Acoustic waveguide with a dissipative inclusion

¹ Inria, Ensta Paris, Institut Polytechnique de Paris, 828 Boulevard des Maréchaux, 91762 Palaiseau, France
² Institut de Mathématiques de Toulouse CNRS UMR 5219, Université Paul Sabatier, F-31062 Toulouse Cedex 9, France
³ Institute of Problems of Mechanical Engineering RAS V.O., Bolshoi pr., 61, St. Petersburg 199178, Russia
⁴ Department of Mathematics and Statistics, University of Helsinki, P.O. Box 68, FI-00014 Helsinki, Finland

^* Corresponding author: lucas.chesnel@inria.fr

Received: 29 June 2022
Accepted: 13 August 2023

Abstract

We consider the propagation of acoustic waves in a waveguide containing a penetrable dissipative inclusion. We prove that as soon as the dissipation, characterized by some coefficient η, is non zero, the scattering solutions are uniquely defined. Additionally, we give an asymptotic expansion of the corresponding scattering matrix when η → 0⁺ (small dissipation) and when η → +∞ (large dissipation). Surprisingly, at the limit η → +∞, we show that no energy is absorbed by the inclusion. This is due to the so-called skin-effect phenomenon and can be explained by the fact that the field no longer penetrates into the highly dissipative inclusion. These results guarantee that in monomode regime, the amplitude of the reflection coefficient has a global minimum with respect to η. The situation where this minimum is zero, that is when the device acts as a perfect absorber, is particularly interesting for certain applications. However it does not happen in general. In this work, we show how to perturb the geometry of the waveguide to create 2D perfect absorbers in monomode regime. Asymptotic expansions are justified by error estimates and theoretical results are supported by numerical illustrations.