  We consider the scattering of elastic waves by highly oscillating anisotropic
periodic media with bounded support. Applying the two-scale homogenization, we
first obtain a constant coefficient second-order partial differential elliptic
equation that describes the wave propagation of the effective or overall wave
field. We study the rate of convergence by introducing complimentary boundary
correctors. To account for dispersion induced by the periodic structure, we
further pursue a higher-order homogenization. We then investigate the rate of
convergence and formally obtain a fourth-order differential equation that
demonstrates the anisotropic dispersion
