Abstract

For a basic model of wave propagation in a matrix containing an infinite periodic array of high-contrast ``soft’’ inclusions,  the macro and micro-scales are coupled reflecting the effect of the inclusion resonances. This leads to two-scale descriptions displaying in an asymptotically explicit way such effects as formation of band gaps due to the resonances. One can obtain refined two-scale type approximations accompanied by rigorous error estimates via asymptotic analysis of a scaled Floquet-Bloch-Gelfand transform combined with an (inverse scaled) Fourier transform. The latter combined transform appears to be a two-scale version of a classical Whittaker-Shannon interpolation. It has several interesting properties, and the approach allows generalisations in various directions beyond the infinite periodic and PDE settings. Based on a joint work with Shane Cooper and Ilia Kamotski (UCL).