Trapping of water waves by freely floating structures in a channel

Abstract

This article is concerned with the existence of rigid freely floating structures capable of supporting trapped modes (time-harmonic water waves of finite energy in an unbounded domain). Under the usual assumptions of linear water-wave theory, a condition guaranteeing the existence of trapped modes is derived, and structures satisfying this geometric condition are shown to exist in a three-dimensional water channel. The sufficient condition arises from the application of variational principles to a conveniently formulated linear spectral problem, the main effort being the construction of a reduction scheme that turns the quadnic operator pencil associated with the original coupled system into a linear self-adjoint spectral problem. An example of floating bodies supporting at least four trapped modes is given.