Fully Nonlinear Model for Water Wave Propagation from Deep to Shallow Waters

A set of fully nonlinear Boussinessq-type equations (BTEs) with improved linear and nonlinear dispersive performance is presented. The highest order of the derivatives is three in the equations, and they use the minimum number of unknowns: the free surface elevation and the horizontal velocity at a certain depth. The equations allow reduction of the errors both in linear frequency dispersion and shoaling below 0.30% for kh≤5, and below 2.2% for kh≤10, with k as the wave number and h as the water depth. The weakly nonlinear performance is also improved for kh≤2. A simple fourth-order explicit numerical scheme is presented to test the linear and nonlinear behavior of the model equations against analytical and experimental results.