  Scattering of random surface gravity waves by small amplitude topography in
the presence of a uniform current is investigated theoretically. This problem
is relevant to ocean waves propagation on shallow continental shelves where
tidal currents are often significant. A perturbation expansion of the wave
action to second order in powers of the bottom amplitude yields an evolution
equation for the wave action spectrum. A scattering source term gives the rate
of exchange of the wave action spectrum between wave components, with
conservation of the total action at each absolute frequency. With and without
current, the scattering term yields reflection coefficients for the amplitudes
of waves that converge, to the results of previous theories for monochromatic
waves propagating in one dimension over sinusoidal bars. Over sandy continental
shelves, tidal currents are known to generate sandwaves with scales comparable
to those of surface waves. Application of the theory to such a real topography
suggests that scattering mainly results in a broadening of the directional wave
spectrum, due to forward scattering, while the back-scattering is generally
weaker. The current may strongly influence surface gravity wave scattering by
selecting different bottom scales with widely different spectral densities due
the sharp bottom spectrum roll-off.
