Diffraction by a defect in an open waveguide: A Mathematical analysis based on a modal radiation condition

We consider the scattering of a time-harmonic acoustic wave by a defect in a twodimensional open waveguide. The scattered wave satisfies the Helmholtz equation in a perturbed layered half-plane. We introduce a modal radiation condition based on a generalized Fourier transform which diagonalizes the transverse contribution of the Helmholtz operator. The uniqueness of the solution is proved by an original technique which combines a property of the energy flux with an argument of analyticity with respect to the generalized Fourier variable. The existence is then deduced classically from Fredholm's alternative by reformulating the scattering problem as a Lippmann-Schwinger equation by means of the Green's function for the layered half-plane.