On Dispersion for Linear Waves in Nonuniform Media

Abstract

Motivated by several well-known examples of linear waves in uniform media, a general definition of a dispersionfree wave valid for both uniform and nonuniform media is set up. Examples of dispersionfree generalizations of the classical wave equation to nonuniform media are given. Although dispersionfree waves are exceptional both in the uniform and in the nonuniform case, it is proved that hyperbolic systems have the property that in the limit of short wavelengths the solutions behave like a linear combination of dispersionfree waves. On that basis it is possible to study the dispersion properties of hyperbolic waves by an asymptotic expansion. Explicit calculations are given for such expansions for a family of generalizations of the classical wave equation to nonuniform media.