Linear Algebraic Method of Solution for the Problem of Mitigation of Wave Energy Near Seashore by Trench-Type Bottom Topography

Abstract

The problem involving mitigation of wave energy by trench-type structure, in particular, a pair of trenches imposed on the seabed in the absence and presence of a vertical wall, is examined for its solution. The problem under consideration leads to multiple series relations involving trigonometric functions. Instead of converting these relations into a system of integral equations, direct algebraic approaches are utilized for solving the reduced system of overdetermined algebraic equations and the corresponding solutions are obtained approximately. Here, the overdetermined system of algebraic equations are solved with the aid of the well-known least-squares (LS) and singular value decomposition (SVD) methods. Results involving the hydrodynamic quantities such as reflection and transmission coefficients related to the single trench problem are derived and are found to be in excellent agreement with the results available in the literature. The present algebraic methods appear to be very direct and quick. The energy balance relation for the given scattering problem is derived and used to check the accuracy of numerical results. Some important results such as the behavior of singularity in flow near each edge of the trenches, surface elevation profiles, and force experienced by the wall are investigated and analyzed through graphs to analyze the transformation of wave energy by a pair of trenches imposed on a seabed. It is observed that creation of a pair of trenches imposed on a seabed helps to reduce the wave load on the wall; consequently, the wall as well as the seashore (in absence of a wall) is protected.