Smoothing of Stokes' discontinuity for the generalized Bessel function

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Abstract
A channel of infinite length and depth and of constant width contains inviscid heavy fluid having a 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 linearized 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 in 1974 is applied to a possible trapped‐mode potential and a comparison, or trial, function. This identity yields the uniqueness of the problem if the trial function has special properties. A number of trial functions possessing these properties are suggested for some sets of parameters of the problem. The potentials are constructed with the help of singular solutions, namely modified Bessel functions and Green's function of the problem. A comparison is given between the bounds obtained here and known bounds and examples of trapped modes.

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