It was pointed out in the Preface that methods of investigation of the uniqueness and solvability for the water-wave problem depend essentially on the type of obstacle in respect to its intersection with the free surface. Among various possibilities, the simplest one is the case in which the free surface coincides with the whole horizontal plane (and so rigid boundaries of the water domain are represented by totally submerged bodies and the bottom of variable topography); we restrict our attention to this case in the present chapter.

We begin with the method of integral equations (Section 2.1), which not only provides information about the unique solvability of the water-wave problem but also serves as one of the most frequently used tools for a numerical solution of the problem. In Section 2.2, various geometric criteria of uniqueness are obtained with the help of auxiliary integral identities. The uniqueness theorem established allows us to prove the solvability of the problem for various geometries of submerged obstacles in Section 2.3. The last section, Section 2.4, contains bibliographical notes.

Method of Integral Equations and Kochin's Theorem

When Green's function is constructed it is natural to solve the water-wave problem by applying integral equation techniques, which is a standard approach to boundary value problems. In doing so, a proof of the solvability theorem for an integral equation is usually based on Fredholm's alternative and the uniqueness of the solution to the boundary value problem.