2016/12/12 by Ory Schnitzer, Schnitzer, Ory · 1 citation
Engineering · Materials Science · Physics and Astronomy · #Acoustic Wave Phenomena Research #Analysis of PDEs (math.AP) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Metamaterials and Metasurfaces Applications #Optics (physics.optics) #Photonic Crystals and Applications
paper · pdf · doi:10.48550/arxiv.1612.03895
openalex publication_date 2016/12/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.