Weakly nonlinear modulation of interfacial gravity-capillary waves

The modified nonlinear envelope equation of interfacial gravity-capillary waves in a two-layer fluid of infinite depths for broader bandwidth with a uniform velocity of the upper fluid is derived. The derivation is made from Zakharov’s integral equation by relaxing the narrow wave bandwidth restriction to make it more applicable for utilization of a realistic sea wave spectrum. From this equation instability regions are drawn in the perturbed wave number space. The modified equation limits the wave bandwidth of a uniform Stokes wave in an excellent agreement with the accurate numerical results. We have also drawn the growth rate of modulational instability for the case of pure capillary waves.