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mathematical relationship between the variables of inter- est in acoustics, often the acoustic pressure or particle velocity and the speed of a wave. The equation relates the temporal and spatial changes to these variables, including dependence on the wavelength and frequency of the wave. As a consequence, the equation is a second-order partial differential equation of pressure or particle velocity and is three-dimensional. The pressure or particle velocity is dependent on three spatial directions and time.
In all areas of physics, solutions to problems involving the wave equation require specifying additional boundary conditions that depend on the geometry of the problem. Only in specific ideal cases with simple conditions and geometry are analytical solutions even possible. However, the wave equation is a powerful and useful tool for inves- tigating the physics involved.
For most real problems of interest, the geometry involved is much too complicated to solve by any other means than by computational methods. For example, if one wanted to simulate the propagation of sound through the ear canal (Puria, 2020), the geometric structure would not be simple and defining real boundary conditions would make the problem too complicated to be solved any other way than by numerical solution of the wave equation.
Propagation of acoustic waves in a variety of environments is well understood and documented, but any real environment is overly complex and prediction of sound fields becomes impossible to solve analytically. For example, one may wish to determine the acoustic pressure field in a large area underwater in the ocean (e.g., Duda et al., 2019) where the environment, boundary conditions and spatial distributions of fluid prop- erties are complicated. To solve a wave equation with such complexity, the problem is reduced to numerical solutions.
There are a bevy of techniques discussed in this article for solution of the wave equation in various situations. Several of these techniques are numerical methods applied to solve the equations directly without approximations, whereas others require a successive approximation of results.
The Emergence of Computational Methods
Since its invention in the 1930s, the digital computer has been used to solve difficult problems in physics. Early uses were in areas of nuclear physics where they
performed simulations on ballistics and particle evo- lution for the development of the atomic bomb.
Monte Carlo Simulation
Several techniques and algorithms were developed at the Los Alamos (NM) National Laboratory by Jon von Neu- mann as part of his work on the atomic bomb, leading to what we now know as Monte Carlo simulations. As one might expect, Monte Carlo applications involve any phe- nomena that could be modeled as random or spontaneous, such as playing games of chance at a casino. Some phenom- ena that are modeled this way include radioactive decay and the random nature of thermal motion (Landau and Price, 1997). Additionally, Monte Carlo simulations can be used to model sound propagation in the atmosphere (Burkatovskaya et. al., 2016) where multiple scattering and the turbulent nature of the atmosphere (Blanc-Benon et al., 2002) can be taken into consideration.
Continued work in computational physics led to the dis- covery of chaotic behavior in nonlinear dynamics where deterministic mechanical systems exhibited seemingly random states of motion. The theoretical underpinnings of mechanics had existed for nearly half a century before computer technology made it possible to make the com- plicated computations needed to simulate the interactions.
Early Use of Computers in Acoustics
An early mention of using computers in acoustics is provided by a talk given at the 62nd meeting of the Acoustical Soci- ety of America by Schroeder (1961) on novel uses in room acoustics. In his abstract, Schroeder spoke of using digital computers to simulate complicated transmission of sound in rooms and simulation of spatial and frequency responses in rooms using Monte Carlo techniques. Schroeder’s insight revolutionized architectural acoustics. Computational methods have proven enormously powerful in predicting acoustic performance of interior spaces and have enhanced the ability of the specialist to design spaces acoustically, such as in concert halls (Sviolja and Xiang, 2020).
The decades of improvement in computer technology and computational performance have allowed greater use of such numerical methods for acoustic wave propagation, scattering, radiation, and other acoustically related phe- nomena. This, in turn, has enhanced discovery and problem solving. Simulations of different phenomena have provided
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