The resonant scattering of topographically trapped, low-mode progressive
by longshore periodic topography is investigated using a multiple-scale
the linear shallow water equations. Coupled evolution equations for the
amplitudes of incident and scattered edge waves are derived for small-amplitude,
periodic depth perturbations superposed on a plane beach.
In ‘single-wave scattering’,
an incident edge wave is resonantly scattered into a single additional
edge wave having the same or different mode number (i.e. longshore wavenumber),
and propagating in the same or opposite direction (forward and backward
respectively), as the incident edge wave. Backscattering into the same
as the incident edge wave, the analogue of Bragg scattering of surface
is a special case. In ‘multi-wave scattering’, simultaneous
forward and backward
resonant scattering results in several (rather than only one) new progressive
waves. Analytic solutions are obtained for single-wave scattering and for
case of multi-wave scattering involving mode-0 and mode-1 edge waves, over
depth regions of both finite and semi-infinite longshore extent. In single-wave
backscattering with small (subcritical) detuning (i.e. departure from exact
the incident and backscattered wave amplitudes both decay exponentially
distance over the periodic bathymetry, whereas with large (supercritical)
detuning the amplitudes oscillate with distance. In single-wave forward
wave amplitudes are oscillatory regardless of the magnitude of the detuning.
solutions combine aspects of single-wave backward and forward scattering.
both single- and multi-wave scattering, the exponential decay rates and
wavenumbers of the edge wave amplitudes depend on the detuning. The results
that naturally occurring rhythmic features such as beach cusps and crescentic
bars are sometimes of large enough amplitude to scatter a significant amount
incident low-mode edge wave energy in a relatively short distance
(O(10) topographic wavelengths).