Trapping of Surface Water Waves by Fixed Bodies in a Channel

Solutions are constructed by the method of matched asymptotic expansions for surface water waves trapped by fixed bodies in a deep channel with vertical walls. In particular, approximate relations between the frequency of trapped waves and the channel width are obtained in terms of simple properties of the bodies. A number of geometries are considered. First, the result of Ursell for a submerged, horizontal, circular cylinder spanning the channel is generalized to cylinders symmetric about a vertical plane perpendicular to the channel walls, but of otherwise arbitrary cross-section. Secondly, both submerged and surface-piercing compact, three-dimensional bodies with appropriate symmetries are considered. Finally, a solution is given for a vertical cylinder extending throughout the channel depth which is equivalent to a two-dimensional problem in acoustics.