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

A channel of infinite length and depth and of constant width contains inviscid heavy fluid having free surface. The fluid is bounded internally by a submerged cylinder which spans the channel and has its generators normal to the sidewalls. The existence of trapped modes, i.e. states with finite energy corresponding to localized fluid oscillations, is well established in the linearised theory of water waves and the modes have been proven to occur at some frequencies for any geometry of the submerged cylinder. The purpose of this work is to find lower bounds for these trapped-mode frequencies. An integral identity suggested by Grimshaw (1974) and involving possible trapped-mode potential and a comparison function (trial potential) is applied. This identity yields uniqueness of the problem if the trial function has special properties. A number of the trial potentials is suggested possessing these properties in some sets of parameters of the problem. The potentials are constructed with the help of singular solutions: modified Bessel functions and Green’s function of the problem. Comparison of the bounds obtained and known examples of trapped modes is given.