Uniformly valid asymptotic wave propagation based upon variational principles

Abstract

We present a uniformly valid asymptotic theory for linear, short-wavelength waves based upon ‘slow’ variational principles in configuration space, mixed space, and momentum space. The variational principles define an eikonal equation, which determines the ray geometry, and transport equations, which determine the amplitude variations in the various projections of phase space. There are no caustics in phase space, i.e. neighboring rays in phase space never cross, as a result of Liouville's theorem for Hamiltonian systems. It is only when the phase-space trajectories are projected onto configuration space, a mixed space, or momentum space, that caustics occur. The essential strategy is to consider a mixed-space or momentum-space variational principle in the vicinity of a configuration-space caustic. We use the two-dimensional Helmholtz equation to illustrate the theory because it constitutes a simple example that captures all the features of the variational technique.