An approximate analysis based on standard perturbation technique together with an application of Green's integral theorem is used in this paper to study the problem of scattering of water waves by a two dimensional thin plate submerged in deep ocean with ice cover. The reflection and transmission coefficients upto first order are obtained in terms of the shape function describing the plate and are studied graphically for different shapes of the plate.

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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.

In this paper, we study the problem of scattering of surface water waves by a thin circular arc shaped porous plate submerged in the deep ocean. The problem is formulated in terms of a hypersingular integral equation of the second kind in terms of an unknown function representing the difference of potential function across the curved barrier. The hypersingular integral equation is then solved by a collocation method after expanding the unknown function in terms of Chebyshev polynomials of the second kind. Using the solution of the hyper-singular integral equation, the reflection coefficient, transmission coefficient and energy dissipation coefficient are computed and depicted graphically against the wave number. Known results for the rigid curved barrier are recovered. It is observed that the porosity of the barrier reduces the reflection and transmission of the waves and enhances the dissipation of wave energy. The reflection coefficient and dissipation of wave energy decreases as the length of the porous curved barrier increases. Also the reflection coefficient is almost independent of the inertial force coefficient of the material of the porous barrier. However, the inertial force coefficient of the material of the porous barrier enhances transmission and reduces dissipation of wave energy.