Wave propagation through a gap in a thin vertical wall in deep water
The problem of oblique scattering of surface water waves by a vertical wall with a gap submerged in infinitely deep water is re-investigated in this paper. It is formulated in terms of two first kind integral equations, one involving the difference of potential across the wetted part of the wall and the other involving the horizontal component of velocity across the gap. The integral equations are solved approximately using one-term Galerkin approximations involving constants multiplied by appropriate weight functions whose forms are dictated by the physics of the problem. This is in contrast with somewhat complicated but known solutions of corresponding deep water integral equations for the case of normal incidence, used earlier in the literature as one-term Galerkin approximation. Ultimately this leads to very closed (numerically) upper and lower bounds of the reflection and transmission coefficients so that their averages produce fairly accurate numerical estimates for these coefficients. Known numerical results for normal incidence and for a narrow gap obtained by other methods in the literature are recovered, thereby confirming the correctness of the method employed here.
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