Diffraction of Elastic Waves

Abstract

An integral solution to the differential equations of elastodynamics is given. The solution for the displacement at any point in an isotropic homogeneous, elastic medium is obtained in terms of the initial distribution of body forces within a given volume and the initial distribution of displacements and stresses over the bounding surface of the volume. These quantities appear in the solutions as retarded displacements, stresses, and body forces, with two types of retardation depending upon the velocities of longitudinal and transverse waves. From the integral solution the diffraction of elastic waves through large apertures bounded by opaque walls in solid materials is formulated. The incident wave motions may be either of longitudinal or transverse polarizations. The expected shear-compression interaction is obtained in the course of the computation. By analogy with optical procedure, Fraunhofer diffraction of elastic waves may be investigated as a special case. In the example of diffraction by a slit, an incident compression wave is diffracted into (1) a compression wave with the “ordinary” spatial distribution similar to that obtained in the corresponding optical problem and (2) a weaker shear wave with an “extraordinary” distribution. In the case of incident shear waves, there is obtained a diffracted shear wave with an ordinary distribution, and a possible weak compression wave with the extraordinary distribution. The presence of the diffracted compression wave depends upon the polarization of the incident shear wave. Opacity in the Kirchhoff sense is a property of materials which are perfectly rigid.