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

Bérangère Delourme (1) , Patrick Joly (2) , Elizaveta Vasilevskaya (1)

1 LAGA - Laboratoire Analyse, Géométrie et Applications
2 POEMS - Propagation des Ondes : Étude Mathématique et Simulation

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.

Mots clés: guided waves; periodic media; spectral theory AMS codes : 78M35; 35J05; 58C40

Domaines: Analyse numérique [math.NA]; Théorie spectrale [math.SP]

Bérangère Delourme, Patrick Joly, Elizaveta Vasilevskaya. Existence of guided waves due to a lineic perturbation of a 3D periodic medium. Applied Mathematics Letters, 2016, ⟨10.1016/j.aml.2016.11.017⟩. ⟨hal-01412396⟩