A set of model equations for water wave propagation is derived by piecewise integration of the primitive equations of motion through N arbitrary layers. Within each layer, an independent velocity profile is determined. With N separate velocity profiles, matched at the interfaces of the layers, the resulting set of equations have N+1 free parameters, allowing for an optimization with known analytical properties of water waves. The optimized two-layer model equations show good linear wave characteristics up to kh ≈8, while the second-order nonlinear behavior is well captured to kh ≈6. The three-layer model shows good linear accuracy to kh ≈14, and the four layer to kh ≈20. A numerical algorithm for solving the model equations is developed and tested against nonlinear deep-water wave-group experiments, where the kh of the carrier wave in deep water is around 6. The experiments are set up such that the wave groups, initially in deep water, propagate up a constant slope until reaching shallow water. The overall comparison between the multi-layer model and the experiment is quite good, indicating that the multi-layer theory has good nonlinear, as well has linear, accuracy for deep-water waves.
On regularization procedures with linear accuracy estimates of approximations
