STABILITY ANALYSIS FROM FOURTH ORDER NONLINEAR EVOLUTION EQUATIONS FOR TWO CAPILLARY GRAVITY WAVE PACKETS IN THE PRESENCE OF WIND OWING OVER WATER.

Subject Areas : International Journal of Mathematical Modelling & Computations

J. Mondal ¹ ( Department of Mathematics,Indian Institute of Engineering Science and Technology, P. O. Botanical Garden, Howrah - 711103, West Bengal, India; )
K. Dhar ² ( Department of Mathematics,IIEST, Howrah - 711103, West Bengal, India. )

Received: 2016-07-13 Accepted : 2016-07-13 Published : 2016-03-01

Keywords:

Abstract :

Asymptotically exact and nonlocal fourth order nonlinear evolution equations are derived for two coupled fourth order nonlinear evolution equations have been derived in deep water for two capillary-gravity wave packets propagating in the same direction in the presence of wind flowing over water.We have used a general method, based on Zakharov integral equation.On the basis of these evolution equations,the stability analysis is made for a uniform capillary gravity wave train in the presence of another wave train having the same group velocity. Instability condition is obtained and graphs are plotted for maximum growth rate of instability and for wave number at marginal stability against wave steepness for some different values of dimensionless wind velocity.

References:

T.B.Benjamin and J.E.Feir, The disintegration of wave trains on deep water, Part I Theory, J.Fluid

 

Mech 27(1967)417.

 

S.Bhattacharyya and K.P. Das, Fourth order nonlinear evolution equations for surface gravity waves

 

in the presence of a thin thermocline, J.Austral, Math. Soc.Sec.B 39(1997)214.

 

S.Debsarma and K.P. Das Fourth order nonlinear evolution equations for gravity-capillary waves in

 

the the presence of a thin thermocline in deep water, ANZIAM J. 43,(2002)513.

 

A.K.Dhar and K.P. Das, A fourth order nonlinear evolution equation for deep water surface gravity

 

waves in the presence of wind blowing over water, Phys. Fluids A 2(5)(1990)778- 783.

 

A.K.Dhar and K.P.Das, Stability analysis from fourth order evolution equation for small but finite

 

amplitude interfacial waves in the presence of a basic current shear, J.Austral. Math.Soc. Ser.B

 

(1994)348-365.

 

A. K. Dhar and K.P. Das, Fourth order nonlinear evolution equation for two Stokes wave trains in

 

deep water, Phys. Fluids A 3(12)(1991)3021.

 

A.K.Dhar and K.P. Das, Effect of capillarity on f ourth-order nonlinear evolution equations for two

 

Stokes wave trains in deep water, J.Indian Inst.Sci. 73(1993)579

 

A. K. Dhar and J.Mondal,,Stability analysis from fourth order evolution equation for counterpropagating

 

gravity wave packets in the presence of wind owing over water, ANZIAM J. 56(E)(2015) pp

 

-49

 

K. B. Dysthe, Note on a modification to the nonlinear Schrdinger equation for application to deep

 

water waves, Proc. R. Soc. Lond.A 369(1979) 105.

 

P.A.E.M.Janssen, On a fourth order envelope equation for deep water waves, J.Fluid Mech.

 

(1983)1.

 

T.Hara and C.C.Mei, Frequency downshift in narrowbanded surface waves under the influence of

 

wind, J.Fluid Mech. 230(1991)429.

 

T.Hara and C.C.Mei, Wind effects on nonlinear evolution of slowly varying gravity-capillary waves,

 

J.Fluid Mech. 267(1994)221.

 

S. J. Hogan, The fourth order evolution equation for deep water gravity capillary waves, Proc.R.Soc.

 

Lond.A 402(1985)359.

 

S. Ahmadi, Applications of Partial Differential Equations in Stability Index and Critical Length in

 

Avalanche Dynamics,IJM2C, 02 - 02 (2012) 137 -144.

 

S. J. Hogan,I.Gruman and M.Stiassine, On the changes in phase speed of one train of water waves

 

in the presence of another., J.Fluid Mech. 192(1988)97.

 

V.P. Krasitskii, On reduced equations in the Hamiltonian theory of weakly nonlinear surface waves,

 

J.Fluid Mech. 273(1994)1.

 

M.S.Longuet-Higgins and O.M.Philips, Phase velocity effects interraction, J.Fluid Mech.

 

(1962)333.