  The propagation of waves through microstructured media with periodically
arranged inclusions has applications in many areas of physics and engineering,
stretching from photonic crystals through to seismic metamaterials. In the
high-frequency regime, modelling such behaviour is complicated by multiple
scattering of the resulting short waves between the inclusions. Our aim is to
develop an asymptotic theory for modelling systems with arbitrarily-shaped
inclusions located on general Bravais lattices. We then consider the limit of
point-like inclusions, the advantage being that exact solutions can be obtained
using Fourier methods, and go on to derive effective medium equations using
asymptotic analysis. This approach allows us to explore the underlying reasons
for dynamic anisotropy, localisation of waves, and other properties typical of
such systems, and in particular their dependence upon geometry. Solutions of
the effective medium equations are compared with the exact solutions, shedding
further light on the underlying physics. We focus on examples that exhibit
dynamic anisotropy as these demonstrate the capability of the asymptotic theory
to pick up detailed qualitative and quantitative features.
