We study the generalized Bragg scattering of surface
waves over a wavy bottom. We
consider the problem in the general context of nonlinear
wave–wave interactions, and
write down and provide geometric constructions for the
Bragg resonance conditions
for second-order triad (class I) and third-order quartet
(class II and class III) wave–
bottom interactions. Class I resonance involving one bottom
and two surface wave
components is classical. Class II resonance manifests bottom
nonlinearity (it involves
two bottom and two surface wave components), and has been
studied in the laboratory. Class III Bragg resonance is new
and is a result of free-surface nonlinearity
involving resonant interaction among one bottom and three
surface wave components. The amplitude of the resonant wave
is quadratic in the surface wave slope
and linear in the bottom steepness, and, unlike the former
two cases, the resonant
wave may be either reflected or transmitted (relative to the
incident waves) depending
on the wave–bottom geometry. To predict the initial
spatial/temporal growth of the
Bragg resonant wave for these resonances, we also provide
the regular perturbation
solution up to third order. To confirm these predictions
and to obtain an efficient
computational tool for general wave–bottom problems
with resonant interactions,
we extend and develop a powerful high-order spectral
method originally developed
for nonlinear wave–wave and wave–body interactions.
The efficacy of the method is
illustrated in high-order Bragg resonance computations in two
and three dimensions.
These results compare well with existing experiments and
perturbation theory for the
known class I and class II Bragg resonance cases, and obtain
and elucidate the new
class III resonance. It is shown that under realistic
conditions with moderate to small
surface and bottom steepnesses, the amplitudes of third-order
class II and class III
Bragg resonant waves can be comparable in magnitude to those
resulting from class
I interactions and appreciable relative to the incident wave.