Abstract | ||
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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-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. |
Year | DOI | Venue |
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2017 | 10.1137/16M110784X | SIAM JOURNAL ON APPLIED MATHEMATICS |
Keywords | Field | DocType |
Bloch waves,periodic media,singular perturbations | Dispersion (optics),Bloch wave,Mathematical analysis,Homogenization (chemistry),WKB approximation,Gravitational singularity,Periodic graph (geometry),Airy function,Wavelength,Mathematics | Journal |
Volume | Issue | ISSN |
77 | 4 | 0036-1399 |
Citations | PageRank | References |
0 | 0.34 | 1 |
Authors | ||
1 |
Name | Order | Citations | PageRank |
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O. Schnitzer | 1 | 0 | 1.35 |