Linear long wave propagation over discontinuous submerged shallow water topography

The dynamics of an isolated long wave passing over underwater obstacles are discussed in this paper within the framework of linear shallow water theory. Areas of practical application include coastal defense against tsunami inundation, harbor protection and erosion prevention with submerged breakwaters, and the construction and design of artificial reefs to use for recreational surfing. Three sea-floor configurations are considered: an underwater shelf, a flat sea-floor with a single obstacle, and a series of obstacles. A piecewise continuous coefficient is used to model the various sea-floor topographies. A simple and easily implementable numerical scheme using explicit finite difference methods is developed to solve the discontinuous partial differential equations. The numerical solutions are verified with the exact analytical solutions of linear wave propagation over an underwater shelf. The scope of this simplified approach is determined by comparison of its results to those of another numerical solution and wave transmission and reflection coefficients from experimental data available in the literature. The efficacy of approximating more complicated continuous underwater topographies by piecewise constant distributions is determined. As an application, a series of underwater obstacles is implemented.