Asymptotics of solutions of the Helmholtz equation in a region of caustic shadow I.

Z. A. Yanson

Zapiski Nauchnykh Seminarov POMI, Mathematical problems in the theory of wave propagation. Part 13, Tome 128 (1983), pp. 172-185

Résumé

A generalization of the method of geometrical optics for finding the solutions of reduced wave equation on a dark side of a caustic is used. The case of inhomogeneous two-dimensional space with analytical velocity $v(x, z)$ of wave propagation is considered. It is shown that the law of changing of amplitude coefficients are determined by two veotorial fields depended on the combination of vectors $\nabla\xi$ and $\nabla\eta$ where $\tau=\xi+i\eta$ is the eikonal. The procedure of analytical continuation of the eikonal equation to complex coordinare space is applied and as a result the system of partial differential equations for $\xi$ and $\eta$ is arised. The method of complex rays for solving this system with initial data on a caustie is used. The method is illustrated by the standard example $v^{-2}(z)=c_0-c_1z$ where $c_0, c_1={\rm const}$, $z$ is the distance from the caustic.