Surface wave scattering by submerged cylinders of arbitrary cross-section


by R. Porter


The reflection of surface waves normally-incident upon an infinite uniform horizontal cylinder of arbitrary cross-section totally-immersed beneath the free surface of a fluid of either finite or infinite depth is considered under the assumptions of linearised theory. The approach involves using Green's identity with Cauchy-Riemann type relations to transfer normal derivatives of certain functions to tangential derivatives of others. Thus, an integral equation of the first kind is derived for an unknown function related to the tangential fluid velocity on the surface of the cylinder, the kernel of the integral equation being complex symmetric and weakly singular. An approximation to the solution is made using Galerkin's method and is shown to converge rapidly as the number of terms in the expansion of the unknown function is increased. Thus for most bodies and wave parameters, just ten terms are sufficient to claim six decimal place accuracy in the reflection and transmission coefficients. This is the case even when the cylinder is an arbitrary thin plate, where special test functions are introduced to correctly model the anticipated singularity in the velocity at the ends of the plate. Results for infinite and finite depth are compared with existing results and new results are also presented showing zeros of transmission for families of submerged obstacle.

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