On frequency bounds for modes trapped near a channel-spanning cylinder

A channel of infinite length and depth and of constant width contains invis cid heavy fluid having a free surface. The fluid is bounded internally by a submerged cylinder which spans the channel and has its generators normal t o the sidewalls. The existence of trapped modes, i.e. states with finite en ergy corresponding to localized fluid oscillations, is well established in the linearized theory of water waves, and the modes have been proven to occ ur at some frequencies for any geometry of the submerged cylinder. The purp ose of this work is to find lower bounds for these trapped-mode frequencies . An integral identity suggested by Grimshaw in 1974 is applied to a possib le trapped-mode potential and a comparison, or trial, function. This identi ty yields the uniqueness of the problem if the trial function has special p roperties. A number of trial functions possessing these properties are sugg ested for some sets of parameters of the problem. The potentials are constr ucted with the help of singular solutions, namely modified Bessel functions and Green's function of the problem. A comparison is given between the bou nds obtained here and known bounds and examples of trapped modes.