Propagation of elastic waves through a lattice of cylindrical cavities

We consider the wave propagation through a doubly-periodic array of cavity cylinders in an isotropic elastic medium using the method of matched asymptotic expansions, based on the assumptions that the scatterer size is much smaller than both the wavelength and the array periodicity. There is no restriction on the wavelength relative to the array periodicity, and hence the method yields explicit approximations that can describe the phenomena associated with periodic media. This is illustrated by reference to square and hexagonal lattices.