ON WAVES DUE TO ROLLING OF A SHIP IN WATER OF FINITE DEPTH

The problem of the generation of waves due to small rolling oscillatio ns of a thin vertical plate partially immersed in uniform finite-depth water is investigated here by utilizing two mathematical methods assu ming the linearised theory of water waves. In the first method, the us e of eigenfunction expansion of the velocity potentials on the two sid es of the plate produces the amplitude of wave motion at infinity in t erms of an integral involving the unknown horizontal velocity across t he gap, and also in terms of another integral involving the unknown di fference of the potential across the plate. These unknown functions sa tisfy two integral equations. Any one of these, when solved numericall y, can be used to compute the amplitude of the wave motion set up at e ither 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 equati on is solved numerically, and its numerical solution is used to comput e the wave amplitude at infinity. The two methods produce almost the s ame 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 im mersed to the order of one-tenth of the bottom depth.