A Dispersive Effective Medium for Wave Propagation in Periodic Composites

Santosa, Fadil; Symes, William W.

doi:10.1137/0151049

Cited by 196 publications

(206 citation statements)

References 10 publications

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“…The fact that the homogenized equations for the envelope v + and v − are Schrödinger equations is a confirmation of the dispersive properties (i.e. the nonlinear character of the effective dispersion relation) of periodic composite materials as already studied in [2,3,13,28].…”

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confidence: 90%

Diffractive behavior of the wave equation in periodic media: weak convergence analysis

Allaire

Palombaro

Rauch

2008

Annali di Matematica

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We study the homogenization and singular perturbation of the wave equation in a periodic media for long times of the order of the inverse of the period. We consider initial data that are Bloch wave packets, i.e., that are the product of a fast oscillating Bloch wave and of a smooth envelope function. We prove that the solution is approximately equal to two waves propagating in opposite directions at a high group velocity with envelope functions which obey a Schrödinger type equation. Our analysis extends the usual WKB approximation by adding a dispersive, or diffractive, effect due to the non uniformity of the group velocity which yields the dispersion tensor of the homogenized Schrödinger equation.

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confidence: 90%

Diffractive behavior of the wave equation in periodic media: weak convergence analysis

Allaire

Palombaro

Rauch

2008

Annali di Matematica

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“…Parnell & Abrahams (2006) or Andrianov et al (2008). While the frequency range of such models can be extended by considering higher order correction terms (Santosa & Symes 1991;Bakhvalov & Eglit 2000;Smyshlyaev & Cherednichenko 2000), the resulting models cannot fully reproduce high-frequency dynamic behaviours characteristic of microstructured materials, such as strong dispersion, the presence of band gaps or negative refraction. The other way to describe this limitation of the traditional homogenization theory would be to say that it is only capable of describing the fundamental Bloch mode at low frequencies.…”

Section: Introductionmentioning

confidence: 99%

High-frequency homogenization for periodic media

2010

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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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“…The diffractive effect comes from the nonlinear character of the effective dispersion relation of periodic materials (see [13], [31] for other instances of this effect). The same is also true of the parabolic or paraxial approximation for waves propagating in a privileged direction (see [6], [28], [34]).…”

Section: Introductionmentioning

confidence: 99%

Diffractive Geometric Optics for Bloch Wave Packets

Allaire

Palombaro

Rauch

2011

Arch Rational Mech Anal

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Abstract. We study, for times of order 1/ε, solutions of wave equations which are O(ε 2 ) modulations of an ε periodic wave equation. The solutions are of slowly varying amplitude type built on Bloch plane waves with wavelength of order ε. We construct accurate approximate solutions of three scale WKB type. The leading profile is both transported at the group velocity and dispersed by a Schrödinger equation given by the quadratic approximation of the Bloch dispersion relation at the plane wave. A ray average hypothesis of small divisor type guarantees stability. We introduce techniques related to those developed in nonlinear geometric optics which lead to new results even on times scales t = O(1). A pair of asymptotic solutions yield accurate approximate solutions of oscillatory initial value problems. The leading term yields H 1 asymptotics when the envelopes are only H 1 .

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A Dispersive Effective Medium for Wave Propagation in Periodic Composites

Cited by 196 publications

References 10 publications

Diffractive behavior of the wave equation in periodic media: weak convergence analysis

Diffractive behavior of the wave equation in periodic media: weak convergence analysis

High-frequency homogenization for periodic media

Diffractive Geometric Optics for Bloch Wave Packets

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