On unique solvability and regularity in the linearized two-dimensional wave resistance problem

We discuss existence, uniqueness, and regularity of the solutions of a boundary value problem in a strip, which is obtained by linearization of the equations of the wave-resistance problem for a cylinder semisubmerged in a heavy fluid of constant depth $H$ and moving at uniform velocity $c$ in the direction orthogonal to its generators. We show that the problem has a unique solution, rapidly decreasing at infinity, for every $c > \sqrt {gH}$, where $g$ is the acceleration of gravity. For $c < \sqrt {gH}$, we prove unique solvability provided $c \ne {c_k}$, where ${c_k}$ is a known sequence monotonically decreasing to zero. In this case, the related flow has in general nontrivial oscillations at infinity downstream.