Polarization approach to the scattering of elastic waves—I. Scattering by a single inclusion

Scattering problems in elastodynamics are formulated in terms of integral equations, whose kernels are obtained from the Green's function for a comparison body. The comparison body will usually be taken as homogeneous and elastic in applications but, at least formally, there is no bar to its being inhomogeneous, viscoelastic and non-local. The novel feature of the formulation is the introduction of a “momentum polarization” to cope with density variations in a way that exactly parallels the stress polarization's correspondence with variations in moduli. To illustrate the use of the equations, scattering by an ellipsoidal inhomogeneity in a generally anisotropic matrix is studied in the Rayleigh limit and an asymptotic formula for its scattering cross-section is given. Detailed results are presented for a spheroidal inhomogeneity in an isotropic matrix, with explicit limiting forms for the scattering cross-sections of penny-shaped cracks, rigid circular discs and rigid needles.

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