  When shallow water flows over uneven bathymetry, the water surface is
modulated. This type of problem has been revisited numerous times since it was
first studied by Lord Kelvin in 1886. Our study analytically examines currents
whose unperturbed velocity profile $U(z)$ follows a power-law $z^q$, flowing
over a three-dimensional uneven bed. This particular form of $U$, which can
model a miscellany of realistic flows, allows explicit analytical solutions.
Arbitrary bed shapes can readily be imposed via Fourier's theorem provided
their steepness is moderate. Three-dimensional vorticity-bathymetry interaction
effects are evident when the flow makes an oblique angle with, a sinusoidally
corrugated bed. Streamlines are found to twist and the fluid particle drift is
redirected away from the direction of the unperturbed current. Furthermore, a
perturbation technique is developed which satisfies the bottom boundary
condition to arbitrary order also for large-amplitude obstructions which
penetrate well into the current profile. This introduces higher-order harmonics
of the bathymetry amplitude. States of resonance for first and higher order
harmonics are readily calculated. Although the method is theoretically
restricted to bathymetries of moderate inclination, a wide variety of steeper
obstructions are satisfactorily represented by the method, even provoking
occurrences of recirculation. All expressions are analytically explicit and
sequential fast Fourier transformations ensure quick and easy computation for
arbitrary three-dimensional bathymetries. A method for separating near and far
fields ensures computational convergence under the appropriate radiation
condition.
