Outgoing Conditions Research Articles

The Helmholtz problem in slowly varying waveguides at locally resonant frequencies

This article aims to present a general study of the Helmholtz problem in slowly varying waveguides. This work is of particular interest at locally resonant frequencies, where a phenomenon close to the tunnel effect for Schrödinger equation in quantum mechanics can be observed. In this situation, locally resonant modes propagate in the waveguide under the form of Airy functions. Using previous mathematical results on the Schrödinger equation, we prove the existence of a unique solution to the Helmholtz source problem with outgoing conditions in such waveguides. We provide an explicit modal approximation of this solution, as well as a control of the approximation error in Hloc1. The main theorem is proved in the case of a waveguide with a monotonously varying profile and then generalized using a matching strategy. We finally validate the modal approximation by comparing it to numerical solutions based on the finite element method.

Cloaking via change of variables for the Helmholtz equation in the whole space

AbstractThis paper is devoted to the study of a cloaking device that is composed of a standard near cloak based on a regularization of the transformation optics, i.e., a change of variables that blows up a small ball to the cloaked region, and a fixed lossy layer for the Helmholtz equation in the whole space of dimension 2 or 3 with the outgoing condition at infinity. We establish a degree of near invisibility, which is independent of the content inside the cloaked region, for this device. We also show that the lossy layer is necessary to ensure the validity of the degree of near invisibility when no constraint on physical properties inside the cloaked region is imposed. © 2010 Wiley Periodicals, Inc.

Diffraction d'une onde électromagnétique par un dièdre infini absorbant

We study the diffraction of electromagnetic waves by an infinite wedge of dielectric material. For this aim we consider the stationary coupled vacuum-dielectric Maxwell equations with an outgoing condition in the vacuum. We show the equivalence of the latter problem to a Caldéron boundary operator system for particular classes of incoming data. We study the solutions of this system in the case of traces of monochromatic plane waves. In particular, we give the asymptotics of the diffracted signal in high frequency regime at a given point with fixed distance to the boundary and away from some incident directions. To cite this article: J.-M. Caron, C. R. Acad. Sci. Paris, Ser. I 337 (2003).