3 - Geometric wave theory

Summary

Geometric wave theory is the natural extension of WKB theory to situations in which the still layer depth H (and therefore the wave speed) is a slowly varying function of both x and y, and possibly even of t, although we will not consider that case here. In fact, even for constant H geometric wave theory is useful because it allows the computation of the structure of normal modes in bounded domains with irregular shapes, i.e., shapes for which there is no simple explicit expression for the normal modes.

The basic assumption of geometric wave theory is that there is a scale separation between the rapidly varying phase of the wavetrain on the one hand, and the slowly varying layer depth and wavetrain parameters such as amplitude and wavenumber on the other. Of course, in bounded domains the domain size must also be large compared to the wavelength. This basic assumption leads to a flexible and generic asymptotic procedure for solving for the wave field. Eventually, with the inclusion of dispersive effects, geometric wave theory becomes the ray-tracing method, which is the Swiss army knife for computing the asymptotic behaviour of small-scale waves in many fields of physics, including GFD.

A peculiarity of the progression from one-dimensional WKB theory to twodimensional geometric wave theory and finally to dispersive ray tracing is that the structure of the theory becomes easier, not harder,asits generality increases.