It is well known, underwater ridges and submerged horizontal cylinders can serve as waveguides for surface water waves. For the wave number k in the direction of the ridge there is only one trapped wave. In this paper, using Green functions we obtain solutions to trapped water waves and the frequencies over an underwater ridge of small height and depth in the shallow water approximation by means of reducing the initial problem to an integral equation and then by applying the method of Neumann series, in order to solve it.

trapped water waves, asymptotics, eigenfunctions, Green function.