Scattering of spherical elastic waves from a small-volume spherical inclusion

Abstract

A theory is presented for the scattering of spherical elastic waves by a small, spherical inclusion. The domain of validity of the treatment is ka′ ≪ 1,\ where k is the wavenumber and a′ is the radius of the spherical inclusion. The treatment is an extension of that of Gritto, Korneev and Johnson (1995), who derived exact expressions for the scattered wavefield from a plane P wave incident on a spherical inclusion in the low-frequency limit. Since neither treatment assumes small perturbations, they are also valid for an arbitrary material property contrast between the spherical inclusion and the background medium. For spherical elastic wavefields propagating through an infinite, homogeneous medium and interacting with a single small, spherical inclusion, the scattered wavefield is obtained directly as a sum of spheroidal motions of degrees 0, 1 and 2 in scatterer-centred coordinates, with nine independent amplitude coefficients which depend on the contrast in material properties and are proportional to the volume of the inclusion. If the inclusion is embedded in a stratified and bounded medium, then these scattering coefficients provide a valid local description of the scattered wavefield in the vicinity of the inclusion. Exploiting this fact, source equivalents in the frequency domain (six moment tensor elements and three single forces) are derived from the scattering coefficients. To first order in material property contrasts, equivalence is established between these source equivalents and the coupled mode theory developed for surface waves and long-period body waves using perturbation theory.