Bie method to analyze wave motion in solids with periodically distributed cavities

Abstract

A three-dimensional boundary integral equation method is developed to analyze reflection and transmission of incident waves by a doubly periodic planar array of spherical cavities in an elastic solid. By taking into account the periodicity of the incident field and the geometrical configuration, the system of integral equations is reduced to a system over the surface of a single cavity in a reference cell. The displacements on the cavity surface, which are obtained by solving the system of boundary integral equations, are used to obtain reflection and transmission coefficients via a novel application of the Betti-Rayleigh reciprocal theorem. The dispersion relation for propagation of horizontally polarized transverse waves in a triply periodic three-dimensional array of spherical cavities is subsequently determined with the aid of the Floquet or Bloch theory, and by using the reflection and transmission coefficients for a single planar array of cavities. The dispersion curves show passing and stopping bands. Numerical results are presented for three directions of wave propagation and two geometrical configurations.