On the absence of trapped water waves near a cliffed cape

Abstract

The water wave problem is considered for a class of semi-infinite domains each having its shore shaped as a cliffed cape. In particular, the free surface of a water domain is supposed to be an infinite sector whose vertex angle is greater than $\pi$, whereas the water layer lying under the free surface is of constant depth. Under these assumptions, it is shown that there are no trapped mode solutions of the problem for all values of a non-dimensional spectral parameter; in other words, no point eigenvalues are embedded in the problem's continuous spectrum.