On waves due to rolling of a ship in water of finite depth (Q1368278)

The problem of the generation of waves due to small rolling oscillations of a thin vertical plate partially immersed in uniform finite-depth water is investigated by utilizing two mathematical methods assuming the linearized theory of water waves. In the first method, the use of eigenfunction expansion of the velocity potentials on the two sides of the plate produces the amplitude of wave motion at infinity in terms of an integral involving the unknown horizontal velocity across the gap, and also in terms of another integral involving the unknown difference of the potentials across the plate. In the second method, the problem is formulated in terms of a hypersingular integral equation involving the difference of the potential functions across the plate. The hypersingular integral equation is solved numerically, and its numerical solution is used to compute the wave amplitude at infinity. The two methods produce almost the same numerical results.