An algebraic method of solution of a water wave scattering problem involving an asymmetrical trench

Abstract

The problem of propagation of obliquely incident surface water waves involving an asymmetric rectangular trench in a channel of finite depth is examined for its solution. The problem under consideration leads to multiple series relations involving trigonometric functions. Instead of converting these relations into systems of integral equations, direct algebraic approaches are utilized and the corresponding solutions are obtained approximately, by way of solving the reduced system of over-determined algebraic equations. The system of over-determined algebraic equations are solved with the aid of the well-known algebraic approaches “Least square” (LS) and “Singular value decomposition” (SVD) methods. The presently determined results involving the reflection and transmission coefficients related to the scattering problem under consideration, agree very well, with the known results obtained recently, where integral equations and their approximate numerical solutions were determined. The effect of system of parameters on hydrodynamics quantities such as reflection and transmission coefficients, wave elevation are analysed and shown through tables and graphs. The present algebraic methods appear to be very direct and quick. A similar techniques are found to be applicable to the problem of scattering of surface water waves by any finite number of asymmetrical placed rectangular trenches. The problem of a pair of asymmetrical trenches is under progress. The energy balance relation for the given problem is derived and used to check the accuracy of numerical results. Some important results such as wave elevation profiles, the singularity behaviour of the flow near trench edges are investigated and analyzed through graphs.