Dispersion of waves in two and three-dimensional periodic media.

We consider the propagation of acoustic time-harmonic waves in homogeneous media containing periodic lattices of spherical or cylindrical inclusions. It is assumed that the wavelength has the order of the periods of the lattice while the radius a of inclusions is small. A new approach is suggested to derive the complete asymptotic expansions of the dispersion relations in two- and three-dimensional cases as a → 0 and first several terms of the expansions are evaluated explicitly. Our method is based on the reduction of the original singularly perturbed (by inclusions) problem to the regular one. The Dirichlet, Neumann, and transmission boundary conditions are considered. In the former case, we estimate the cutoff wavelength λ max supported by the periodic medium in two and three dimensions. The effective wave speed is obtained as a function of the wave frequency, the filling fraction of the inclusions, and the physical properties of the constituents of the mixture. Dependence of the asymptotic formulas obtained in the paper on geometric and material parameters is illustrated by graphs. [ABSTRACT FROM AUTHOR]