Localization of surface waves by small perturbations of the boundary of a semisubmerged body

It is shown under the condition of symmetry that, by means of formation of a thin groove on the planar surface of a body parallel to the liquid's horizon in a cylindrical channel, we can achieve the following effect in the linear problem concerning the waves on water: on every arbitrarily short interval $(0,d)$ of the continuous spectrum, any prescribed number of the eigenvalues is formed giving rise to “localized” solutions, i.e., belonging to a Sobolev space.