Mathematical analysis of guided water waves

In this article, the authors consider the problem of three-dimensional linear water waves that are guided along the coast. The problem is reduced to a two-dimensional eigenvalue problem for a family of unbounded selfadjoint operators with noncompact resolvent. Questions are investigated concerning the existence, number, and properties of the guided modes, with particular attention given to low-and high-frequency behavior. The authors point out a large variety of results that greatly depend on the geometry of the bottom relief.