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Modulational instability of obliquely interacting capillary-gravity waves over infinite depth

Manna S., Dhar A. K.

Archives of Mechanics

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2021

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Vol. 73, nr 5-6

583--598

EN

Two coupled time-dependent two dimensional nonlinear Schrödinger equations have been derived using multiscale expansion for two nonlinearly interacting capillary-gravity waves over an infinite depth of water. These equations are then utilised to discuss the modulational (Benjamin-Feir) instability of two Stokes wavetrains due to unidirectional and bidirectional perturbations. It is found from the graphs and the three-dimensional contour plots that the rate of growth of instability for two wave packets interacting obliquely is higher than the instance of modulation of one wave packet. We have likewise examined the influence of capillarity on modulational instability.

2

The effect of randomness on the stability of capillary gravity waves in the presence of air flowing over water

Majumder D. P., Dhar A. K.

International Journal of Applied Mechanics and Engineering

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2015

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Vol. 20, no. 4

835--855

EN

A nonlinear spectral transport equation for the narrow band Gaussian random surface wave trains is derived from a fourth order nonlinear evolution equation, which is a good starting point for the study of nonlinear water waves. The effect of randomness on the stability of deep water capillary gravity waves in the presence of air flowing over water is investigated. The stability is then considered for an initial homogenous wave spectrum having a simple normal form to small oblique long wave length perturbations for a range of spectral widths. An expression for the growth rate of instability is obtained; in which a higher order contribution comes from the fourth order term in the evolution equation, which is responsible for wave induced mean flow. This higher order contribution produces a decrease in the growth rate. The growth rate of instability is found to decrease with the increase of spectral width and the instability disappears if the spectral width increases beyond a certain critical value, which is not influenced by the fourth order term in the evolution equation.

3

Fourth order nonlinear evolution equation for interfacial gravity waves in the presence of air flowing over water and a basic current shear

Majumder D. P., Dhar A. K.

International Journal of Applied Mechanics and Engineering

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2015

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Vol. 20, no. 3

517--530

EN

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.

4

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

Dhar A. K., Mondal J.

International Journal of Applied Mechanics and Engineering

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2015

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Vol. 20, no. 2

267--282

EN

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.

5

Stability of small but finite amplitude interfacial capillary gravity waves for perturbations in two horizontal directions

Majumder D. P., Dhar A. K.

International Journal of Applied Mechanics and Engineering

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2011

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Vol. 16, no 2

425-434

EN

By exact numerical computation Yuen (1984) obtained regions of type-I instability for waves propagating at the interface of two superposed fluids of infinite thickness in which the upper fluid has a constant streaming velocity. In the present paper it is shown that the long wavelength part of these instability regions can be obtained analytically from a fourth order nonlinear evolution equation for small but finite amplitude interfacial capillary gravity waves in the presence of air flowing over water.

6

Stability analysis from fourth order nonlinear evolution equation for two stokes wave trains in deep water in the presence of air flowing over water

Majumder D. P., Dhar A. K.

International Journal of Applied Mechanics and Engineering

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2009

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Vol. 14, no 4

989-1008

EN

Fourth order nonlinear evolution equations are derived for two Stokes wave trains in deep water in the presence of air flowing over water. The importance of the fourth order term in the evolution equation was pointed out by Dysthe (1979). Stability analysis is then made for uniform two Stokes wave trains in the presence of air flowing over water. From these evolution equations the expressions for the maximum growth rate of instability, the wave number at marginal stability and the wave number separation of fastest growing side band are derived and graphs are plotted for the above three expressions against the wave steepness. Significant improvements can be achieved from the results obtained from the two coupled third order nonlinear Schrödinger equations.

7

Stability analysis from fourth order evolution equation for deep water capillary-gravity waves in the presence of air flowing over water

Majumder D. P., Dhar A. K.

International Journal of Applied Mechanics and Engineering

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2009

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Vol. 14, no 2

433-442

EN

Fourth order nonlinear evolution equations, which are a good starting point for the study of nonlinear water waves as first pointed out by Dysthe (1979) and later elaborated by Janssen (1983), are derived for deep water capillary-gravity waves in the presence of air flowing over water. Stability analysis is then made for a uniform Stokes capillary gravity wave train. Graphs are plotted for the maximum growth rate of instability, the frequency at marginal stability and the frequency separation for fastest growing side-band component as a function of wave steepness. Significant deviations are noticed from the results obtained from the third-order evolution equation, which is the nonlinear Schrödinger equation.

8

The effect of randomness on stability of surface gravity waves from fourth order nonlinear evolution equation

Dhar A. K., Das K. P.

International Journal of Applied Mechanics and Engineering

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2001

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Vol. 6, no 1

11-34

EN

A fourth order nonlinear evolution equation, which is a good starting point for the study of nonlinear water waves of wave-steepness up to 0.25, is used here to investigate the effect of randomness on stability of deep-water surface gravity waves in the presence of wind blowing over water. A spectral transport equation for narrow band Gaussian surface wave is derived. With the use of this transport equation stability analysis is made for an initial homogeneous wave spectrum having a very simple normal form to small oblique long wave length perturbations for a range of spectral widths. An expression for the growth rate of instability is obtained, in which higher order contribution comes from only one of the fourth order terms in the evolution equation, which is responsible for wave-induced mean flow. This higher order contribution in this expression for growth rate of instability produces a decrease in the growth rate. The growth rate of instability is found to decrease with the increase of spectral width and ultimately the instability disappears if the spectral width increases beyond a certain critical value, which is not influenced by the fourth order terms in the evolution equation.

9

A fourth order evolution equation for capillary-gravity waves including the effects of wind input and shear in the water current

Dhar A. K., Das K. P.

Applied Mechanics and Engineering

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1999

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Vol. 4, no 1

5-24

EN

A fourth order nonlinear evolution equation is derived for capillary gravity waves in deep water including the effect of a surface drift current in the water and shear in the air flow. From this evolution equation instability conditions are derived for a uniform capillary-gravity wave train. Graphs are plotted showing the maximum growth rate of instability and instability regions for weakly damped (linearly) and weakly growing (linearly) waves for some different values of friction velocity of the flow. From these graphs it is found that the effect of the wind input and shear in water current is to produce a decrease in the growth rate for weakly damped (linearly) waves and to produce an increase in growth rate for weakly growing (linearly) waves. The shear in the water current and the wind input are found to produce a shrinkage in the instability regions.