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Unformatted text preview: Chapter 2 Linear and Nonlinear Waves Our exploration of the vast mathematical continent that is partial differential equa- tions will begin with simple first order equations. In applications, first order partial dif- ferential equations are most commonly used to describe dynamical processes, and so time, t , is one of the independent variables. Our discussion will focus on dynamical models in a single space dimension, bearing in mind that most of the methods can be extended to higher dimensional situations. First order partial differential equations and systems model a wide variety of wave phenomena, including transport of pollutants in fluids, flood waves, acoustics, gas dynamics, glacier motion, chromatography, traffic flow, and a variety of bi- ological and ecological systems, [ 145 ]. As always in mathematical analysis, one must first master the linear versions before venturing into the nonlinear wilderness. One basic solution technique relies on an inspired change of variables, that comes from rewriting the equation in a moving coordinate frame. This naturally leads to the fun- damental concept of characteristic curves, along which signals and physical disturbances propagate. The resulting method of characteristics solves a first order linear partial dif- ferential equation by reducing it to one or more first order nonlinear ordinary differential equations. In the nonlinear regime, the most important new phenomenon is the possible break- down of solutions in finite time, resulting in the formation of discontinuous shock waves. A familiar example is the supersonic boom produced by an airplane that breaks the sound barrier. Signals continue to propagate along characteristic curves, but now they may cross each other, precipitating the onset of a shock discontinuity. The characterization of the ensuing shock dynamics is not specified by the partial differential equation alone, but relies on additional physical information, to be specified by the choice of a conservation law and causality condition. A full-fledged analysis of shock dynamics becomes quite challenging, and only the basics will be developed here. Having attained a basic understanding of first order wave dynamics, we then focus our attention on the first of three fundamental second order partial differential equations, known as the wave equation, which is used to model waves and vibrations in a violin string, a column of air in a clarinet, or an elastic bar. Its multi-dimensional versions serve to model vibrations of membranes, solid bodies, water waves, electromagnetic waves, including light, radio and micro-waves, acoustic waves, and many other physical phenomena. The one- dimensional wave equation is one of a small handful of physically relevant partial differential equations that has an explicit solution formula, originally discovered by the eighteenth century French mathematician (and encyclopedist) Jean dAlembert. His solution is the result of being able to factorize the second order wave equation into a pair of first 1/19/12 12 c circlecopyrt 2012 Peter J. Olver x...
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