Fourth‐order nonlinear evolution equation for two Stokes wave trains in deep water

1991 ◽  
Vol 3 (12) ◽  
pp. 3021-3026 ◽  
Author(s):  
A. K. Dhar ◽  
K. P. Das
Keyword(s):  
Evolution Equation ◽  
Fourth Order ◽  
Deep Water ◽  
Nonlinear Evolution Equation ◽  
Nonlinear Evolution ◽  
Stokes Wave ◽  
Wave Trains

scholarly journals Effect of Capillarity on Fourth Order Nonlinear Evolution Equation for Two Stokes Wave Trains in Deep Water in the Presence of Air Flowing Over Water

2015 ◽  
Vol 20 (2) ◽  
pp. 267-282
Author(s):  
A.K. Dhar ◽  
J. Mondal
Keyword(s):  
Fourth Order ◽  
Deep Water ◽  
Water Waves ◽  
Nonlinear Evolution ◽  
Wave Train ◽  
Nonlinear Evolution Equations ◽  
Wave Number ◽  
Stokes Wave ◽  
Instability Wave ◽  
Starting Point

Abstract Fourth order nonlinear evolution equations, which are a good starting point for the study of nonlinear water waves, are derived for deep water surface capillary gravity waves in the presence of second waves in which air is blowing over water. Here it is assumed that the space variation of the amplitude takes place only in a direction along which the group velocity projection of the two waves overlap. A stability analysis is made for a uniform wave train in the presence of a second wave train. Graphs are plotted for the maximum growth rate of instability wave number at marginal stability and wave number separation of fastest growing sideband component against wave steepness. Significant improvements are noticed from the results obtained from the two coupled third order nonlinear Schrödinger equations.

scholarly journals Fourth Order Nonlinear Evolution Equation For Interfacial Gravity Waves In The Presence Of Air Flowing Over Water And A Basic Current Shear

2015 ◽  
Vol 20 (3) ◽  
pp. 517-530
Author(s):  
D.P. Majumder ◽  
A.K. Dhar
Keyword(s):  
Evolution Equation ◽  
Gravity Waves ◽  
Fourth Order ◽  
Water Waves ◽  
Nonlinear Evolution Equation ◽  
Nonlinear Evolution ◽  
Wave Train ◽  
Wave Number ◽  
Marginal Stability ◽  
Starting Point

Abstract A fourth order nonlinear evolution equation, which is a good starting point for the study of nonlinear water waves as first pointed out by Dysthe (1979) is derived for gravity waves propagating at the interface of two superposed fluids of infinite depth in the presence of air flowing over water and a basic current shear. A stability analysis is then made for a uniform Stokes gravity wave train. Graphs are plotted for the maximum growth rate of instability and for wave number at marginal stability against wave steepness for different values of air flow velocity and basic current shears. Significant deviations are noticed from the results obtained from the third order evolution equation, which is the nonlinear Schrödinger equation.

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