  We study the problem of propagation of linear water waves in a deep water in
the presence of a critically submerged body (i.e. the body touching the water
surface). Assuming uniqueness of the solution in the energy space, we prove the
existence of the solution which satisfies the radiation conditions at infinity
as well as, additionally, at the cusp point where the body touches the water
surface. This solution is obtained by the limiting absorption procedure.
  Next we introduce a relevant scattering matrix and analyse its properties.
Under a geometric condition introduced by Maz'ya, see \cite{M1}, we show that
the method of multipliers applies to cusp singularities, thus proving a new
important property of the scattering matrix, which may be interpreted as the
absence of a version of "full internal reflection". This property also allows
us to prove uniqueness and existence of the solution in the functional spaces
$H^2_{loc}\cap L^\infty $ and $H^2_{loc}\cap L^p $, $2<p<6$, provided a
spectral parameter in the boundary conditions on the surface of the water is
large enough.
  This description of the solution does not rely on the radiation conditions or
the limiting absorption principle. This is the first result of this type known
to us in the theory of linear wave problems in unbounded domains.
