Cover Page

Wiley Series in Acoustics, Noise and Vibration

Computational AcousticsBergmanJanuary 2018
Wind Farm NoiseHansenFebruary 2017
The Effects of Sound on PeopleCowan  May 2016
Engineering Vibroacoustic Analysis:
Methods and ApplicationsHambric et al.,  April 2016
Formulas for Dynamics, Vibration and AcousticsBlevinsNovember 2015



David R. Bergman

Exact Solution Scientific Consulting LLC, Morristown, NJ, USA


Series Preface

This book series will embrace a wide spectrum of acoustics, noise, and vibration topics from theoretical foundations to real‐world applications. Individual volumes will range from specialist works of science to advanced undergraduate and graduate student texts. Books in the series will review scientific principles of acoustics, describe special research studies, and discuss solutions for noise and vibration problems in communities, industry, and transportation.

The first books in the series include those on Biomedical Ultrasound; Effects of Sound on People, Engineering Acoustics, Noise and Vibration Control, Environmental Noise Management; and Sound Intensity and Windfarm Noise. Books on a wide variety of related topics.

The books I edited for Wiley—Encyclopedia of Acoustics (1997), The Handbook of Acoustics (1998), and Handbook of Noise and Vibration Control (2007)—included over 400 chapters written by different authors. Each author had to restrict their chapter length on their special topics to no more than about 10 pages. The books in the current series will allow authors to provide much more in‐depth coverage of their topic.

The series will be of interest to senior undergraduate and graduate students, consultants, and researchers in acoustics, noise, and vibration and in particular those involved in engineering and scientific fields, including aerospace, automotive, biomedical, civil/structural, electrical, environmental, industrial, materials, naval architecture, and mechanical systems. In addition the books will be of interest to practitioners and researchers in fields such as audiology, architecture, the environment, physics, signal processing, and speech.

Malcolm J. Crocker
Series Editor


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.