On the rigorous foundation of short-wave asymptotics

Abstract

AbstractCertain physical theories are short-wave limits of more general theories. Thus ray optics is the short-wave limit of wave optics, and classical mechanics is the short-wave limit of wave mechanics. In principle it must be possible to deduce the former from the latter theories by a rigorous mathematical limiting process; in fact the arguments found in the literature are formal, plausible and non-rigorous. (We are here concerned with linear wave equations and time-periodic phenomena.) For some wave equations there are, however, a few explicit rigorous canonical solutions relating to simple geometrical configurations, e.g. to conics in two dimensions for the equations of acoustics, and for these the asymptotics can be found rigorously. For more general configurations the solution of a typical boundary-value problem can be reduced to the solution of a Fredholm integral equation of the second kind.