Steepness effect on modulation instability of the nonlinear wave train

2007 ◽  
Vol 19 (1) ◽  
pp. 014105 ◽  
Author(s):  
Wen-Son Chiang ◽  
Hwung-Hweng Hwung
Keyword(s):  
Nonlinear Wave ◽  
Modulation Instability ◽  
Wave Train

Drifting Rogue Packets

2018 ◽  
Author(s):  
Amin Chabchoub ◽  
Norbert Hoffmann ◽  
Nail Akhmediev ◽  
Takuji Waseda
Keyword(s):  
Modulation Instability ◽  
Wave Train ◽  
Modulation Frequency ◽  
Phase Speed ◽  
Periodic Wave ◽  
Unstable Wave ◽  
Extreme Waves ◽  
Wave Groups ◽  
The Waves ◽  
Good Agreement

Modulation instability (MI) is one possible mechanism to explain the formation of extreme waves in uni-directional and narrow-banded seas. It can be triggered, when side-bands around the main frequency are excited and subsequently follow an exponential growth. In physical domain this dynamics translates to periodic pulsations of wave groups that can reach heights up to three times the initial amplitude of the wave train. It is well-known that these periodic wave groups propagate with approximately half the waves phase speed in deep-water. We report an experimental study on modulationally unstable wave groups that propagate with a velocity that is higher than the group velocity since the modulation frequency is complex. It is shown that when this additional velocity to the wave groups is small a good agreement with exact nonlinear Schrödinger (NLS) models, that describe the nonlinear stage of MI, is reached. Otherwise a significant deviation is observed that could be compensated when increasing accuracy of the water wave modeling beyond NLS.

Supercritical dead water: effect of nonlinearity and comparison with observations

2016 ◽  
Vol 803 ◽  
pp. 436-465 ◽  
Author(s):  
John Grue ◽  
Daniel Bourgault ◽  
Peter S. Galbraith
Keyword(s):  
Internal Wave ◽  
Nonlinear Wave ◽  
Wave Train ◽  
Linear Method ◽  
Field Measurements ◽  
Wave Resistance ◽  
Three Dimensions ◽  
Surface Velocity ◽  
Dead Weight ◽  
Dead Water

Supercritical ship internal wave wakes with $Fr=U/c_{0}\sim 4{-}12$ (where $U$ is the ship speed and $c_{0}$ is the linear internal long-wave speed) are calculated by a strongly nonlinear two-layer model in three dimensions, accounting for the complex ship geometry, and compared with field measurements. The degree of nonlinearity, defined by the ship draught ($d_{0}$) to average depth of pycnocline ($h_{0}$) ratio, is explored in the range $d_{0}/h_{0}\sim 0.1{-}1.2$, comparing nonlinear and linear calculations. For $d_{0}/h_{0}\sim 1$, the wave amplitude far downstream is overpredicted by up to 50 % by the linear method. The nonlinear trough amplitudes decay algebraically in the lateral coordinate with decay exponents in the range 0.16–0.61. The nonlinear leading trough is systematically somewhat ahead of the classical asymptotic pattern, and its amplitude and forerunning slope are appreciable, while the linear counterparts are very small. The calculated and asymptotic patterns are close for large $Fr$. Field measurements in a Canadian fjord of the internal wave wake of a 221 m long cargo ship of dead weight 43 000 tonnes moving at $Fr=6.6$ document a series of waves of 1–2 m isopycnal displacements at an off-track distance of 700 m. The nonlinear computations of a similar ship predict a wave train of height 1.5 m at a similar position, matching the field observation, whereas linear calculations give a wave train of height 2.3 m. Two- and three-layer theoretical models predict speeds that match the observed speeds of the second and third wave troughs. The observed leading signature of the diverging wave wake is moving at supercritical speed. This may be explained by the position and speed of the nonlinear wave slope moving ahead of the leading trough. Nonlinear computations of the surface velocity and strain rate compare well with measurements in the Loch Linnhe experiment of Watson et al. (J. Geophys. Res., vol. 97 (C6), 1992, pp. 9689–9703). The calculated nonlinear wave resistance of the observed cargo ship is comparable to the frictional force for $Fr\sim 4{-}6$, exceeds the surface wave resistance and increases the total drag by 40 %. A linear force prediction is useless when $d_{0}/h_{0}\sim 1$. The results show that nonlinearity of the dead water wake depends on the relative ship volume as well as the relative ship draught, with the pycnocline depth as the relevant length scale.

scholarly journals The Effects on Water Particle Velocity of Wave Peaks Induced by Nonlinearity under Different Time Scales

2021 ◽  
Vol 9 (7) ◽  
pp. 748
Author(s):  
Aifeng Tao ◽  
Shuya Xie ◽  
Di Wu ◽  
Jun Fan ◽  
Yini Yang
Keyword(s):  
Particle Velocity ◽  
Modulation Instability ◽  
Wave Train ◽  
Quadratic Function ◽  
Offshore Structures ◽  
Stokes Wave ◽  
Wave Load ◽  
Freak Waves ◽  
Water Particle ◽  
Generation Mechanisms

The water particle velocity of the wave peaks is closely related to the wave load borne by offshore structures. It is of great value for marine disaster prevention to study the water particle velocity of nonlinear extreme waves represented by Freak waves. This study applies the High-order Spectral Method (HOS) numerical model to analyze the characteristics and influencing factors of the water particle velocity of Freak wave peak with two different generation mechanisms under the initial condition of a weakly modulated Stokes wave train. Our results show that the water particle velocity of the wave peak increases linearly with wave height and initial wave steepness in the evolution stage of modulation instability. While in the later stage, the relationship becomes exponential. Under the condition of similar wave heights, the deformation degrees of Freak waves with different generation mechanisms are distinct, the deformation degree of modulation instability stage is smaller than that of the later stage. The water particle velocity of the wave peaks increases with the deformation degrees. Furthermore, the correlation between wave peak height and water particle velocity is a quadratic function. This provides a theoretical basis for further understanding of nonlinear waves and the prediction of marine disasters.

Nonlinear deep-water waves: theory and experiment. Part 2. Evolution of a continuous wave train

1977 ◽  
Vol 83 (1) ◽  
pp. 49-74 ◽  
Author(s):  
Bruce M. Lake ◽  
Henry C. Yuen ◽  
Harald Rungaldier ◽  
Warren E. Ferguson
Keyword(s):  
Schrödinger Equation ◽  
Nonlinear Wave ◽  
Nonlinear Schrödinger Equation ◽  
Deep Water ◽  
Water Waves ◽  
Schrodinger Equation ◽  
Wave Train ◽  
Wave Packets ◽  
Nonlinear Schrodinger Equation ◽  
Nonlinear Schrödinger

Results of an experimental investigation of the evolution of a nonlinear wave train on deep water are reported. The initial stage of evolution is found to be characterized by exponential growth of a modulational instability, as was first discovered by Benjamin ' Feir. At later stages of evolution it is found that the instability does not lead to wave-train disintegration or loss of coherence. Instead, the modulation periodically increases and decreases, and the wave train exhibits the Fermi–Pasta–Ulam recurrence phenomenon. Results of an earlier study of nonlinear wave packets by Yuen ' Lake, in which solutions of the nonlinear Schrödinger equation were shown to provide quantitatively correct descriptions of the properties of nonlinear wave packets, are applied to describe the experimentally observed wave-train phenomena. A comparison between the laboratory data and numerical solutions of the nonlinear Schrödinger equation for the long-time evolution of nonlinear wave trains is given.

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