SCATTERING OF WATER-WAVES BY A SUBMERGED THIN VERTICAL WALL WITH A GAP

A train of surface water waves normally incident on a thin vertical wa ll completely submerged in deep water and having a gap, experiences re flection by the wall and transmission through the gaps above and in th e wall. Using Havelock's expansion of water wave potential, two differ ent integral equation formulations of the problem are presented. While the first formulation involves multiple integral equations which are solved here by reducing them to a singular integral equation with Cauc hy kernel in a double interval, the second formulation involves a firs t-kind singular integral equation in a double interval with a combinat ion of logarithmic and Cauchy kernel, the solution of which is obtaine d by utilizing the solution of a singular integral equation with Cauch y kernel in (0, infinity) and also in a double interval. The reflectio n coefficient is evaluated by both the methods.