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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