Helmholtz Equation with Artificial Boundary Conditions in a Two-Dimensional Waveguide

Abstract

We consider a time-harmonic acoustic wave propagation problem in a two-dimensional water waveguide confined between a horizontal surface and a locally varying bottom. We formulate a model based on the Helmholtz equation coupled with nonlocal Dirichlet-to-Neumann boundary conditions imposed on two artificial boundaries. We establish the well-posedness of the associated variational problem, under the assumption of a downsloping bottom, by showing stability estimates in appropriate function spaces. The outcome of some numerical experiments with a code implementing a standard/Galerkin finite element approximation of the variational formulation of the model are also presented.