Waves in slowly varying band-gap media

TL;DR

This paper develops an asymptotic method to analyze high-frequency waves in slowly varying periodic media, capturing complex phenomena near band-gap edges and providing accurate reflection descriptions.

Contribution

It introduces a novel asymptotic approach combining WKB and high-frequency homogenization to study wave behavior near band-gap edges in 1D media.

Findings

Asymptotic solutions match numerical results with high accuracy.

Near band-gap edges, solutions resemble Airy functions modulated by Bloch eigenfunctions.

The method accurately predicts wave reflection in layered media.

Abstract

This paper is concerned with the asymptotic description of high-frequency waves in locally periodic media. 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 describe an asymptotic approach for overcoming this difficulty, and take a first step by studying in detail the simplest case of 1D Helmholtz waves. The method entails matching adiabatically propagating Bloch waves, captured by a multiple-scale Wentzel-Kramers-Brillouin (WKB) approximation, with complementary multiple-scale solutions spatially localised about dispersion singularities. Within the latter regions the Bloch wavenumber is nearly critical; this allows their homogenisation, following the method of high-frequency homogenisation (HFH), over a naturally arising scale intermediate between the periodicity (wavelength) and the macro-scale. Analogously to a classical turning-point analysis, we show that close to a spatial band-gap edge the solution is an Airy function, only that it is modulated on the short scale by a standing-wave Bloch eigenfunction. By carrying out the asymptotic matching between the WKB and HFH solutions we provide a detailed description of Bloch-wave reflection from a band gap. Finally, we implement the asymptotic theory for a layered medium and demonstrate excellent agreement with numerical computations.