On the solvability of a two-dimensional water-wave radiation problem

The existence of a unique weak solution for the two-dimensional water-wave radiation problem arising when a floating rigid body oscillates on the free surface is established for all but a discrete set of oscillation frequencies. The body boundary is taken to be of C ∗ 1 , α C_ * 1,α class (see Sec. 2) and the body boundary condition is satisfied in the L 2 {L^2} -sense. The proof relies on an expansion theorem (Athanassoulis [1]) and on the property of the associated water-wave multipoles being a Riesz basis of L 2 ( − π , 0 ) {L^2} ( { - π ,0} ) , a fact which is established in the present paper. Under stronger geometrical restrictions on the body boundary it is proved, using a method due to Ursell [10], that the weak solution is actually a classical one; that is, the velocity field is continuous up to ana including the body boundary.