The detailed mathematical study of the recent paper by Sajjadi et al. [21] is presented. The mathematical development considered by them, for unsteady growing monochromatic waves is also extended to Stokes waves. The present contribution also demonstrates agreement with the pioneering work of Belcher and Hunt [1] which is valid in the limit of the complex part of the wave phase speed  It is further shown that the energy-transfer parameter and the surface shear stress for a Stokes wave revert to a monochromatic wave when the second harmonic is excluded. Furthermore, the present theory can be used to estimate the amount of energy transferred to each component of nonlinear surface waves on deep water from a turbulent shear flow blowing over it. Finally, it is demonstrated that in the presence of turbulent eddy viscosity, the Miles [13] critical-layer does not play an important role. Thus, it is concluded that in the limit of zero growth rate, the effect of the wave growth arises from the elevated critical-layer by finite turbulent diffusivity, so that the perturbed flow and the drag force are determined by the asymmetric and sheltering flow in the surface shear layer and its matched interaction with the upper region.

wave generation, Air-Sea interactions, surface water waves, turbulence model.