  We show existence and uniqueness for a linearized water wave problem in a two
dimensional domain $G$ with corner, formed by two semi-axis $\Gamma_1$ and
$\Gamma_2$ which intersect under an angle $\alpha\in (0,\pi ]$. The existence
and uniqueness of the solution is proved by considering an auxiliary mixed
problem with Dirichlet and Neumann boundary conditions. The latter guarantees
the existence of the Dirichlet to Neumann map. The water wave boundary value
problem is then shown to be equivalent to an equation like $v_{tt}+g\Lambda
v=P$ with initial conditions, where $t$ stands for time, $g$ is the
gravitational constant, $P$ means pressure, and $\Lambda$ is the Dirichlet to
Neumann map. We then prove that $\Lambda$ is a positive self-adjoint operator.
