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Published online by Cambridge University Press: 05 June 2012
People routinely employ a variety of numerical methods to analyze waves in materials with complex constitutive equations and boundary conditions. Indeed, it may appear to you that the number of numerical solution methods is equal to or greater than the number of numerical analysts. However, most of these algorithms are based on a few simple concepts. Here we shall discuss these concepts by examining several algorithms. We shall solve the linear wave equation with a numerical algorithm derived from the method of characteristics, and we shall solve the equations describing nonlinear waves with a numerical algorithm derived from the method of finite differences. Through these examples, you will become acquainted with the fundamental issues in the numerical analysis of waves. You will need to understand these issues to successfully select and apply any good commercial software package to analyze your wave-propagation problems.
We can only obtain analytic solutions to the equations describing nonlinear waves for a very limited set of geometries, constitutive equations, and boundary conditions. For example, the Riemann integrals yield a solution for a one-dimensional simple wave in an infinite volume of nonlinear-elastic material; however, obtaining analytic solutions for the interaction of two simple waves ranges from difficult to impossible depending upon the constitutive equations of the medium. Without the advent of the digital computer and sophisticated methods of numerical analysis, advances in the field of nonlinear wave motion and the development of constitutive theories for dynamic loading of materials would have stagnated over the past 40 years.
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