The Helmholtz Problem in Slowly Varying Waveguides at Locally Resonant Frequencies

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