  The detailed mathematical study of the recent paper by Sajjadi, Hunt and
Drullion (2014) is pre- sented. The mathematical developement considered by
them, for unsteady growing monochro- matic waves is also extended to Stokes
waves. The present contribution also demonstrates agree- ment with the
pioneering work of Belcher and Hunt (1993) which is valid in the limit of the
complex part of the wave phase speed \c_i \downarrow 0. It is further shown
that the energy-transfer parameter and the surface shear stress for a Stokes
wave reverts 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 (1957) 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 is
determined by the asymmetric and sheltering flow in the surface shear layer and
its matched interaction with the upper region.
