Wave scattering by a step of arbitrary profile.
by R. Porter and D. Porter (to be published in J. Fluid Mech)
The two-dimensional scattering of water waves over a finite region of
arbitrarily varying topography linking two semi-infinite regions of
constant depth is considered. Unlike many approaches to this problem,
the formulation employed is {\it exact}, utilising simple combinations
of Green's functions appropriate to water of constant depth and the
Cauchy-Riemann equations to derive a system of coupled integral equations
for components of the fluid velocity at certain locations. Two cases
arise, depending on whether the deepest point of the topography does
or does not lie below the lower of the semi-infinite horizontal bed
sections. In each, the reflected and transmitted wave amplitudes are
related to the incoming wave amplitudes by a scattering matrix which
is defined in terms of inner products involving the solution of the
corresponding integral equation system.
This solution is approximated by using the variational method in
conjunction with a judicious choice of trial function which correctly
models the fluid behaviour at the free surface and near the joins of
the varying topography with the constant depth sections, which may
not be smooth. The numerical results are remarkably accurate, with
just a two-term trial function giving three decimal places of accuracy
in the reflection and transmission coefficents in most cases, whilst
increasing the number of terms in the trial function results in rapid
convergence. The method is applied to a range of examples.
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