Hypersingular Integral Equation Formulation of the Problem of Water Wave Scattering by A Circular Arc Shaped Impermeable Barrier Submerged in Water of Finite Depth

Abstract

Summary In this article, we study the problem of scattering of water waves by a thin impermeable circular arc shaped barrier submerged in ocean of finite depth under the assumption of linearised theory of water waves. The problem is formulated in terms of a hypersingular integral equation of an unknown function representing the difference of potential function across the curved barrier. The hypersingular integral equation is then solved by using two numerical methods. The first method is BEM where the domain and range of integral equation are discretised into small line segments and the unknown function satisfying the integral equation is assumed to be constant in each small subinterval. This reduces the integral equations to a system of algebraic equations which is then solved to obtain the unknown function in each sub-interval. The second method is collocation method where the unknown function is expanded in terms of Chebyshev polynomials of second kind. Choosing the collocation points suitably, the integral equation is reduced to a system of algebraic equations which is then solved to obtain the unknown function satisfying the hypersingular integral equation. The physical quantities of interest viz, the reflection coefficient, transmission coefficients, which are expressed in terms of the solution of the hypersingular integral equation, are computed by both the methods. The comparison of the reflection coefficient by the two methods shows reasonably good agreement. The reflection coefficient is depicted graphically against the wave number. The graphical results show that the size, position of the barrier and the depth of the water region has some effect on the reflected and transmitted wave.