Radiation conditions at the top of a rotational cusp in the theory of water-waves

Abstract

We study the linearized water-wave problem in a bounded domain (e.g. a finite pond of water) of R 3 , having a cuspidal boundary irregularity created by a submerged body. In earlier publications the authors discovered that in this situation the spectrum of the problem may contain a continuous component in spite of the boundedness of the domain. Here, we proceed to impose and study radiation conditions at a point O of the water surface, where a submerged body touches the surface (see Fig. 1). The radiation conditions emerge from the requirement that the linear operator associated to the problem be Fredholm of index zero in relevant weighted function spaces with separated asymptotics. The classification of incoming and outgoing (seen from O) waves and the unitary scattering matrix are introduced. −Δxϕ(x )= f (x) ,x ∈ Ω, homogeneous Neumann conditions ∂ν ϕ(x )=0( ∂ν is the normal derivative) on the boundary except for the water surface Γ, where a Steklov type spectral condition ∂νϕ(x )= λϕ(x )i s posed withλ ∈ C as a spectral parameter. At O the two tangential boundary components create a thinning water blanket, which is cuspidal in the vertical direction. This geometry makes the asymptotic behaviour of the solutions ϕ as x →O quite complicated. We shall consider the above problem in the weak formulation, which transforms it into a more standard spectral problem of an unbounded operator in a Hilbert space. As a consequence of the geometry,