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Title: High-frequency homogenization for periodic media
Authors: Craster, RV
Kaplunov, J
Pichugin, AV
Keywords: Floquet-Bloch waves;Stop bands;Photonics;High-frequency long waves;Homogenization
Issue Date: 2010
Publisher: The Royal Society
Citation: Proceedings of the Royal Society A: Mathematical, Physical & Engineering Sciences, 466(2120), 2341 - 2362, 2010
Abstract: An asymptotic procedure based upon a two-scale approach is developed for wave propagation in a doubly periodic inhomogeneous medium with a characteristic length scale of microstructure far less than that of the macrostructure. In periodic media, there are frequencies for which standing waves, periodic with the period or double period of the cell, on the microscale emerge. These frequencies do not belong to the low-frequency range of validity covered by the classical homogenization theory, which motivates our use of the term ‘high-frequency homogenization’ when perturbing about these standing waves. The resulting long-wave equations are deduced only explicitly dependent upon the macroscale, with the microscale represented by integral quantities. These equations accurately reproduce the behaviour of the Bloch mode spectrum near the edges of the Brillouin zone, hence yielding an explicit way for homogenizing periodic media in the vicinity of ‘cell resonances’. The similarity of such model equations to high-frequency long wavelength asymptotics, for homogeneous acoustic and elastic waveguides, valid in the vicinities of thickness resonances is emphasized. Several illustrative examples are considered and show the efficacy of the developed techniques.
Description: This article is available open access through the publisher’s website at the link below. Copyright @ 2010 The Royal Society.
ISSN: 1364-5021
Appears in Collections:Dept of Mathematics Research Papers
Mathematical Sciences

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