Interaction of water waves with three-dimensional periodic topography
by R. Porter and D. Porter (to appear in J. Fluid Mech)
The scattering and trapping of water waves by three-dimensional submerged
topography, infinite and periodic in one horizontal coordinate and
of finite extent in the other, is considered under the assumptions
of linearised theory. The mild-slope approximation is used to reduce
the governing boundary value problem to one involving a form of the
Helmholtz equation in which the coefficient depends on the topography
and is therefore spatially-varying.
Two problems are considered: the scattering by the topography of
parallel-crested obliquely-incident waves and the propagation of trapping
modes along the periodic topography. Both problems are formulated in
terms of `domain' integral equations which are solved numerically.
Trapped waves are found to exist over any periodic topography which is
`sufficiently' elevated above the unperturbed bed level. In particular,
every periodic topography wholly elevated above that level supports
trapped waves. Fundamental differences are shown to exist between
these trapped waves and the analogous Rayleigh-Bloch waves which exist
on periodic gratings in acoustic theory.
Results computed for the scattering problem show that, remarkably, there
exist zeros of transmission at discrete wavenumbers for any periodic
bed elevation and for all incident wave angles. One implication of this
property is that total reflection of an incident wave of a particular
frequency will occur in a channel with a single symmetric elevation
on the bed. The zeros of transmission in the scattering problem are
shown to be related to the presence of a `nearly-trapped' mode in the
corresponding homogeneous problem.
The scattering of waves by multiple rows of periodic topography is also
considered and it is shown how Bragg resonance -- well-established in
scattering of waves by two-dimensional ripple beds -- occurs in modes
other than the input mode.
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