Solution of the problem of propagation of water waves over a pair of asymmetrical rectangular trenches

Abstract

The problem of propagation of obliquely incident surface water waves over a pair of asymmetrical rectangular trenches in a channel of finite depth is examined for its solution. The mathematical problem is handled for its numerical solution with the aid of a system of singular integral equations of first kind. The resulting integral equations are solved numerically by using suitably designed polynomial approximations of the unknown functions. System of linear algebraic equations is obtained by utilizing the zeros of Chebyshev polynomial of first kind as collocation points and hence the occurrence of ill-conditioned matrix is avoided. The effectiveness of the pair of trenches is studied by analyzing the physical quantities namely the reflection and transmission coefficients. As a special limiting case, the results for a single trench problem are derived and are found to be in excellent agreement with the results available in the literature. The effect of various parameters are analyzed through different graphs for a pair of asymmetrical trenches. The energy balance relation is derived and used to check the accuracy of numerical results.