(2005). In these physical experiments of Hammack et al., as well as in the numerical simulation
of Fuhrman and Madsen (2006), a linear wavemaker method was used to generate the (nonlinear) short-crested waves. The nonlinear model, and the physical experiment, responded by releasing spurious free harmonics due to the fact that third-order components in the wave generation are neglected. This resulted in modulations in the computational domain and in the physical experiment. Fuhrman and Madsen showed that inclusion of the third-order wave components in the wave generation reduces significantly the first-harmonic spurious modulations. selleck kinase inhibitor This shows that wavemaker theory should take higher order harmonic steering into account when dealing with highly
nonlinear waves. The appearance of spurious free waves can also be expected in embedded wave generation methods if the force function is derived for a linear(ized) wave model. Wei and Kirby (1998) used a selleck screening library numerical filtering method proposed by Shapiro (1970) in order to reduce the effects of the spurious free waves. They conclude that the method is cumbersome to write and inconvenient to code in the program. Instead of using higher order steering or numerical filtering, we propose to use an adjustment for nonlinear wave generation that is motivated by Dommermuth (2000). Dommermuth remarked that nonlinear dispersive selleck inhibitor wave models can be initialized with linear wave fields if the flow field is given sufficient time to adjust. For the initial value problem which he investigated, he introduced an adjustment scheme
in time that allows the natural development of nonlinear self-wave (locked modes) and wave-wave (free modes) interactions. To implement this idea in nonlinear wave models, the higher-order terms, denoted by F , are multiplied by a slowly increasing function from 0 to 1 in a time interval T a, leading to the adjustment F˜ given by F˜=[1−exp(−(t/Ta)n)]Ffor some positive power n. In his examples, the optimal length of the time interval Ta should be larger than two times the period of the longest waves in the simulation. For embedded wave generation, which takes place in time during the whole simulation, we modify the adjustment accordingly: the influxed waves are propagated away from the influx position by a spatially dependent increase of the nonlinear terms of the equation. Specifically, consider embedded influxing in a nonlinear Hamiltonian model with force functions (14) and with additional nonlinear (higher order) terms N 1 and N 2, given by ∂tη=Dgϕ+G1+N1∂tϕ=−gη+G2+N2The adjustment scheme in space uses a characteristic function χ(x,La)χ(x,La) that gradually grows from 0 to 1 in a transition zone with length L a; multiplying the nonlinear terms to N 1 and N 2 with this function results in equation(22) ∂tη−Dgϕ−G1=χN1 equation(23) ∂tϕ+gη−G2=χN2∂tϕ+gη−G2=χN2Fig.