Cover
Title Page
Series Preface
1 Introduction
2 Computation and Related Topics
1. 2.1 Floating‐Point Numbers
2. 2.2 Computational Cost
3. 2.3 Fidelity
4. 2.4 Code Development
5. 2.5 List of Open‐Source Tools
6. 2.6 Exercises
7. References
3 Derivation of the Wave Equation
1. 3.1 Introduction
2. 3.2 General Properties of Waves
3. 3.3 One‐Dimensional Waves on a String
4. 3.4 Waves in Elastic Solids
5. 3.5 Waves in Ideal Fluids
6. 3.6 Thin Rods and Plates
7. 3.7 Phonons
8. 3.8 Tensors Lite
9. 3.9 Exercises
10. References
4 Methods for Solving the Wave Equation
1. 4.1 Introduction
2. 4.2 Method of Characteristics
3. 4.3 Separation of Variables
4. 4.4 Homogeneous Solution in Separable Coordinates
5. 4.5 Boundary Conditions
6. 4.6 Representing Functions with the Homogeneous Solutions
7. 4.7 Green’s Function
8. 4.8 Method of Images
9. 4.9 Comparison of Modes to Images
10. 4.10 Exercises
11. References
5 Wave Propagation
1. 5.1 Introduction
2. 5.2 Fourier Decomposition and Synthesis
3. 5.3 Dispersion
4. 5.4 Transmission and Reflection
5. 5.5 Attenuation
6. 5.6 Exercises
7. References
6 Normal Modes
1. 6.1 Introduction
2. 6.2 Mode Theory
3. 6.3 Profile Models
4. 6.4 Analytic Examples
5. 6.5 Perturbation Theory
6. 6.6 Multidimensional Problems and Degeneracy
7. 6.7 Numerical Approach to Modes
8. 6.8 Coupled Modes and the Pekeris Waveguide
9. 6.9 Exercises
10. References
7 Ray Theory
1. 7.1 Introduction
2. 7.2 High Frequency Expansion of the Wave Equation
3. 7.3 Amplitude
4. 7.4 Ray Path Integrals
5. 7.5 Building a Field from Rays
6. 7.6 Numerical Approach to Ray Tracing
7. 7.7 Complete Paraxial Ray Trace
8. 7.8 Implementation Notes
9. 7.9 Gaussian Beam Tracing
10. 7.10 Exercises
11. References
8 Finite Difference and Finite Difference Time Domain
1. 8.1 Introduction
2. 8.2 Finite Difference
3. 8.3 Time Domain
4. 8.4 FDTD Representation of the Linear Wave Equation
5. 8.5 Exercises
6. References
9 Parabolic Equation
1. 9.1 Introduction
2. 9.2 The Paraxial Approximation
3. 9.3 Operator Factoring
4. 9.4 Pauli Spin Matrices
5. 9.5 Reduction of Order
6. 9.6 Numerical Approach
7. 9.7 Exercises
8. References
10 Finite Element Method
1. 10.1 Introduction
2. 10.2 The Finite Element Technique
3. 10.3 Discretization of the Domain
4. 10.4 Defining Basis Elements
5. 10.5 Expressing the Helmholtz Equation in the FEM Basis
6. 10.6 Numerical Integration over Triangular and Tetrahedral Domains
7. 10.7 Implementation Notes
8. 10.8 Exercises
9. References
11 Boundary Element Method
1. 11.1 Introduction
2. 11.2 The Boundary Integral Equations
3. 11.3 Discretization of the BIE
4. 11.4 Basis Elements and Test Functions
5. 11.5 Coupling Integrals
6. 11.6 Scattering from Closed Surfaces
7. 11.7 Implementation Notes
8. 11.8 Comments on Additional Techniques
9. 11.9 Exercises
10. References
Index
End User License Agreement

List of Tables

Chapter 02
1. Table 2.1 Binary representation of integers
2. Table 2.2 Binary representations of fractions
3. Table 2.3 Four alternate representations of the first 16 (base 10) integers
4. Table 2.4 Exponent and mantissa widths and exponent bias for floating‐point numbers
5. Table 2.5 Machine epsilon for floating‐point types
6. Table 2.6 Size of various numbers in bytes
7. Table 2.7 Open‐source alternatives to MATLAB, MAPLE, or Mathematica
8. Table 2.8 List of C/C++ numeric libraries and other items
9. Table 2.9 List of tools related to finite and boundary element analysis
Chapter 03
1. Table 3.1 Three representations of the second‐order wave equation
Chapter 04
1. Table 4.1 Associated Legendre polynomials
2. Table 4.2 List of eigenfunctions and eigenvalues for three ideal cases
3. Table 4.3 Modified Green’s functions for a single boundary
4. Table 4.4 Locations and relative phase of images for three cases
Chapter 06
1. Table 6.1 Hermite polynomials
2. Table 6.2 Mode degeneracy for 2‐dim and 3‐dim examples
3. Table 6.3 Boundary conditions for relaxation
Chapter 08
1. Table 8.1 Indices for neighboring points contributing to Laplacian in 3 dim
2. Table 8.2 Stencils for spatial derivative
3. Table 8.3 Index sets for boundary points
4. Table 8.4 Stencils for finite differences in time
5. Table 8.5 Example estimates of the number of sample points required for setting up an FD grid
Chapter 09
1. Table 9.1 Common choices for operator splitting used to develop the parabolic equation

List of Illustrations

Chapter 03
1. Figure 3.1 Description of the tangent plane at a point on a curved manifold
Chapter 04
1. Figure 4.1 Sample mode, , for two soft (left), two hard (center), and mixed (right) boundary conditions
2. Figure 4.2 Image method defined
3. Figure 4.3 Multiple images for satisfying two boundary conditions
4. Figure 4.4 Images for a source in a right‐angle corner
5. Figure 4.5 Images for a point source in a rectangular enclosure
6. Figure 4.6 Contour plot of |p| built from images
7. Figure 4.7 Contour plot of |p| built from modes
Chapter 05
1. Figure 5.1 Group velocity and phase velocity curves for a normal mode problem
2. Figure 5.2 Incident, reflected, and refracted waves at an interface between two media
Chapter 06
1. Figure 6.1 The Garrettt–Munk sound speed profile (left) and its effective potentials (right)
2. Figure 6.2 Effect potentials for the linear sound speed profile at various strengths
3. Figure 6.3 Effective potentials for an ideal refractive waveguide
4. Figure 6.4 First three modes for the quadratic potential
5. Figure 6.5 Modes of the linear potential for pressure release (top) and hard (bottom) boundary conditions
6. Figure 6.6 Perturbation applied too linear sound speed and soft boundary conditions
7. Figure 6.7 Definition of relaxation variables and lattice points
8. Figure 6.8 Relaxation applied to linear sound speed and soft boundary conditions
9. Figure 6.9 Setup for the Pekeris waveguide
10. Figure 6.10 Waveguide with irregular boundary
11. Figure 6.11 Setup for coupled mode
Chapter 07
1. Figure 7.1 Illustration showing quantities defined in the paraxial ray trace
2. Figure 7.2 Ray trace for the linear sound speed profile
3. Figure 7.3 Sample ray trace for the linear wind profile
4. Figure 7.4 Sample ray trace for the Kormilitsin profile
5. Figure 7.5 Ray trace for the refractive waveguide
6. Figure 7.6 Ray trace for a linear refractive sound speed profile and hard reflective ground as
7. Figure 7.7 Three‐dimensional ray trace for the Garrett–Munk profile
Chapter 08
1. Figure 8.1 Sample 2‐dim grid with rectangular boundary
2. Figure 8.2 Example of irregular boundary illustrating interior, boundary, and near points
3. Figure 8.3 Cutting of a cubic point lattice by a sphere
4. Figure 8.4 Local normal derived from edge vectors of a triangular patch
5. Figure 8.5 Identification of interior and boundary points and local normal vectors for implementing Neumann boundary conditions in 2 dim
6. Figure 8.6 Example 5 × 5 grid
7. Figure 8.7 Leapfrog method on 1 + 1 dim lattice
8. Figure 8.8 Location on lattice where variables are evaluated
9. Figure 8.9 Time steps for a 2 + 1 dim implementation
Chapter 09
1. Figure 9.1 Computational stencil for the Crank–Nicolson method
Chapter 10
1. Figure 10.1 One‐dimensional patches and basis elements defined
2. Figure 10.2 Sample tiling of a square with right triangle
3. Figure 10.3 Sample tiling of a square with right triangles
4. Figure 10.4 Example of a random tiling of a square using Gmsh
5. Figure 10.5 Cube cut into five tetrahedra
6. Figure 10.6 A cube in 3‐dim tiled with tetrahedra using Gmsh
7. Figure 10.7 Two‐dimensional rooftop basis defined
8. Figure 10.8 Mapping of a generic triangle (left) into the standard triangle (center) and to a square (right)
9. Figure 10.9 Example of Dunavant quadrature points in simplex space, order = 12
Chapter 11
1. Figure 11.1 Definition of terms for the exterior problem
2. Figure 11.2 Real part of Green’s function
3. Figure 11.3 Variables used for evaluating singular integrals
4. Figure 11.4 Evaluation of
5. Figure 11.5 Contour plot of Figure 11.4
6. Figure 11.6 Example of far field bi‐static scattering from a sphere

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This edition first published 2018
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Library of Congress Cataloging-in-Publication Data

Names: Bergman, David R., author.
Title: Computational acoustics : theory and implementation / David R. Bergman.
Description: Hoboken, NJ : John Wiley & Sons, 2018. | Includes bibliographical references and index. |
Identifiers: LCCN 2017036469 (print) | LCCN 2017046745 (ebook) | ISBN 9781119277330 (pdf) | ISBN 9781119277279 (epub) | ISBN 9781119277286 (cloth)
Subjects: LCSH: Sound‐waves–Measurement. | Sound‐waves–Computer simulation. | Sound‐waves–Mathematical models.
Classification: LCC QC243 (ebook) | LCC QC243 .B38 2018 (print) | DDC 534.0285–dc23
LC record available at https://lccn.loc.gov/2017036469

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1
Introduction

Computers have become an invaluable tool in science and engineering. Over time their use has evolved from a device to aid in complex lengthy calculations to a self‐contained discipline or field of study. Coursework in science and engineering often involves learning analytic techniques and exact solutions to a sizable collection of problems. Although educational, these are rarely useful beyond the classroom. On the other side of the spectrum is the practical experimental approach to investigating nature and developing engineering solutions to practical everyday problems. Most readers are familiar with the nonideal nature of things that, in many cases, prevents one from seeing the utility of the theoretical approach. In science theorist and experimentalist see nature from a different perspective in a quest for understanding its laws but agree that the facts can only be found through observation, as patterns in data acquired via well‐planned and executed experiments designed to isolate certain degrees of freedom, to replicate an ideal circumstance to the best of our ability. Experiments can be costly but are the only mechanism for determining scientific truth. While the scientist works to create ideal circumstances to verify a fundamental law or hypothesis, an engineer must design and build with nonideal conditions in mind. In many cases the only approach available is trial and error. This requires the resources to build new versions of a device or invention every time it fails, each prototype being built and used to see what will happen and how it will fail and to learn from the experience. In this regard, computational science offers a path toward testing prototypes in a virtual environment. When executed carefully this approach can save time and money, prevent human injury or loss of life, and reduce impact to the environment.

As computers became larger, faster, and more efficient, the size of the tasks that could be performed also became larger, evolving from modeling the stress on a single beam to that found throughout the structure of a building, ship, or aircraft under dynamic loading. In recent times the use of computer‐based modeling and simulation has gained a certain credibility in fields where there is no possible experimental method available and theory does not offer a suitable path forward in exploring the consequences of natural law, in particular the fields of numerical relativity (NR) and computational fluid mechanics (CFM). In these fields of study, the computer has become the laboratory, offering us the ability to experiment on systems we cannot build in the physical world.

Over the past decade or so, we are perhaps seeing the emergence of a new class of scientist or scientific specialist, along with the experimentalist and theorist, the numericist. Just as an experimentalist needs to be aware of the science behind the inner working of their probes and detectors, the numericist must understand the limitations imposed by working with finite precision, or a discrete representation of the real number system. The computer is the device we used to probe our virtual world, and discrete mathematics imposes constraints on the precision of the probe. In moving from the world of smooth operations on the continuum to discrete representations of the same, we lose some basic kernels of truth we rely on as common sense. Namely, certain operations are abelian. More precisely there are certain procedures that when carried out by hand produce the same results regardless of the order in which the steps are performed but when executed on a computer could lead to different results for different implementations. The development of computational procedures requires attention to this fact, a new burden for the numericist, and understanding of the impact of this behavior on expected results.

This text focuses on the application of computational methods to the fields of linear acoustics. Acoustics is broadly defined as the propagation of mechanical vibration in a medium. Several aspects of this make acoustics an interesting field of study. First is the need for a medium to support the acoustic phenomenon, which unlike light propagates in free space at constant speed relative to all inertial observers. Another point of interest is that there are as many types of acoustic phenomena as there are media, from longitudinal pressure waves in a fluid to S and P waves in seismology. The material properties of the medium determine the number and type of acoustic waves that may be created and observed. We typically think of acoustics as a macro phenomenon, the result of bulk movement of the medium. However, as we probe nature at smaller scales, this type of phenomenon is precisely what is creating the acoustic phenomenon in solids and similarly particle collisions in fluids. The acoustic phenomenon is seen at small scales in lattice vibrations in crystals. Here the acoustic field is quantized and the quanta are referred to as phonons. This model is the result of an attempt to understand a phenomenon that exists at scales too large to be described by the fundamental process and too small to be a purely classical phenomenon.

The goal of this text is to introduce to the reader those numerical methods associated with the development of computational procedures for solving problems in acoustics and understanding linear acoustic propagation and scattering. The intended audience are students and professionals who are interested in the ingredients needed for the development of these procedures. The presentation of the material in this text is unique in the sense that it focuses on modeling paradigms first and introduce the numerical methods appropriate to that modeling paradigm rather than offer them in a preliminary chapter or appendix. Along the way, implementation issues that readers should be aware of are discussed. Examples are provided along with suggested exercises and references. The intent is to be pedagogical in the approach to presenting information so that readers who are new to the subject can begin experimenting. Classic methods and approaches are featured throughout the text while additional comments are included that highlight modern advances and novel modeling approaches that have appeared in the literature.

Since the intended audience consists of upper‐level undergraduate students, graduate students, or professionals interested in this discipline, expected prerequisites to this material are:

An introductory course that covers acoustics or fluid dynamics
Familiarity with ordinary differential and partial differential equations, perhaps a course in mathematical methods for scientists and engineers
Some exposure to programming in a high‐level language such as Maple, Mathematica, and MATLAB or its open‐source counterparts, SCILAB and Octave

The key feature of the presentation contained in this text is that it serves to bridge the gap between theory and implementation. The main focus is on techniques for solving the linear wave equation in homogeneous medium as well as inhomogeneous and anisotropic fluid medium for modeling wave propagation from a source and scattering from objects. Therefore, the starting point for much of this text will be the standard wave equation or the Helmholtz equation.

The transition from equations to computer procedures is not always a straightforward path. High‐level programming languages come with easy‐to‐use interfaces for solving differential equations, matrix equations, and performing signal processing. Beyond these are professional software packages designed to allow users to build and run specific types of simulations using common modeling paradigms. Examples include ANSYS, FEMLAB, and FEKO, just to name a few. An understanding of the math, physics, and numerics is required to evaluate and interpret the results, but low‐level programming is not necessary. Why learn these techniques? Specialized software can be very expensive, in fact cost prohibitive for students or those engaging in self‐study. Many software companies offer personal or student versions of their software at a severely discounted price and with a restricted user license. If the reader is using this text for coursework in computational acoustics at a college or university, chances are student licenses for some professional software packages are made available through the campus bookstore. If not, it is easy to find this information online. Open‐source versions of professional software exist and are worth trying. The downside to this is that bugs exist and due to certain constraints a fix may not be available in a hurry. Also, some open‐source tools are not compatible with all operating systems. Readers who like programming and are amenable to the open‐source philosophy can always contribute their fixes and upgrades (read the license). Pure curiosity drives most scientists and engineers to want to know what’s going on in any system, and this is a driver for developing homegrown algorithms even when libraries are available.

A brief description of each chapter is provided. Chapter 2 introduces topics related to numerics, computers, and algorithm development. These topics include binary representation of numbers, floating‐point numbers, and O(N) analysis, to name a few. Chapter 3 contains a survey of the linear wave equation and its connection to the supporting medium, from elastic bodies to fluids. In this chapter the linear wave equation for acoustics in a moving medium is introduced and discussed in detail. Chapter 4 introduces a variety of mathematical techniques and methods for solving the wave equation and describing the general behavior of the acoustic field. Chapter 5 discusses a variety of topics related to the analysis of acoustic waves: dispersion, refraction, attenuation, and Fourier analysis. After these chapters the structure of the text focuses on specific modeling techniques. In Chapter 6 normal modes are discussed. The wave equation is solved for a variety of 1‐dimensional (1‐dim) refractive profiles using exact methods, perturbation theory, and the numerical technique of relaxation. The chapter closes with a brief description of coupled modes and their use in modeling acoustics in realistic environments. Chapter 7 provides an introduction to ray theory and ray tracing techniques. Exact solutions to 1‐dim problems are discussed along with methods of developing ray trace procedures that account for 3‐dim propagation without simplifying assumptions. Numerical techniques are also discussed and the Runge–Kutta method is introduced. In Chapter 8 the finite difference (FD) and finite difference time domain (FDTD) technique are discussed in theory and applied to the wave equation in the frequency and time domains. Following the FD method, Chapter 9 discusses the parabolic equation and its application to modeling sound in ducted environments. Chapter 10 provides an introduction to the finite element method (FEM), introducing numerical techniques required for building an FEM model of the acoustic field in the frequency domain. The last chapter, Chapter 11, is dedicated to the boundary element method (BEM). This chapter discusses the integral equation form of the Helmholtz equation and its discretization into a matrix equation. The exterior and interior problems are discussed, but attention is spent on developing models of the scattering cross section of hard bodies. This chapter introduces techniques for dealing with singular integrals.

Computational Acoustics	Bergman	January 2018
Wind Farm Noise	Hansen	February 2017
The Effects of Sound on People	Cowan	May 2016
Engineering Vibroacoustic Analysis:
Methods and Applications	Hambric et al.,	April 2016
Formulas for Dynamics, Vibration and Acoustics	Blevins	November 2015

COMPUTATIONAL ACOUSTICS

THEORY AND IMPLEMENTATION

Series Preface

1
Introduction