Asymptotic theory of wave-propagation

Abstract

A general method is presented for finding asymptotic solutions of problems in wave-propagation. The method is applicable to linear symmetric-hyperbolic partial differential equations and to the integro-differential equations for the electromagnetic field in a dispersive medium. These equations may involve a large parameter λ. In the electromagnetic case λ is a characteristic frequency of the medium. The parameter may also appear in initial data or in the source terms of the equations, in a variety of different ways. This gives rise to a variety of different types of asymptotic solutions. The expansion procedure is a “ray method”, i.e., all the functions that appear in the expansion satisfy ordinary differential equations along certain space-time curves called rays. In general, these rays do not lie on characteristic surfaces, but may, for example, fill out the interior of a characteristic hypercone. They are associated with an appropriately defined “group velocity”. In subsequent papers the ray method developed here will be applied to the analysis of transients, Cerenkov radiation, transition radiation, and other phenomena of wave-propagation.

An interesting by-product of the ray method is the conclusion, derived in section 6.3, that the theory of relativity imposes no restriction on the speed of energy transport in anisotropic media.