In the context of acoustics, single polarization electromagnetism and elastic plates we consider microstructured media that have an
underlying periodic structure and we develop an asymptotic continuum
model that captures the essential microstructural behaviour entirely
in a macroscale setting. The asymptotics are based upon a two-scale
approach and are valid even at high frequencies when the wavelength
and microscale length are of the same order. The general theory is
illustrated via one- and two-dimensional model problems that can have
zero-frequency stop bands that preclude conventional averaging and
homogenization theories. Localized defect modes created by material or shape
variations are also modelled using the theory and compared to
numerical simulations.