K. P. Das

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

Fourth‐order nonlinear evolution equations, which are a good starting point for the study of nonlinear water waves as first pointed out by Dysthe [Proc. R. Soc. London Ser. A 369, 105 (1979)] and later elaborated by Janssen [J. Fluid Mech. 126, 1 (1983)], are derived for a deep‐water surface gravity wave packet in the presence of a second wave packet. 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. Stability analysis is made for a uniform Stokes wave train in the presence of a second wave train. Graphs are plotted for maximum growth rate of instability and for wave number at marginal stability against wave steepness. Significant deviations are noticed from the results obtained from the third‐order evolution equations which consist of two coupled nonlinear Schrodinger equations.

A fourth‐order evolution equation for deep water surface gravity waves in the presence of wind blowing over water

The stability of a train of nonlinear surface gravity waves in deep water in the presence of wind blowing over water is considered. An evolution equation is derived for the wave envelope that is correct to fourth order in the wave steepness. The importance of the fourth‐order term in the evolution equation was pointed out by Dysthe [Proc. R. Soc. London Ser. A 369, 105 (1979)]. From this evolution equation the expressions for the maximum growth rate of the instability and the frequency at marginal stability are derived, and graphs are plotted for those two expressions against the wave steepness.

A higher-order nonlinear evolution equation for broader bandwidth gravity waves in deep water

From Zakharov’s integral equation a nonlinear evolution equation for broader bandwidth gravity waves in deep water is obtained, which is one order higher than the corresponding equation derived by Trulsen and Dysthe [“A modified nonlinear Schrodinger equation for broader bandwidth gravity waves on deep water,” Wave Motion 24, 281 (1996)]. The instability regions in the perturbed wave-number space for a uniform Stokes wave obtained from this equation are in surprisingly good agreement with the exact results obtained by McLean et al. [“Three dimensional instability of finite amplitude water waves,” Phys. Rev. Lett. 46, 817 (1981)].

Fourth-order nonlinear evolution equations for a capillary-gravity wave packet in the presence of another wave packet in deep water

Starting from the Zakharov integral equation, two coupled fourth-order nonlinear equations have been derived for the evolution of the amplitudes of two capillary-gravity wave packets propagating in the same direction. The two evolution equations are used to investigate the stability of a uniform capillary-gravity wave train in the presence of another having the same group velocity. The relative changes in phase velocity of each uniform wave train due to the presence of the other one have been shown in figures for different wave numbers. The condition of instability of a wave of greater wavelength in the presence of a wave of shorter wavelength is obtained. It is observed that the instability region for a surface gravity wave train in the presence of a capillary-gravity wave train expands with the increase of wave steepness of the capillary-gravity wave train. It is found that the presence of a uniform capillary-gravity wave train causes an increase in the growth rate of instability of a uniform surface gr...

A Korteweg–de Vries equation modified by viscosity for waves in a channel of uniform but arbitrary cross section

A KdV equation modified by viscosity is derived for weakly nonlinear long waves propagating in a channel of uniform but arbitrary cross section. The case of high Reynolds number is considered, and the method of matched asymptotic expansion is employed. The equation derived here is found to be similar to the corresponding equation for a two‐dimensional layer of liquid derived by previous authors. The only difference is that the dispersive, nonlinear, and viscous terms are multiplied by constants dependent on the cross section geometry of the channel.

Modulational instability in crossing sea states over finite depth water

Nonlinear evolution equations are derived in a situation of crossing sea states characterized by water waves having two different spectral peaks. The nonlinear evolution equations derived here are valid for any water depth except for shallow water depth case. These evolution equations are then employed to study the instability properties of two Stokes wave trains considering both unidirectional and bidirectional perturbations. Figures have been plotted showing the growth rate of instability for various depths of water and for different values of the angle of interaction of the two wave systems. All the figures serve as an evidence to the fact that freak waves can be formed as a result of modulational instability in crossing sea states over finite depth water. It is observed that the growth rate of instability in crossing sea states situation over finite depth water is much higher than that for infinite depth case and it increases with the decrease of the depth of water.

The Korteweg–de Vries equation modified by viscosity for waves in a two‐layer fluid in a channel of arbitrary cross section

Using a two‐layer fluid model, Korteweg–de Vries (KdV) equations modified by viscosity are derived that describe weakly nonlinear long waves propagating along a channel of uniform but arbitrary cross section. Equations are deduced for both surface waves and internal waves. The case of high Reynolds number is considered, and the method of matched asymptotic expansion is employed. The coefficients of the KdV equation, which depend on the geometry of the channel cross section, are determined exactly for a rectangular cross section. Some particular cases including the Boussinesq limit are considered.

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

Asymptotically exact nonlocal fourth-order nonlinear evolution equations are derived for two counterpropagating capillary-gravity wave packets on the surface of water of infinite depth. On the basis of these equations a stability analysis is made for a uniform standing capillary-gravity wave for longitudinal perturbation. The instability conditions and an expression for the maximum growth rate of instability are obtained. Significant deviations are noticed between the results obtained from third-order and fourth-order nonlinear evolution equations.

Wind-forced modulations in crossing sea states over infinite depth water

The present work is motivated by the work of Leblanc [“Amplification of nonlinear surface waves by wind,” Phys. Fluids 19, 101705 (2007)] which showed that Stokes waves grow super exponentially under fair wind as a result of modulational instability. Here, we have studied the effect of wind in a situation of crossing sea states characterized by two obliquely propagating wave systems in deep water. It is found that the wind-forced uniform wave solution in crossing seas grows explosively with a super-exponential growth rate even under a steady horizontal wind flow. This is an important piece of information in the context of the formation of freak waves.

Ion‐acoustic solitons in a cylindrical plasma‐loaded waveguide

Nonlinear behavior of ion‐acoustic waves propagating along a highly nonisothermal magnetized plasma filled cylindrical waveguide is investigated. A Korteweg–de Vries equation is derived to describe the weakly nonlinear behavior of the waves. The effects of a finite boundary on the propagation of ion‐acoustic solitons are analyzed.