Geometrical theory of diffraction for three-D elastodynamics

Abstract

Keller’s geometrical theory of diffraction is applied to three‐dimensional elastodynamics, in particularly to diffraction of longitudinal waves by a crack. The theory provides useful approximations for large frequencies and/or large distances from the edge of the crack. For the class of problems considered in this paper, the canonical solution is provided by the fields describing diffraction by a semi‐infinite crack of a plane longitudinal wave which is incident under an arbitrary angle with the edge of the crack. The formal solution to the canonical problem is obtained by means of integral transform techniques in conjunction with an application of the Wiener–Hopf method to a set of coupled equations. The pertinent asymptotic expressions for the diffracted field are evaluated, and the diffraction coefficients which enter the geometrical theory are determined. As an example, the three‐dimensional problem of diffraction of a point‐source field by a semi‐infinite crack is worked out in detail.