An evaluation of a model equation for water waves

The aim of this paper is to assess how well the partial differential equation [Note: Equation omitted. See the image of page 458 for this equation.] describes the propagation of surface water waves in a channel. In (*) the variables are all scaled, with x proportional to the horizontal coordinate along the channel, t proportional to the elapsed time, and $\eta $ proportional to the vertical displacement of the surface of the water from its equilibrium position. The parameter $\lambda $ is a non-negative constant. A numerical scheme has been developed to solve (*) in the domain {(x,t):x,t > 0}, subject to the initial condition $\eta $(x,0)$\equiv $ 0 and to the boundary condition $\eta $(0,t)=h(t) The specified function h corresponds to a given displacement of the free surface at one end of the channel. The numerical scheme, which introduces some novel ideas for the approximation of solutions of certain partial differential equations, is explicit, unconditionally stable, and has fourth-order accuracy in both the spatial and temporal variables. The errors inherent in the integration procedure are rigorously analysed, and convergence tests of the computer code are presented. A comparison is made between the predictions of the theoretical model and the results of laboratory experiments. The experiments, which were performed at a fixed wavelength $\lambda $ and at a number of wave amplitudes a, covered a range of the parameter S(=a$\lambda ^{2}/d^{3}$ of more than two orders of magnitude. Here d is the depth of the undisturbed water. The model was found to give quite a good description of the spatial and temporal development of periodically generated waves over a wide range of the parameter S. It was noteworthy that, at least on laboratory scales, an allowance for dissipative effects was crucial in obtaining good agreement between experimental observations and the predictions of a theoretical model. At the larger wave amplitudes used in the experiments there were important differences between the forecast of the model and the empirical results. Possible reasons for these discrepancies are discussed.