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

Summary 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 here by utilizing two mathematical methods assuming the linearised 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 potential across the plate. These unknown functions satisfy two integral equations. Any one of these, when solved numerically, can be used to compute the amplitude of the wave motion set up at either infinity on the two sides of the plate for various values of the wave number.In the second method, the problem is formulated in terms of a hypersingular integral equation involving the difference of the potential function 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. The results are illustrated graphically, and a comparison is made with the deep-water result. It is observed that the deep-water result effectively holds good if the plate is partially immersed to the order of one-tenth of the bottom depth.This work was initiated when the first Author was visiting Mathematics Department, Indian Institute of Science, Bangalore. It was partially supported by DST, and by CSIR. The authors take this opportunity to thank the Managing Editor for his suggestions to improve the paper in the present form.