Water wave interaction with a circular arc shaped porous barrier submerged in a water of finite depth

In this paper, we study the problem of scattering of water waves by a thin circular arc-shaped porous barrier submerged in ocean of finite depth. By judicious application of Green’s integral theorem, the problem is formulated in terms of a hypersingular integral equation of second kind where the unknown function represents the difference of potential function across the curved barrier. The hypersingular integral equation is then solved by two methods. The first method is Boundary Element method where the domain and range of the integral equation are discretised into small line segments. Assuming the unknown function satisfying the integral equation to be constant in each small line segment, the hypersingular integral equation is reduced to a system of algebraic equations. This system of equations is then solved to obtain the unknown function in each subinterval. Making the subinterval finer, the process is continued till the solution converges to a desired degree of accuracy. The second method is based on using 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. Using the solution of the hypersingular integral equation, obtained by both the methods, the reflection coefficient, transmission coefficient and energy dissipation coefficient are computed and depicted graphically against the wave number. It was observed that the reflection, transmission and energy dissipation coefficients obtained by using the solution of hypersingular integral equation by the two methods are in good agreement. In addition, the reflection coefficient obtained by the present method found to match with the known results in the literature. From the graphs, the effect of the porous barrier on the reflected and transmitted waves and energy dissipation are studied. It was observed that the porosity of the barrier has some effect on the wave propagation.