The Maslov Method and the Asymptotic Fourier Transform: Caustic Analysis

摘要

Representation of a high frequency field by geometrical optics fails in the vicinity of a caustic. This report presents a systematic approach, due to Maslov, which consists essentially of operating in the phase-space M (position-wavevector) instead of the space X (position). An integral representation is described that may be evaluated numerically, analytically, or by applying known techniques to obtain a uniform asymptotic expression. A generalization of the procedure, known as the GO-AFT approach, is explained which circumvents the phase space description. To illustrate this approach, it is applied to a general homogeneous medium problem (one involving a cusp caustic) and to a general linear-layer problem (a plane wave incident on a linear layer). Comparisons with other methods for computing the field near to a caustic are made. Problems concerning indirect and direct asymptotics and caustics in the presence of a diffracting half-plane are also considered.