The Helmholtz Problem in Slowly Varying Waveguides at Locally Resonant Frequencies

36 Pages Posted: 4 Mar 2022

See all articles by Eric Bonnetier

Eric Bonnetier

affiliation not provided to SSRN

Angèle Niclas

Ecole Centrale de Lyon

Laurent Seppecher

Ecole Centrale de Lyon

Grégory Vial

Ecole Centrale de Lyon

Abstract

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 fre- quencies, where a phenomenon close to the tunnel effect for Schrödinger equa- tion in quantum mechanics can be observed. In this situation, locally resonant modes propagate in the waveguide under the form of Airy functions. Using pre- vious 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 solu- tion, as well as a control of the approximation error in H 1 loc . 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 ap- proximation by comparing it to numerical solutions based on the finite element method.

Keywords: Helmholtz equation, waveguide, resonances 2020 MSC: 78M35, 34E20, 35J05

Suggested Citation

Bonnetier, Eric and Niclas, Angèle and Seppecher, Laurent and Vial, Grégory, The Helmholtz Problem in Slowly Varying Waveguides at Locally Resonant Frequencies. Available at SSRN: https://ssrn.com/abstract=4049395 or http://dx.doi.org/10.2139/ssrn.4049395

Eric Bonnetier

affiliation not provided to SSRN ( email )

No Address Available

Angèle Niclas (Contact Author)

Ecole Centrale de Lyon ( email )

36 avenue Guy de Collongue
Ecully, 69134
France

Laurent Seppecher

Ecole Centrale de Lyon ( email )

36 avenue Guy de Collongue
Ecully, 69134
France

Grégory Vial

Ecole Centrale de Lyon ( email )

36 avenue Guy de Collongue
Ecully, 69134
France

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