  We investigate the occurrence of anomalous diffusive transport associated
with acoustic wave fields propagating through highly-scattering periodic media.
Previous studies had correlated the occurrence of anomalous diffusion to either
the random properties of the scattering medium or to the presence of localized
disorder. In this study, we show that anomalous diffusive transport can occur
also in perfectly periodic media and in the absence of disorder. The analysis
of the fundamental physical mechanism leading to this unexpected behavior is
performed via a combination of deterministic, stochastic, and fractional-order
models in order to capture the different elements contributing to this
phenomenon. Results indicate that this anomalous transport can indeed occur in
perfectly periodic media when the dispersion behavior is characterized by
anisotropic (partial) bandgaps. In selected frequency ranges, the propagation
of acoustic waves not only becomes diffusive but its intensity distribution
acquires a distinctive L\'evy alpha-stable profile having pronounced
heavy-tails. In these ranges, the acoustic transport in the medium occurs
according to a hybrid transport mechanism which is simultaneously propagating
and anomalously diffusive. We show that such behavior is well captured by a
fractional diffusive transport model whose order can be obtained by the
analysis of the heavy tails.
