From zero transmission to trapped modes in waveguides

Lucas Chesnel (1) , Vincent Pagneux (2)

1 CMAP - Centre de Mathématiques Appliquées - Ecole Polytechnique
2 LAUM - Laboratoire d'Acoustique de l'Université du Mans

We consider the time-harmonic scattering wave problem in a 2D waveguide at wavenumber k such that one mode is propagating in the far field. In a first step, for a given k, playing with one scattering branch of finite length, we demonstrate how to construct geometries with zero transmission. The main novelty in this result is that the symmetry of the geometry is not needed: the proof relies on the unitary structure of the scattering matrix. Then, in a second step, from a waveguide with zero transmission, we show how to build geometries supporting trapped modes associated with eigenvalues embedded in the continuous spectrum. For this second construction, using the augmented scattering matrix and its unitarity, we play both with the geometry and the wavenumber. Finally, the mathematical analysis is supplemented by numerical illustrations of the results.

Mots clés: Waveguides; zero transmission; trapped modes; scattering matrix; asymptotic analysis

Domaines: Equations aux dérivées partielles [math.AP]

Lucas Chesnel, Vincent Pagneux. From zero transmission to trapped modes in waveguides. Journal of Physics A: Mathematical and Theoretical, 2019. ⟨hal-01952980v2⟩