Scattering of elastic waves by periodic arrays of spherical bodies

We develop a formalism for the calculation of the frequency band structure of a phononic crystal consisting of non-overlapping elastic spheres, characterized by Lam'e coefficients which may be complex and frequency dependent, arranged periodically in a host medium with different mass density and Lam'e coefficients. We view the crystal as a sequence of planes of spheres, parallel to and having the two dimensional periodicity of a given crystallographic plane, and obtain the complex band structure of the infinite crystal associated with this plane. The method allows one to calculate, also, the transmission, reflection, and absorption coefficients for an elastic wave (longitudinal or transverse) incident, at any angle, on a slab of the crystal of finite thickness. We demonstrate the efficiency of the method by applying it to a specific example.