WATER WAVE SCATTERING BY INCLINED BARRIER SUBMERGED IN FINITE DEPTH WATER Mridula Kanoria 1 and B.N. Mandal 2 1 Department of Applied Mathematics, Calcutta University,India 2 Physics and Applied Mathematics Unit, Indian Statistical Institute,India 1. Introduction : Water wave scattering problems involving thin barrier of arbitrary shape sub- merged in finite depth water are generally tackled by some approximate techniques assuming linear theory. The technique of hypersingular integral equation is demonstrated here in tackling the problem of water wave scattering by an inclined thin barrier submerged in finite depth water. The corresponding deep water problem was earlier investigated by Parson and Martin[1]. An appropriate use of Green’s integral theorem produces a representation of the velocity potential describing the irrotational motion in the fluid region, in terms of the unknown dicontinuity of the potential across the submerged barrier. Utilization of the boundary condition on the barrier gives rise to an integro-differential equation for the discontinuity, which is interpreted as equivalent to a hypersingular integral equation. This is solved numerically by approximating the discontinuity in terms of a finite series involving Chebysev polyno- mials of the second kind followed by a collocation method. The reflection and transmission coefficients are then estimated numerically using this solution. The force and moment(about the origin) acting on the barrier per unit width are also estimated for various positions of the barrier. Comparison is made with available deep water results. It is observed that if the mid point of the barrier is submerged to the order of one-tenth of the bottom depth, then the deep water results effectively hold good, and in that case, the finite depth problem can be modelled as deep-water problem. 2. Mathematical formulation : Let a thin straight inclined barrier Γ be submerged in water of uniform finite depth h, and let d be the depth of its mid point below the mean free surface and the co- ordinate system be so chosen that the position of Γ is described by y = d + ta cos θ,x = ta sin θ(-1 ≤ t ≤ 1,d>a cos θ, h > d + a cos θ, 0 ≤ θ ≤ 90 0 ,θ being the angle of inclination of the barrier with the vertical). A train of surface water waves of amplitude b 0 and circular frequency σ is incident from the direction of x = -∞ on the barrier. The incident wave potential Re{φ 0 (x, y)e -iσt } with φ 0 (x, y)= gb 0 σ cosh k 0 (h - y)e ik 0 x cosh k 0 h where k 0 is the unique positive real root of the transcendental equation k tanh kh = K, with K = σ 2 /g, g being the gravity. The ensuing motion in the fluid region described by velocity potential Re{φ(x, y)e -iσt } where φ(x, y) satisfies ∇ 2 φ =0, 0 ≤ y ≤ h Kφ + ∂φ ∂y =0 on y =0, ∂φ ∂n = 0 on Γ where ∂ ∂n denotes normal derivatives on Γ, r 1/2 ∇φ is bounded as r → 0 where r is the distance from the submerged edges of Γ, 1