Waves in Slowly Varying Band-Gap Media

Abstract

This paper is concerned with waves in locally periodic media, in the high-frequency limit where the wavelength is commensurate with the period. A key issue is that the Bloch-dispersion curves vary with the local microstructure, giving rise to hidden singularities associated with band-gap edges and branch crossings. We suggest an asymptotic approach for overcoming this difficulty, which we develop in detail in the case of time-harmonic waves in one dimension. The method entails matching adiabatically propagating Bloch waves, captured by a two-variable Wentzel--Kramers--Brillouin (WKB) approximation, with complementary multiple-scale solutions spatially localized about dispersion singularities. The latter solutions, obtained following the method of high-frequency homogenization (HFH), hold over dynamic length scales intermediate between the periodicity (wavelength) and the macro-scale. In particular, close to a spatial band-gap edge the solution is an Airy function modulated on the short scale by a standing...