On the scattering of SH waves from a point source by a sphere

Summary. The canonical problem of the scattering of SH waves from a harmonic point source in a homogeneous elastic medium, by an inviscid fluid-filled cavity in the medium, is examined in some detail. Partial wave-expansions are derived for the incident, secondary and total wave potentials. These expansions are converted by the Watson transformation to contour integrals which for high frequencies may be asymptotically evaluated by steepest descents, the Fock method, by residue series expressions or numerically. In general the analysis and results for this canonical problem are similar to those for an incident plane wave but the exact contour-integral representations differ substantially in the lit region. The appropriate integral representation of the incident or total wave potential in the lit region is found to depend on whether or not the incident ray has a turning point. This change of representation is also required in the study of PP and SS phases from a point source inside a sphere but has been overlooked elsewhere. Only some of the potentials may be usefully exactly expressed as the contour integral of a single simple integrand along a simple contour while some of the other potentials may be approximately expressed in this way. These contour integrals may be numerically evaluated to find the potentials. In the lit region, asymptotic evaluation of the integrals by steepest descents gives the second-order ray approximations for the incident and reflected waves. These approximations break down near the shadow boundary, where the incident and reflected waves coalesce. In this region combined waves are evaluated by the Fock method to give the Fresnel term and an additional term. The shift of the half-amplitude shadow boundary into the geometrical shadow is noted. Finally the diffracted wave-potential in the deep shadow and the multiply-diffracted wave potential in both the lit and shadow regions are evaluated by residue series expressions and agree with the ray theory of diffraction.

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