Dispersion, nonlinearity, and viscosity in shallow-water waves

The roles of frequency dispersion, nonlinearity, and laminar viscosity in the evolution of long waves over distances of many wavelengths in constant water depth are investigated with numerical solutions of the Boussinesq equations. Pronounced frequency doubling and trebling is predicted, and the initial evolution to a wave shape with a pitched-forward front face and peaky crests is followed by development of a steep rear face and a nearly symmetric crest/trough profile. While reducing overall energy levels, laminar viscosity acts to prolong cycling of third moments and to inhibit the onset of disordered evolution characteristic of nonlinear, inviscid systems. Preliminary laboratory results show some qualitative similarities to the numerical simulations. However, these laboratory experiments were not suitable for detailed model-data comparisons because dissipation in the flume could not be accounted for with either laminar or quadratic damping models. More carefully controlled experiments are required to assess the importance of viscosity (and the accuracy of the Boussinesq model) in the evolution of nonlinear waves over distances of many wavelengths.