Uniqueness below a cut-off frequency for the two-dimensional linear water-wave problem

The two-dimensional velocity potential associated with the scattering of linear waves in water of finite but non-uniform depth is shown to be unique for Khmax ≤ 1. Here hmax is the maximum depth of the fluid and K = ω2/g, where ω is the angular frequency and g is the acceleration due to gravity. Uniqueness is established by proving that the homogeneous boundary value problem for the velocity potential φ has only the trivial solution φ = 0 when Khmax ≤ 1. It is first shown that there is a nodal line in the fluid on which φ = 0 and that this line has one end on the mean free surface and asymptotes to a horizontal line as x → ∞. The line forms the lower boundary of a fluid region contained between itself and the free surface. The potential in this region is shown to be identically equal to zero for Kdmax < 1, where dmax is the maximum depth of the layer, by bounding the potential energy in the region by a fraction of the kinetic energy. By analytic continuation the potential is zero in the whole fluid for this range of frequencies and as dmax < hmax, uniqueness is established for Khmax ≤ 1. The critical value of Kdmax = 1 corresponds to a cut-off frequency below which waves cannot propagate in a uniform layer of depth dmax with φ = 0 on the lower boundary. The analysis is extended to prove uniqueness for the same range of frequencies when any number of non-bulbous, surface-piercing bodies are in the fluid or a single surface-piercing or submerged body with arbitrary shape.