearfield Holography 260 Problems 264 Chapter 8. Green Functions & the Helmholtz Integral 266 8.1 Introduction 266 8.2 Green's Theorem 266 8.3 The Interior Helmholtz Integral Equation 267 8.4 HIE for Radiation Problems (Exterior Domain) 275 8.5 HIE for Scattering Problems 277 8.6 Green Functions & the Inhomogeneous Wave Equation 279 8.7 Simple Source Formulation 282 8.8 The Dirichlet and Neumann Green Functions 287 8.9 Construction by Eigenfunction Expansion 292 8.10 Evanescent Neumann & Dirichlet Green Functions 296 8.11 Arbitrarily Shaped Bodies 303 8.12 Conformal NAH for Arbitrary Geometry 306 Problems 308 Index 311 Intended a both a textbook and a reference, Fourier Acoustics develops the theory of sound radiation uniquely from the viewpoint of Fourier Analysis. This powerful perspective of sound radiation provides the reader with a comprehensive and practical understanding which will enable him or her to diagnose and solve sound and vibration problems in the 21st Century. As a result of this perspective, Fourier Acoustics is able to present thoroughly and simply, for the first time in book form, the theory of nearfield acoustical holography, an important technique which has revolutionised the measurement of sound. Relying little on material outside the book, Fourier Acoustics will be invaluable as a graduate level text as well as a reference for researchers in academia and industry.

Key Features
* The physics of wave propogation and sound vibration in homogeneous media
*Acoustics, such as radiation of sound, and radiation from vibrating surfaces
*Inverse problems, such as the theory of nearfield acoustical holography
*Mathematics of specialized functions, such as spherical harmonics Intended a both a textbook and a reference, Fourier acoustics develops the theory of sound radiation uniquely from the viewpoint of Fourier analysis. This powerful perspective of sound radiation provides the reader with a comprehensive and practical understanding which will enable him or her to diagnose and solve sound and vibration problems in the 21st century. As a result of this perspective, Fourier Acoustics is able to present thoroughly and simply, for the first time in book form, the theory of nearfield acoustical holography, an important technique which has revolutionised the measurement of sound ... Fourier acoustics will be invaluable as a graduate level text as well as a reference for researchers in academia and industry. Key features: the physics of wave propagation and sound vibration in homogeneous media; acoustics, such as radiation of sound, and radiation from vibrating surfaces; inverse problems, such as the theory of nearfield acoustical holography; and mathematics of specialized functions, such as spherical harmonics At the heart of Fourier acoustics is the Fourier transform which includes the concepts of the Fourier series and the Hankel transform.