Hamiltonian Formulation of Water Waves

Abstract

This report describes the formulation, numerical implementation and application of a weakly nonlinear wave model for finite depth based on a Hamiltonian formulation (see Radder, 1992). Due to the type of non-linearity explicitly accounted for in the expansion of the kernel of the Hamiltonian density (sum of kinetic and potential energy per unit surface area), the model is valid for waves of small, but finite amplitude and fairly long wave length (compared to the water depth) in roughly the same sense many 'Boussinesq-type' models are. There are, however, a few significant differences. Firstly, the Hamiltonian density in the present formulation is always positive definite: a condition necessary to ensure good dynamical behavior of the model equations for numerical computation. Secondly, the dispersion equation obtained from the linearized version of the equations is exact. This property results in a better modelling of the phase relations (hence the wave asymmetry at a given location) of the superharmonic field which evolves from the primary wave system due to surface non-linearity. More importantly, it is possible to remove the restriction of waves being long through a proper inclusion of the 'short-wave' non-linearity in the expansion of the kernel. This results in a uniformly valid model unlike most of the weakly nonlinear models which are valid over either deep or shallow water. Further, it is discussed in the text that even for long waves the 'short-wave' non-linearity becomes locally important near the crest of a wave as the surface curvature increases. Implementation of 'short-wave' non-linearity is therefore considered as one of the first priorities i n the future developments of the model. Two numerical models have been developed: a time-domain model and a pseudo-spectral model based on the sinc-series for the global approximations. The numerical code based on the sinc-series requires less computing time and gives the option of choosing higher-order interpolation for computing derivatives and integrals over local intervals. An incident train of sinusoidal waves has been represented in the computation by a packet of sinusoidal waves of finite length with the leading edge a few wave-lengths behind the bar. Details of the geometry and significance of this test can be found in chapter 7. In spite of a practical disadvantage due to the way the input conditions must be specified in the code (we have an initial-value problem) these models can now be used to study nonlinear evolutions of non-breaking waves over varying depth. Attempts to introduce effects of wave breaking have not met with much success yet. In the first phase, an integral criterion has been implemented to determine if the instantaneous surface shape should lead to breaking. Application of the criterion to computed surface elevation for conditions observed to have given rise to mild breaking in laboratory tests shows that breaking stage is not reached. It is believed that this difference between the computed results and the laboratory observations is caused by the omission of the 'short-wave' non-linearity (under prediction of the surface steepness), rather than the failure of the integral criterion. This is another aspect which underscores the importance of implementation of 'short-wave' non-linearity. Finally to conclude, comparison of the computed results with the experimental measurements and in a wider context with several 'Boussinesq-type' models (see Dingemans, 1994a) provides strong motivation for further developments of the model. This is further supported by the inherent theoretical appeal of the formulations used. As summarized in chapter 9, the major recommendations for future work are: • The formulation of a boundary-value problem instead of an initial-value problem. This greatly enhances the practical usefulness of the program. • Inclusion of short-wave non-linearity. For applicability over the full range of deep to shallow water of a nonlinear wave model, inclusion of short-wave non-linearity is needed. Furthermore, it has been shown that for wave breaking also the inclusion of short-wave nonlinearity is needed to obtain more realistic breaker indices. • Further study of wave breaking characteristics after inclusion of short-wave non-linearity. • Effect of steeper bottom slopes on the wave behavior.