An experimental and numerical study of the spatial evolution of unidirectional nonlinear water-wave groups

Abstract

Spatial evolution of nonlinear narrow-spectrum deep-water wave groups is studied experimentally in a wave tank. The experimental results are compared with the computations based on the unidirectional Zakharov equation and the Dysthe model. The very good agreement between the computational results based on both models with the experiments prompted an attempt to perform simulations for a wider initial spectral width, that formally violate the assumptions adopted in the derivation of the Dysthe model. The accuracy of the results based on the Dysthe model is checked against the solutions of the Zakharov equation, which is free of restrictions on the spectral width. Conclusions regarding the domain of validity of the Dysthe model are drawn.

Experimental and numerical study of spatial and temporal evolution of nonlinear wave groups

“…Such a modification of the Dysthe model was carried out by Lo and Mei (1985) who obtained a version of the equation that describes the spatial evolution of the group envelope. Numerical computations based on the Dysthe model for unidirectional wave groups propagating in a long wave tank indeed provided good agreement with experiments and exhibit the front-tail asymmetry, Shemer et al (2002). The spatial version of the Dysthe equation was also derived by Kit and Shemer (2002) from the spatial form of the Zakharov equation (Shemer et al, 2001 that is free of any restrictions on the spectrum width.…”

“…Kit and She-mer (2002) have demonstrated that this modification can be easily derived by expanding the dispersion term in the Zakharov equation into the Taylor series. The effect of each one of the additional (as compared to the NLS equation) terms in the Dysthe model was studied in Shemer et al (2002). They demonstrated that for steep waves all these terms are essential and contribute significantly to the accuracy of the solution.…”

“…The spectral width of the signal given by Eq. (18), calculated as in Shemer et al (2002), is ω/ω 0 =0.054<ε, thus satisfying the narrow spectrum limit of the Dysthe model. For these experimental conditions, the dimensionless depth k 0 h≈5, so that the condition for the validity of the Dysthe model, k 0 h=O(1/ε) is also satisfied.…”

“…Alternatively, the variation of the complex envelope in time, a(t), can be specified at a prescribed location x=x 0 , and the variation of a(t) along the tank is then studied in the spatial formulation using the appropriately modified model equations. It should be stressed that the spatial formulation is routinely applied in the experiment-related studies (Lo and Mei, 1985;Shemer et al, 2002), since the wave gauges provide information on the temporal variation of the surface elevation at fixed locations. The experimental approach of the present study makes it possible to measure the variation with time of the instantaneous complex group envelope along the tank, as well as the variation of the surface elevation with time at any location within the tank.…”

On the existence of steady-state resonant waves in experiments

“…The four resonant waves are obtained theoretically (Liao 2011;Xu et al 2012;Liu & Liao 2014) by means of the HAM (Liao 1992(Liao , 1997(Liao , 2004(Liao , 2012. They are chosen in such a way that the related dimensionless scaled variable X = 2 k 0 x introduced by Shemer et al (2002) reaches 1.5 at the ninth wave gauge, corresponding to sufficiently high nonlinearity of the waves considered, where k 0 denotes the wavenumber of the component with the largest amplitude. A large enough number of wave components (with more than 95 % of the total wave energy) were generated so that the corresponding wavefields measured in the basin are close to the theoretical ones.…”