On adiabatic normal modes in a wedge shaped sea

We study a two-dimensional problem that is a model for sound propagation in a narrow water wedge near the shore of a sea. We explicitly construct a solution to the Helmholtz equation that is asymptotically a normal wave propagating along “water” wedge to the “shore”. The solution satisfies the Helmholtz equation in the quadrant one side of which is “the surface of the water”, and the second is perpendicular to it, starts at the top of the wedge and goes into the “bottom”. Boundary conditions at wedge boundaries and at infinity in the “bottom” are satisfied.