Asymptotic evaluation of high‐frequency fields near a caustic: An introduction to Maslov's method

Abstract

It is well known that the geometrical optics approximation, generally valid for high‐frequency fields, fails in the vicinity of a caustic. A systematic procedure of V. P. Maslov that remedies this situation will be reviewed in this paper. Maslov's method makes use of a representation of the geometrical optics field in the phase space M = X × K, where a point m = (x, k) is a pair of a position vector x ε X and a wave vector k ε K. A Lagrangian submanifold of M, Λ, that lies in the dispersion surface and is a union of the phase space trajectories selected by the initial conditions is constructed. It can be considered as a global representation of the phase. The phase space amplitudes (half densities) satisfy transport equations defined along those trajectories in Λ. Since trajectories in M never form a caustic, a globally defined amplitude can be established on Λ. The field on X is related to the resultant field on Λ by the “canonical operator,” an operator introduced by Maslov. It generates an integral form of the solution near a caustic that can be evaluated analytically, numerically, or with uniform asymptotic techniques. Away from the caustic it recovers the geometrical optics field. Alternatively, the phase space field can be projected on a hybrid space Y where some of the space coordinates have been replaced by the corresponding wave vector components. For any caustic point in X, one such hybrid space Y where this projection does not encounter a caustic exists. A geometrical optics field results in Y that is related to the original in X by an asymptotic Fourier transform. The solution in X near a caustic can be represented as the Fourier transform to X of that hybrid space geometrical optics solution. These techniques are illustrated with two simple but revealing problems: continuation of the field through a fold caustic in a linear layer medium and through a caustic with a cusp point in a homogeneous medium.