A Fourier-Boussinesq method for nonlinear water waves

A Boussinesq method is derived that is fully dispersive, in the sense that the error of the approximation is small for all 0⩽kh<∞ (k the magnitude of the wave number and h the water depth). This is made possible by introducing the generalized (2D) Hilbert transform, which is evaluated using the fast Fourier transform. Variable depth terms are derived both in mild-slope form, and in augmented mild-slope form including all terms that are linear in derivatives of h. A spectral solution is used to solve for highly nonlinear steady waves using the new equations, showing that the fully dispersive behavior carries over to nonlinear waves. A finite-difference-FFT implementation of the method is also described and applied to more general problems including Bragg resonant reflection from a rippled bottom, waves passing over a submerged bar, and nonlinear shoaling of a spectrum of waves from deep to shallow water.