Gravity-capillary waves modulated by linear shear flow in arbitrary water depth*

Considering that the fluid is inviscid and incompressible and the flow is irrotational in a fixed frame of reference and using the multiple scale analysis method, we derive a nonlinear Schrodinger equation (NLSE) describing the evolution dynamics of gravity-capillary wavetrains in arbitrary constant depth. The gravity-capillary waves (GCWs) are influenced by a linear shear flow (LSF) which consists of a uniform flow and a shear flow with constant vorticity. The modulational instability (MI) of GCWs with the LSF is analyzed using the NLSE. The MI is effectively modified by the LSF. In infinite depth, there are four asymptotes which are the boundaries between MI and modulational stability (MS) in the instability diagram. In addition, the dimensionless free surface elevation as a function of time for different dimensionless water depth, surface tension, uniform flow and vorticity is exhibited. It is found that the decay of free surface elevation and the steepness of free surface amplitude change over time, which are greatly affected by the water depth, surface tension, uniform flow and vorticity.