Fourth-order nonlinear evolution equations for counterpropagating capillary-gravity wave packets on the surface of water of infinite depth

2002 ◽  
Vol 14 (7) ◽  
pp. 2225 ◽  
Author(s):  
Suma Debsarma ◽  
K. P. Das
Keyword(s):  
Gravity Wave ◽  
Fourth Order ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Wave Packets ◽  
Nonlinear Evolution Equations ◽  
Infinite Depth

scholarly journals Nonlinear Evolution Equations for Broader Bandwidth Wave Packets in Crossing Sea States

2014 ◽  
Vol 2014 ◽  
pp. 1-9 ◽  
Author(s):  
S. Debsarma ◽  
S. Senapati ◽  
K. P. Das
Keyword(s):  
Growth Rate ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Modulational Instability ◽  
Wave Packets ◽  
Nonlinear Evolution Equations ◽  
Wave Spectra ◽  
Surface Gravity Wave ◽  
Freak Waves ◽  
Wave Trains

Two coupled nonlinear equations are derived describing the evolution of two broader bandwidth surface gravity wave packets propagating in two different directions in deep water. The equations, being derived for broader bandwidth wave packets, are applicable to more realistic ocean wave spectra in crossing sea states. The two coupled evolution equations derived here have been used to investigate the instability of two uniform wave trains propagating in two different directions. We have shown in figures the behaviour of the growth rate of instability of these uniform wave trains for unidirectional as well as for bidirectional perturbations. The figures drawn here confirm the fact that modulational instability in crossing sea states with broader bandwidth wave packets can lead to the formation of freak waves.

scholarly journals Fourth order nonlinear evolution equations for gravity-capillary waves in the presence of a thin thermocline in deep water

2002 ◽  
Vol 43 (4) ◽  
pp. 513-524 ◽  
Author(s):  
Suma Debsarma ◽  
K.P. Das
Keyword(s):  
Fourth Order ◽  
Deep Water ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Capillary Wave ◽  
Capillary Waves ◽  
Nonlinear Evolution Equations ◽  
Marginal Stability ◽  
Stability And Instability ◽  
The Stability

AbstractFor a three-dimensional gravity capillary wave packet in the presence of a thin thermocline in deep water two coupled nonlinear evolution equations correct to fourth order in wave steepness are obtained. Reducing these two equations to a single equation for oblique plane wave perturbation, the stability of a uniform gravity-capillary wave train is investigated. The stability and instability regions are identified. Expressions for the maximum growth rate of instability and the wavenumber at marginal stability are obtained. The results are shown graphically.

scholarly journals Fourth-order nonlinear evolution equations for surface gravity waves in the presence of a thin thermocline

1997 ◽  
Vol 39 (2) ◽  
pp. 214-229 ◽  
Author(s):  
Sudebi Bhattacharyya ◽  
K. P. Das
Keyword(s):  
Growth Rate ◽  
Fourth Order ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Maximum Growth ◽  
Single Equation ◽  
Maximum Growth Rate ◽  
Nonlinear Evolution Equations ◽  
Wave Number ◽  
Coupled Equations

AbstractTwo coupled nonlinear evolution equations correct to fourth order in wave steepness are derived for a three-dimensional wave packet in the presence of a thin thermocline. These two coupled equations are reduced to a single equation on the assumption that the space variation of the amplitudes takes place along a line making an arbitrary fixed angle with the direction of propagation of the wave. This single equation is used to study the stability of a uniform wave train. Expressions for maximum growth rate of instability and wave number at marginal stability are obtained. Some of the results are shown graphically. It is found that a thin thermocline has a stabilizing influence and the maximum growth rate of instability decreases with the increase of thermocline depth.

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