Journal
PROCEEDINGS OF THE ROYAL SOCIETY A-MATHEMATICAL PHYSICAL AND ENGINEERING SCIENCES
Volume 466, Issue 2120, Pages 2341-2362Publisher
ROYAL SOC
DOI: 10.1098/rspa.2009.0612
Keywords
Floquet-Bloch waves; stop bands; photonics; high-frequency long waves; homogenization
Categories
Funding
- NSERC (Canada)
- EPSRC (UK) [EP/H021302]
- University of Alberta
- EPSRC [EP/H021302/1] Funding Source: UKRI
- Engineering and Physical Sciences Research Council [EP/H021302/1] Funding Source: researchfish
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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.
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