Propagation of elastic waves through a lattice of cylindrical cavities

We consider 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 periodicity, and hence the method yields explicit approximations that describe the phenomena associated with periodic media. This is illustrated with square and hexagonal lattices.