Two finite element numerical models based on the one-dimensional and unidirectional versions of the time dependent nonlinear mild-slope equation, Nadaoka et al. (1994), are presented herein. The models are applied to simple test cases of wave propagation over different depths - varying between deep to shallow waters - and their results are presented and discussed. The performance of the one-dimensional finite element model shows some numerical instabilities due to the nonlinear term and some problems have occurred near boundaries. Numerical filtering associated with the nonlinear term is used in order to control the nonlinear instability. Comparisons of the performance of different filters are presented and discussed. Some problems have occurred with the unidirectional wave model due to the simplification first adopted for the r/~= term, i.e., ŋxxx=-C-1ŋxxx. Results from the full treatment of this dispersive term are also presented and discussed.
The most important physical effects associated with the transformation of waves in coastal regions can be described by a simple linear scalar elliptic equation first introduced by Berkhoff (1972) or, by the more general time-dependent linear form due to Smith and Sprinks (1975). These models are suitable to the study of the combined refraction-diffraction of surface waves over uneven bottoms and can capture backscattered waves as well. Therefore they are adequate to describe the wave field outside and inside harbours and sheltered zones. However, these equations do not account for nonlinear effects that are particularly important in the nearshore region. Strong interactions of waves with the seabed topography, wave skewness and asymmetry of the wave profile and generation of other harmonics are some examples of the importance of nonlinear effects. In the nearshore zone the Boussinesq-type models can provide a quite accurate description of the nonlinear wave phenomena.
