Fourier Acoustics (eBook)

Cover 1
Contents 6
Preface 12
Chapter 1. Fourier Transforms & Special Functions
1.1 Introduction 16
1.2 The Fourier Transform 16
1.3 Fourier Series 19
1.4 Fourier–Bessel (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
2.14 Vibration & Radiation: Finite, Simply Supported Plate
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
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
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
8.11 Arbitrarily Shaped Bodies 303
8.12 Conformal NAH for Arbitrary Geometry 306
Problems 308
Index 311