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. 1. Introduction. A uniform waveguide may be defined as a propagative medium whose physical features are invariant in one longitudinal direction so that waves can propagate in this direction and remain confined to a limited region in the orthogonal transverse direction(s). Such a waveguide is said to be open when the cross-section is unbounded; the confinement is then due to a particular arrangement of the inho-mogeneities which allows an evanescent behavior of guided waves in the transverse direction(s). In the present paper, we are concerned with the modeling and mathematical analysis of the junction of two different open waveguides, which covers many physical applications in areas such as electromagnetism (junction of optical fibers, or between a fiber and an integrated optical device), acoustics (immersed junction of pipelines), elastodynamics (seismic waves in two layered media separated by a rift), and hydrodynamics (water waves guided by a varying cross-section ocean trench). Figure 1.1 illustrates two examples of junctions which will be considered in the paper, called abrupt and thick, depending on whether the part which contains the variable cross-section has a zero thickness or not. The physical problem we are interested in can be expressed in a very simple way: consider an incident guided wave on one side of the junction; what happens to this wave when it meets the junction? It seems clear that the interaction of this incident wave with the junction will produce three kinds of waves: these are a reflected guided wave which …