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Existence of guided waves due to a lineic perturbation of a 3D periodic medium

Bérangère Delourme 1 Patrick Joly 2 Elizaveta Vasilevskaya 1
2 POEMS - Propagation des Ondes : Étude Mathématique et Simulation
Inria Saclay - Ile de France, UMA - Unité de Mathématiques Appliquées, CNRS - Centre National de la Recherche Scientifique : UMR7231
Abstract : In this note, we exhibit a three dimensional structure that permits to guide waves. This structure is obtained by a geometrical perturbation of a 3D periodic domain that consists of a three dimensional grating of equi-spaced thin pipes oriented along three orthogonal directions. Homogeneous Neumann boundary conditions are imposed on the boundary of the domain. The diameter of the section of the pipes, of order ε > 0, is supposed to be small. We prove that, for ε small enough, shrinking the section of one line of the grating by a factor of √ µ (0 < µ < 1) creates guided modes that propagate along the perturbed line. Our result relies on the asymptotic analysis (with respect to ε) of the spectrum of the Laplace-Neumann operator in this structure. Indeed, as ε tends to 0, the domain tends to a periodic graph, and the spectrum of the associated limit operator can be computed explicitly.
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Submitted on : Thursday, December 8, 2016 - 12:46:52 PM
Last modification on : Friday, December 3, 2021 - 11:34:05 AM
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Bérangère Delourme, Patrick Joly, Elizaveta Vasilevskaya. Existence of guided waves due to a lineic perturbation of a 3D periodic medium. Applied Mathematics Letters, Elsevier, 2016, ⟨10.1016/j.aml.2016.11.017⟩. ⟨hal-01412396⟩



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