Waves in slowly varying band-gap media

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Title: Waves in slowly varying band-gap media
Authors: Schnitzer, O
Item Type: Journal Article
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 match- ing adiabatically propagating Bloch waves, captured by a two-variable Wentzel–Kramers–Brillouin (WKB) approximation, with complementary multiple-scale solutions spatially localised about disper- sion singularities. The latter solutions, obtained following the method of high-frequency homogeni- sation (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-wave Bloch eigenfunction. Asymptotically matching the WKB and HFH solutions in this scenario yields a detailed description of Bloch-wave reflection from a band gap, which is shown to be in excellent agreement with numerical computations for a layered medium
Issue Date: 29-Aug-2017
Date of Acceptance: 29-Jun-2017
URI: http://hdl.handle.net/10044/1/49827
DOI: https://dx.doi.org/10.1137/16M110784X
ISSN: 0036-1399
Publisher: Society for Industrial and Applied Mathematics
Start Page: 1516
End Page: 1535
Journal / Book Title: SIAM Journal on Applied Mathematics
Volume: 77
Issue: 4
Copyright Statement: © 2017, Society for Industrial and Applied Mathematics
Keywords: Science & Technology
Physical Sciences
Mathematics, Applied
Bloch waves
periodic media
singular perturbations
0102 Applied Mathematics
Applied Mathematics
Publication Status: Published
Appears in Collections:Mathematics
Faculty of Natural Sciences

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