Abstract
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.
Highlights
The study of evolution of weakly nonlinear surface gravity waves in crossing sea states has attracted considerable interest in recent years
The problem of weakly nonlinear interaction between two wave systems propagating in two different directions has been studied by some authors ([5,6,7]) as a possible mechanism resulting in freak wave generation
In Onorato et al [5] and in Shukla et al [6], it is concluded that freak waves can be formed due to weakly nonlinear interaction in crossing sea states
Summary
The study of evolution of weakly nonlinear surface gravity waves in crossing sea states has attracted considerable interest in recent years. The reason is that, at fourth order, the wave-induced mean flow terms appear and these terms modify considerably the growth rate of instability ([8]) Keeping this in view, Gramstad and Trulsen [9] have derived two coupled modified nonlinear Schrodinger (hereafter referred to as CMNLS) equations that describe the evolution of two two-dimensional narrow band wave systems with different directions of propagation. We have presented the fourth order nonlinear evolution equations in crossing sea states for narrow band wave packets These equations are in a form different from those derived by Gramstad and Trulsen [9], in the sense that the two equations do not involve the wave-induced mean flow velocity potential explicitly. Shrinkage is observed in the unstable region in the perturbed wavenumber plane when crossing sea states with broader bandwidth wave packets is considered
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