Asymptotics of the frequency of a surface wave trapped by a slightly inclined barrier in a liquid layer

A two-dimensional statement of the scattering problem for an oblique incident surface wave by an obstacle in the form of a submerged barrier is considered. If the barrier is vertical, the discrete spectrum of the problem is shown to be empty, but for an inclined barrier an eigenvalue appears below the threshold of the continuous spectrum and the corresponding trapped mode decays exponentially in the direction perpendicular to the obstacle. The behaviour of the eigenvalue is analyzed for small values of the angle of inclination from the vertical.