Reflection and transmission of an obliquely incident wave by an array of spherical cavities

A plane wave is incident on a doubly periodic array of spherical cavities in an elastic solid. The cavities are of equal radius d, and their centers are located in a single plane, the x₁x₂plane, at positions x₁= ma, x₂= nb. The propagation vector of a plane, time-harmonic, incident longitudinal wave is located in the X1, X₃plane. The scattering problem is formulated rigorously by taking advantage of the geometrical periodicity. The reflected and transmitted longitudinal and transverse wave motions may be expressed as superpositions of an infinite number of wave modes, each with its own cutoff frequency. Reflection and transmission coefficients have been defined as integrals over a single cavity in terms of the displacement components and auxiliary surface traction terms on the surface of the cavity. The system of singular integral equations for the displacement components has been solved numerically by the boundary integral equation method. Curves show the reflection and transmission coefficients for the reflected and transmitted longitudinal and transverse waves as functions of the frequency.