Fully nonlinear extensions of Boussinesq equations are derived to simulate surface wave propagation in coastal regions. By using the velocity at a certain depth as a dependent variable (Nwogu 1993), the resulting equations have significantly improved linear dispersion properties in intermediate water depths when compared to standard Boussinesq approximations. Since no assumption of small nonlinearity is made, the equations can be applied to simulate strong wave interactions prior to wave breaking. A high-order numerical model based on the equations is developed and applied to the study of two canonical problems: solitary wave shoaling on slopes and undular bore propagation over a horizontal bed. Results of the Boussinesq model with and without strong nonlinearity are compared in detail to those of a boundary element solution of the fully nonlinear potential flow problem developed by Grilli et al. (1989). The fully nonlinear variant of the Boussinesq model is found to predict wave heights, phase speeds and particle kinematics more accurately than the standard approximation.

Shoaling and breaking of solitary waves is computed on slopes 1:100 to 1:8 using an experimentally validated fully nonlinear wave model based on potential flow equations. Characteristics of waves are computed at and beyond the breaking point, and geometric self-similarities of breakers are discussed as a function of wave height and bottom slope. No wave breaks for slopes steeper than 12. A breaking criterion is derived for milder slopes, based on values of a nondimensional slope parameter S o. This criterion predicts both whether waves will break or not and which type of breaking will occur (spilling, plunging, or surging). Empirical expressions for the breaking index and for the depth and celerity at breaking are derived based on computations. All results agree well with laboratory experiments. The NSW equations fail to predict these results with sufficient accuracy at the breaking point. Pre-breaking shoaling rates follow a more complex path than previously realized. Post-breaking behaviors exhibit a rapid (non-dissipative) decay, also observed in experiments, associated with a transfer of potential energy into kinetic energy. Wave celerity decreases in this zone of rapid decay.

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Shoaling of solitary waves on both gentle (1:35) and steeper slopes (-<1:6.50) is analyzed up to breaking using both a fully nonlinear wave model and high-accuracy laboratory experiments. For the mildest slope, close agreement is obtained between both approaches up to breaking, where waves become very asymmetric and breaking indices reach almost twice the value for the largest stable symmetric wave. Bottom friction does not seem to affect the results at all. Wave celerity decreases during shoaling and slightly increases before breaking. At breaking, the crest particle velocity is almost horizontal and reaches 90% of the crest celerity, which is two to three times larger than the bottom velocity. The nonlinear shallow water (NSW) equations and the Boussinesq approximation both fail to predict these results. Finally, shoaling rates for various wave heights and bottom slopes differ from the predictions of Green's or Boussinesq shoaling laws. On the mildest slope, shoaling rates roughly follow a "two-zone" model proposed earlier but on steeper slopes reflection becomes significant and wave heights change little during shoaling.

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