It was demonstrated in Section 3.1 that in the presence of a surface-piercing obstacle the water-wave problem is solvable for an arbitrary right-hand-side term in the Neumann condition on the obstacle's surface. However, there is an uncertainty about the set of frequencies providing the solvability. According to the proof given in Subsection 3.1.1, a sequence νn → ∞ (n = 1, 2, …) possibly exists such that for these exceptional values the solvability could be violated for some data given on the obstacle's surface. In particular, this must occur for values νn that are point eigenvalues of the water-wave problem embedded in the continuous spectrum (the latter is known to be the whole positive half-axis as is shown in the Examples section of the Introduction). If a value of the spectral parameter ν belongs to the point spectrum, then the homogeneous problem possesses a nontrivial solution with finite energy, or in other words, there is no uniqueness of solution for the nonhomogeneous problem.

In this chapter (see Section 4.1), we give examples of such non-uniqueness for the two-dimensional and axisymmetric problems, and so the exceptional values of ν do exist at least for some obstacle geometries. Moreover, for every ν > 0 a certain family of obstacles exhibiting the non-uniqueness property can be obtained. An essential point in all these examples is the presence of an isolated portion of the free surface inside the obstacle where the eigenmode waves are trapped.