Two-dimensional phononic crystals and scattering of elastic waves by an array of voids

We study the diffraction of plane elastic waves by a grating of circular holes in an elastic matrix, using a multipole formulation to determine scattering matrices, for both the scalar (out-of-plane shear) and vector (plane-strain) problems. We use these matrices in a recurrence procedure to analyse the reflection and transmission properties of a stack containing an arbitrary number of such gratings. We establish the form of symmetry properties that may be used to verify the elastodynamic scattering matrices and comment on the filtering properties of grating stacks and their relationship to phononic band diagrams.