Radiation conditions for the linear water‐wave problem in periodic channels

Abstract

We study the well‐posedness of the linearized water‐wave problem in a periodic channel with fixed or freely floating compact bodies. Floquet–Bloch–Gelfand‐transform techniques lead to a generalized spectral problem with quadratic dependence on a complex parameter, and the asymptotics of the solutions at infinity can be described using Floquet waves. These are constructed from Jordan chains, which are related with the eigenvalues of the quadratic pencil and which are calculated in the paper in some typical cases. Posing proper radiation conditions requires a careful study of spaces of incoming and outgoing waves, especially in the threshold situation. This is done with the help of a certain skew‐Hermitian form q, which is closely related to the Umov–Poynting vector of energy transportation. Our radiation conditions make the problem operator into a Fredholm operator of index zero and provides natural (energy) classification of outgoing/incoming waves. They also lead to a novel, most natural properties and interpretation of the scattering matrix, which becomes unitary and symmetric even at threshold.