  The propagation of acoustic or elastic waves in artificial crystals,
including the case of phononic and sonic crystals, is inherently anisotropic.
As is known from the theory of periodic composites, anisotropy is directly
dictated by the space group of the unit cell of the crystal and the rank of the
elastic tensor. Here, we examine effective velocities in the long wavelength
limit of periodic acoustic and elastic composites as a function of the
direction of propagation. We derive explicit and efficient formulas for
estimating the effective velocity surfaces, based on second-order perturbation
theory, generalizing the Christofell equation for elastic waves in solids. We
identify strongly anisotropic sonic crystals for scalar acoustic waves and
strongly anisotropic phononic crystals for vector elastic waves. Furthermore,
we observe that under specific conditions, quasi-longitudinal waves can be made
much slower than shear waves propagating in the same direction.
