Propagation of acoustic waves in 3D periodic media: Dispersion, clusters, and local gaps

We present a theoretical study of the propagation of acoustic waves in a 3D infinite medium containing a periodic array of small identical inclusions of arbitrary shape with transmission conditions on their interfaces. The inclusion size a is much smaller than the array period. We present the dispersion relation and show that there are exceptional frequencies for which the solution is a cluster of waves propagating in several different directions. Different clusters may contain waves with the same direction, and the frequencies of the waves depend on the clusters but not on the direction of waves. We show that global gaps do not exist if a is small enough. The notion of local gaps which depends on the choice of the wavevector k, is introduced and discussed. The location of local gaps for a medium with a simple cubic lattice of identical inclusions is determined.