THE SCATTERING OF WATER-WAVES BY 2 SEMI-INFINITE OPPOSED VERTICAL WALLS

In this paper we examine the transmitted and reflected waves generated by plane surface water waves incident on two semi-infinite vertical walls. Cartesian coordinates (x, y, z), z measured vertically upwards, are chosen so that the walls lie along y = 0, x > a and y = -h, x < 0, and the water is assumed to have constant mean depth d. We cast the problem into a matrix Wiener-Hopf functional equation, which for a less-than-or-equal-to 0 (the barriers overlapped) can be solved by subtracting an infinite sum of poles. It is possible to find a product decomposition of the Wiener-Hopf kernel, whose elements contain an infinite dimensional constant vector satisfying an L2-system of algebraic equations. For a > 0 the problem has added complications because of the presence of exponentially growing terms in the matrices to be factored, but these terms may be suppressed if we first post-multiply by a certain entire matrix function. This is a technique introduced by the authors to analyse the scattering by staggered ducts and related physical models. The elements of the entire matrix are determined by the solution of a Fredholm integral equation, which may be rewritten as another algebraic system of equations. Of particular importance is the determination of the transmitted scattered field, and the associated energy flux, and these quantities are computed from the exact solution for a range of values of the physical parameters.