Speed of sound in periodic elastic composites

We consider the low-frequency limit (homogenization) for propagation of sound waves in periodic elastic medium (phononic crystals). Exact analytical formulas for the speed of sound propagating in a three-dimensional periodic arrangement of liquid and gas or in a two-dimensional arrangement of solids are derived. We apply our formulas to the well-known phenomenon of the drop of the speed of sound in mixtures. For air bubbles in water we obtain a perfect agreement with the recent results of coherent potential approximation obtained by M. Kafesaki, R. S. Penciu, and E. N. Economou [Phys. Rev. Lett. 84, 6050 (2000)] if the filling of air bubbles is far from close packing. When air spheres almost touch each other, the approximation gives 10 times lower speed of sound than the exact theory does.