Solution of a singular integral equation and itsapplication to water wave problems

Abstract

In the present paper, the solution of a singular integral equation with logarithmic kernel in two disjoint intervals (0, a)4(b, infinity), (a, b are finite) is obtained by using function theoretic method. The two cases are considered when the unknown function satisfying the integral equation is unbounded or bounded at both nonzero finite end points of the interval. In the latter case, two solvability conditions are to be satisfied in order that the solution of the integral equation exists. We have used these two solvability conditions to evaluate the amplitude of waves at infinity for the three well-known water wave problems. These are: (i) scattering of water waves by a vertical plate submerged in deep water, (ii) generation of waves due to a line source in front of a vertical plate submerged in deep water, and (iii) generation of waves due to rolling of a submerged vertical plate.