| Based on Euler's motion equations, a set of fully nonlinear Boussinesq equations up to the order of O(μ~2,ε~2,μ~2) (where e is the ratio of wave amplitude to water depth and μ~2. is the ratio of water depth to wave length) in terms of the velocity at an arbitrary water level is derived. By adding some "deep-water terms", the resulting set of equations has significantly improved linear dispersion properties in the water of intermediate depth and extended the range of its applicability to μ≈1.0. This is confirmed by comparisons of the simulated and measured time series of the regular waves propagating on a submerged bar. The moving shoreline is treated numerically by replacing the solid beach by a permeable beach characterized by containing a slot. Run-up of nonbreaking waves is verified against the analytical solution for nonlinear shallow water waves. The inclusion of wave breaking is fulfilled by introducing an eddy term, which is spatially and temporally tied to the front face of the wave, in the momentum equation to serve as the breaking wave force term to dissipate wave energy in the surf zone. The model is applied to cross-shore motions of regular waves including various types of breaking on plane sloping beaches. Comparisons of the model results comprising spatial distribution of wave height and mean water level with physical experiments are presented. As an important hydrodynamic phenomenon in the nearshore zone, the cross-shore surf beat is numerically studied in this paper with the model, which resolves the primary wave motion as well as the long waves. Mutual interaction between short waves and long waves is inherent in the model. The model study is based on incident bichromatic wave groups considering a range of mean frequencies, group frequencies, modulation rates. The cross-shore variation in the amplitudes is also investigated. The model results are compared with laboratory experiments from the literature and the agreement is generally found to be very good.
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