Diffractive Geometric Optics for Bloch Wave Packets

Allaire, Grégoire; Palombaro, Mariapia; Rauch, Jeffrey

doi:10.1007/s00205-011-0452-9

Cited by 30 publications

(46 citation statements)

References 32 publications

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“…These assumptions are also needed in our analysis to ensure unique solvability of the Bloch equation. The work of Allaire et al [29] goes further than us in that they prove the enveloping function converges to that predicted by the two-scale analysis, and that they prove the enveloping function obeys a Schrödinger type equation as expected from the paraxial approximation. At the level of our analysis the dispersion of the enveloping function is absent, so further work needs be done to account for it.…”

Section: Introductionmentioning

confidence: 80%

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High-frequency homogenization for travelling waves in periodic media

Harutyunyan

Milton

2016

Proc. R. Soc. A.

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We consider high-frequency homogenization in periodic media for travelling waves of several different equations: the wave equation for scalar-valued waves such as acoustics; the wave equation for vector-valued waves such as electromagnetism and elasticity; and a system that encompasses the Schrödinger equation. This homogenization applies when the wavelength is of the order of the size of the medium periodicity cell. The travelling wave is assumed to be the sum of two waves: a modulated Bloch carrier wave having crystal wavevector [Formula: see text] and frequency ω 1 plus a modulated Bloch carrier wave having crystal wavevector [Formula: see text] and frequency ω 2. We derive effective equations for the modulating functions, and then prove that there is no coupling in the effective equations between the two different waves both in the scalar and the system cases. To be precise, we prove that there is no coupling unless ω 1=ω 2 and [Formula: see text] where Λ=(λ1λ2…λ d ) is the periodicity cell of the medium and for any two vectors [Formula: see text] the product a⊙b is defined to be the vector (a 1 b 1,a 2 b 2,…,a d b d ). This last condition forces the carrier waves to be equivalent Bloch waves meaning that the coupling constants in the system of effective equations vanish. We use two-scale analysis and some new weak-convergence type lemmas. The analysis is not at the same level of rigour as that of Allaire and co-workers who use two-scale convergence theory to treat the problem, but has the advantage of simplicity which will allow it to be easily extended to the case where there is degeneracy of the Bloch eigenvalue

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Section: Introductionmentioning

confidence: 80%

“…As mentioned in the introduction this effective equation fails to capture dispersion which is captured in the approach of Allaire et al [29].…”

Section: Simplifying the Effective Equationmentioning

confidence: 99%

High-frequency homogenization for travelling waves in periodic media

Harutyunyan

Milton

2016

Proc. R. Soc. A.

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“…The most interesting effects occur when one has a saddle point: then the effective equation is hyperbolic and there are associated characteristic directions. One may also employ homogenization techniques for travelling waves at other points in the dispersion diagram [22][23][24][25][26].…”

Section: To Ananya and Krishna To Alaa And Ismailmentioning

confidence: 99%

Front Matter

2017

Dynamics of Lattice Materials

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“…Bloch decomposition, a spectral decomposition for differential operators with periodic coefficients, is a classical tool to study wave propagation in periodic media. For a relatively recent example, Allaire et al [1] considered the problem of propagation of waves packets through a periodic medium, where the period is assumed small compared to the size of the envelope of the wave packet. In this work the authors construct solutions built upon Bloch plane waves having a slowly varying amplitude.…”

Section: Introductionmentioning

confidence: 99%

Bloch Theory and Spectral Gaps for Linearized Water Waves

Craig¹,

Gazeau²,

Lacave³

et al. 2018

SIAM J. Math. Anal.

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The system of equations for water waves, when linearized about equilibrium of a fluid body with a varying bottom boundary, is described by a spectral problem for the Dirichlet -Neumann operator of the unperturbed free surface. This spectral problem is fundamental in questions of stability, as well as to the perturbation theory of evolution of the free surface in such settings. In addition, the Dirichlet -Neumann operator is self-adjoint when given an appropriate definition and domain, and it is a novel but very natural spectral problem for a nonlocal operator. In the case in which the bottom boundary varies periodically,this spectral problem admits a Bloch decomposition in terms of spectral band functions and their associated band-parametrized eigenfunctions. In this article we describe this analytic construction in the case of a spatially periodic bottom variation from constant depth in two space dimensional water waves problem, giving a description of the Dirichlet -Neumann operator in terms of the bathymetry b(x) and a construction of the Bloch eigenfunctions and eigenvalues as a function of the band parameters. One of the consequences of this description is that the spectrum consists of a series of bands separated by spectral gaps which are zones of forbidden energies. For a given generic periodic bottom profile b(x) = εβ(x), every gap opens for a sufficiently small value of the perturbation parameter ε.

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Diffractive Geometric Optics for Bloch Wave Packets

Cited by 30 publications

References 32 publications

High-frequency homogenization for travelling waves in periodic media

High-frequency homogenization for travelling waves in periodic media

Front Matter

Bloch Theory and Spectral Gaps for Linearized Water Waves

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