Computational Acoustics

David R. Bergman, PhD is Owner and Chief Scientist, Exact Solution Scientific Consulting LLC. He has a PhD in physics with a specialization in General Relativity and High Energy Theory. Among other things, he has developed simulations for testing algorithms used in acoustics, modeled electromagnetic remote sensing devices, and modeled underwater and aero-acoustic propagation, acoustic propagation in transducer layers, and performed mechanical vibrational analysis in bio mechanical systems.

• Series Preface ix1 Introduction 12 Computation and Related Topics 52.1 Floating-Point Numbers 52.1.1 Representations of Numbers 52.1.2 Floating-Point Numbers 72.2 Computational Cost 92.3 Fidelity 112.4 Code Development 122.5 List of Open-Source Tools 162.6 Exercises 17References 173 Derivation of the Wave Equation 193.1 Introduction 193.2 General Properties of Waves 203.3 One-Dimensional Waves on a String 233.4 Waves in Elastic Solids 263.5 Waves in Ideal Fluids 293.5.1 Setting Up the Derivation 293.5.2 A Simple Example 303.5.3 Linearized Equations 313.5.4 A Second-Order Equation from Differentiation 333.5.5 A Second-Order Equation from a Velocity Potential 343.5.6 Second-Order Equation without Perturbations 363.5.7 Special Form of the Operator 363.5.8 Discussion Regarding Fluid Acoustics 403.6 Thin Rods and Plates 413.7 Phonons 423.8 Tensors Lite 423.9 Exercises 48References 484 Methods for Solving the Wave Equation 494.1 Introduction 494.2 Method of Characteristics 494.3 Separation of Variables 564.4 Homogeneous Solution in Separable Coordinates 574.4.1 Cartesian Coordinates 584.4.2 Cylindrical Coordinates 594.4.3 Spherical Coordinates 614.5 Boundary Conditions 634.6 Representing Functions with the Homogeneous Solutions 674.7 Green’s Function 704.7.1 Green’s Function in Free Space 704.7.2 Mode Expansion of Green’s Functions 724.8 Method of Images 764.9 Comparison of Modes to Images 814.10 Exercises 82References 825 Wave Propagation 855.1 Introduction 855.2 Fourier Decomposition and Synthesis 855.3 Dispersion 885.4 Transmission and Reflection 905.5 Attenuation 965.6 Exercises 97References 976 Normal Modes 996.1 Introduction 996.2 Mode Theory 1006.3 Profile Models 1016.4 Analytic Examples 1056.4.1 Example 1: Harmonic Oscillator 1056.4.2 Example 2: Linear 1086.5 Perturbation Theory 1106.6 Multidimensional Problems and Degeneracy 1186.7 Numerical Approach to Modes 1206.7.1 Derivation of the Relaxation Equation 1206.7.2 Boundary Conditions in the Relaxation Method 1256.7.3 Initializing the Relaxation 1276.7.4 Stopping the Relaxation 1286.8 Coupled Modes and the Pekeris Waveguide 1296.8.1 Pekeris Waveguide 1296.8.2 Coupled Modes 1316.9 Exercises 135References 1357 Ray Theory 1377.1 Introduction 1377.2 High Frequency Expansion of the Wave Equation 1387.2.1 Eikonal Equation and Ray Paths 1397.2.2 Paraxial Rays 1407.3 Amplitude 1447.4 Ray Path Integrals 1457.5 Building a Field from Rays 1607.6 Numerical Approach to Ray Tracing 1627.7 Complete Paraxial Ray Trace 1687.8 Implementation Notes 1707.9 Gaussian Beam Tracing 1717.10 Exercises 173References 1748 Finite Difference and Finite Difference Time Domain 1778.1 Introduction 1778.2 Finite Difference 1788.3 Time Domain 1888.4 FDTD Representation of the Linear Wave Equation 1938.5 Exercises 197References 1979 Parabolic Equation 1999.1 Introduction 1999.2 The Paraxial Approximation 1999.3 Operator Factoring 2019.4 Pauli Spin Matrices 2049.5 Reduction of Order 2059.5.1 The Padé Approximation 2079.5.2 Phase Space Representation 2089.5.3 Diagonalizing the Hamiltonian 2099.6 Numerical Approach 2109.7 Exercises 212References 21210 Finite Element Method 21510.1 Introduction 21510.2 The Finite Element Technique 21610.3 Discretization of the Domain 21810.3.1 One-Dimensional Domains 21810.3.2 Two-Dimensional Domains 21910.3.3 Three-Dimensional Domains 22210.3.4 Using Gmsh 22310.4 Defining Basis Elements 22510.4.1 One-Dimensional Basis Elements 22610.4.2 Two-Dimensional Basis Elements 22710.4.3 Three-Dimensional Basis Elements 22910.5 Expressing the Helmholtz Equation in the FEM Basis 23210.6 Numerical Integration over Triangular and Tetrahedral Domains 23410.6.1 Gaussian Quadrature 23410.6.2 Integration over Triangular Domains 23510.6.3 Integration over Tetrahedral Domains 23910.7 Implementation Notes 24010.8 Exercises 240References 24111 Boundary Element Method 24311.1 Introduction 24311.2 The Boundary Integral Equations 24411.3 Discretization of the BIE 24911.4 Basis Elements and Test Functions 25311.5 Coupling Integrals 25411.5.1 Derivation of Coupling Terms 25411.5.2 Singularity Extraction 25611.5.3 Evaluation of the Singular Part 26011.5.3.1 Closed-Form Expression for the Singular Part of K 26011.5.3.2 Method for Partial Analytic Evaluation 26111.5.3.3 The Hypersingular Integral 26611.6 Scattering from Closed Surfaces 26711.7 Implementation Notes 26911.8 Comments on Additional Techniques 27111.8.1 Higher-Order Methods 27111.8.2 Body of Revolution 27211.9 Exercises 273References 273Index 275