On the Low-Frequency Interaction between a Central Dyadic Wave and a Spherical Cavity

Abstract

The dyadic fundamental solution of the linearized theory of elasticity in a homogeneous and isotropic medium is considered as a point-generated incident dyadic field. This central field is disturbed by a small spherical cavity within the medium of propagation. This two-center interaction problem provides a general dyadic scattering problem which includes the classical elasticity problems as a special case, obtained via contractions with specific directions. An analytic solution of the low-frequency approximations can be obtained via a generalization from the vector displacement field to a dyadic displacement field. Using this generalization the zeroth- and the first-order approximation of the near field are obtained explicitly. In the far field the leading nonvanishing approximation for the radial and the tangential scattering amplitudes as well as that for the scattering cross-section are also evaluated. The analytic results obtained are then compared against relative results in classical elasticity obtained via the boundary element method and an amazing coincidence is observed when the radius of the sphere is small with respect to the wavelengh of the central incident field.