Mathematical analysis of the junction of two acoustic open waveguides

The present paper concerns the scattering of a time-harmonic acoustic wave by the junction of two open uniform waveguides, where the junction is limited to a bounded region. We consider a two-dimensional problem for which wave propagation is described by the scalar Helmholtz equation. The main difficulty in the modeling of the scattering problem lies in the choice of conditions which characterize the outgoing behavior of a scattered wave. We use here modal radiation conditions which extend the classical conditions used for closed waveguides. They are based on the generalized Fourier transforms which diagonalize the transverse contributions of the Helmholtz operator on both sides of the junction. We prove the existence and uniqueness of the solution, which seems to be the first result in this context. The originality lies in the proof of uniqueness, which combines a natural property related to energy fluxes with an argument of analyticity with respect to the generalized Fourier variable.