Multiple wave scattering by submerged obstacles in an infinite channel of finite depth. I. Streamlines

Abstract

A finite number of rigid bodies of finite extent are submerged and kept fixed in an infinite, homogeneous layer of finite depth of an inviscid fluid of constant density. The layer is bounded from above by a free surface and from below by a horizontal impermeable bottom. An incident harmonic wave of given frequency propagating in the fluid layer is partially reflected by the obstacles, while part is transmitted after multiple scattering. The problem is to determine the resulting motion of the fluid in the frame of the linearized theory. A finite Fourier transform technique is used to this end, in combination with the methods of separation of variables, fundamental solutions and boundary collocation. This yields different expressions for the solution in different parts of the flow domain, each of which depends on a system of orthogonal functions. The obtained solution exhibits smoothness at any order in the fluid domain. It satisfies rigorously the field equation and all the boundary conditions, with the exception of the condition at the obstacle boundaries which is satisfied pointwise by boundary collocation. The method deals invariably with any finite number of obstacles, and its accuracy is clearly defined. Numerical applications with one and two circular obstacles are presented and the effects of the level of submergence are discussed. Plots are provided for the resulting system of streamlines.