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 nonoverlapping elastic spheres, charact erized by Lame coefficients which may be complex and frequency dependent, a rranged periodically in a host medium with different mass density and Lame coefficients. We view the crystal as a sequence of planes of spheres, paral lel to and having the two-dimensional periodicity of a given crystallograph ic plane, and obtain the complex band structure of the infinite crystal ass ociated with this plane. The method allows one to calculate, also, the tran smission, reflection, and absorption coefficients for an elastic wave (long itudinal 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.