Introduction

In the previous chapters we have studied waves which can be represented either as exact solutions of the field equations (as in the case of simple waves) or as propagating surfaces of discontinuities, for which exact transport laws can be obtained. These classes of waves, although very important for testing the mathematical structure of the theory, have a limited range of applicability. In particular, simple waves are restricted to one dimensional propagation into a constant state whereas propagating surfaces can model only impulsive waves. This leaves out the vast class of harmonic waves propagating into an arbitrary nonuniform state, which comprises most of the interesting applications. In order to treat the latter case it is necessary, in general, to resort to perturbation methods. Most of the perturbation methods used for this purpose are based on the so-called geometrical optics approximation, or variants thereof such as the high-frequency expansion or the method of multiple scales.

The idea underlying these methods is that there are at least two widely different length (or time) scales in the problem, the length L characteristic of the variation of the background state into which the wave is propagating, and the mean wavelength λ of the wavetrain, with the ordering λ « L. One then introduces a parameter ɛ = λ/L into the problem (usually through the appearance of one or several rapidly varying phase functions and also through other stretched variables, depending on ɛ) and seeks asymptotic solutions of the field equations as appropriate power series in ɛ.