On the Calculation of the Morse Index and the Extension of Ray Formulas Beyond Caustics

Abstract

Being the simplest and illustrative from the physical point of view, the ray method is extensively used for computations of short wave fields of a different physical nature: acoustic, electrodynamic, and elastodynamic. However it is not applicable in neighborhoods of caustics, where ray amplitudes get singular. Although caustics may appear in quantities in complex inhomogeneous media, they have zero measure, and therefore there are subdomains free of caustics in inhomogeneous media where ray formulas can be used for computations of wave fields. To this end it is necessary to calculate the phase jumps caused by the transition of rays through caustics. Mathematically, we must calculate the Morse index for a ray, i.e., the number of focal points (counting their multiplicity) on the ray between the source and the observation point. In the article, this problem is considered and a complete solution to it is given in the case of two space variables. Namely, a complex-valued function of the arc length along a ray is constructed and the increment of its argument between the source and the observation point, computed modulo 2π, gives the Morse index for that ray in the two cases where the field of rays is produced by a point source or is generated by an initially given wave front. Bibliography: 8 titles.