  We investigate the behavior of waves in a periodic medium containing small
soft inclusions or cavities of arbitrary shape, such that the homogeneous
Dirichlet conditions are satisfied at the boundary. The leading terms of Bloch
waves, their dispersion relations, and cutoff frequencies are rigorously
derived. Our approach reveals the existence of exceptional wave vectors for
which Bloch waves are comprised of clusters of perturbed plane waves that
propagate in different directions. We demonstrate that for these exceptional
wave vectors, no Bloch waves propagate in any one specific direction.
