Approximations to water wave scattering by steep topography.
R. Porter & D. Porter, 2005, J. Fluid Mech. (submitted)
A new method is developed for approximating the scattering of linear
surface gravity waves on water of varying quiescent depth in two
dimensions. A conformal mapping of the fluid domain into a uniform
rectangular strip transforms steep and discontinuous bed profiles
into relatively slowly-varying, smooth functions in the transformed
free surface condition. By analogy with the mild-slope approach
used extensively in unmapped domains, an approximate solution of the
transformed problem is sought in the form of a modulated propagating
wave which is determined by solving a second-order ordinary differential
equation. This can be achieved numerically, but an analytic solution
in the form of a rapidly convergent infinite series is also derived and
provides simple explicit formulae for the scattered wave amplitudes. Small
amplitude and slow variations in the bedform that are excluded from
the mapping procedure are incorporated in the approximation by a
straightforward extension of the theory. The error incurred in using
the method is established by means of a rigorous numerical investigation
and it is found that remarkably accurate estimates of the scattered wave
amplitudes are given for a wide range of bedforms and frequencies.
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