Propagation of obliquely incident water waves over a trench

The diffraction of obliquely incident surface waves by an asymmetric trench is investigated using linearized potential theory. A numerical solution is constructed by matching particular solutions for each subregion of constant depth along vertical boundaries ; the resulting matrix equation is solved numerically. Several cases where the trench-parallel wavenumber component in the incident-wave region exceeds the wavenumber for freely propagating waves in the trench are investigated and are found to result in large reductions in wave transmission ; however, reflection is not total owing to the finiteness of the obstacle. Results for one case are compared with data obtained from a small-scale wave-tank experiment. An approximate solution based on plane-wave modes is derived and compared with the numerical solution and, in the long-wave limit, with a previous analytic solution. 1. Introduction The problem of the diffraction of incident waves by a finite obstacle in an otherwise infinite and uniform domain remains of general interest in linear wave theory. Several geometries of interest can be schematized by domains divided into separate regions by vertical geometrical boundaries, with the fluid depth being constant in each subdomain. Representative two-dimensional problems, with the boundary geometry uniform in the direction normal to the plane of interest, include those of elevated rectangular sills, fixed or floating rectangular obstacles at the water surface, and submerged trenches. The approach to the solution of problems of this type has typically been to construct solutions for each constant-depth subdomain in terms of eigenfunction expansions of the velocity potential ; the solutions are then matched at the vertical boundaries, resulting in sets of linear integral equations which must be truncated to a finite number of terms and solved numerically. One of the earliest solutions of this type was given by Takano (1960), who studied the cases of normal wave incidence on an elevated sill and fixed obstacle at the surface. In this study, we employ a modification of Takano’s method, discussed in $3. Newman (19653) also employed an integral-equation approach to study reflection and transmission of waves normally incident on a single step between finite- and infinite-depth regions. A variational approach, developed by Schwinger to study discontinuitiesin waveguides (see Schwinger & Saxon 1968) has been used by Miles (1967), to study Newman’s single-step problem, and by Mei & Black (1969), who studied the symmetric elevated sill and a floating rectangular cylinder. Lassiter (1 972), using the variational approach, studied waves normally incident on a rectangular trench where the water depths before and after the trench are constant but not necessarily equal, referred to here as the asymmetric case. Lee &