Abstract
The Miles' quasi laminar theory of waves generation by wind in finite depth h is presented. In this context, the fully nonlinear Green–Naghdi model equation is derived for the first time. This model equation is obtained by the non perturbative Green–Naghdi approach, coupling a nonlinear evolution of water waves with the atmospheric dynamics which works as in the classic Miles' theory. A depth-dependent and wind-dependent wave growth γ is drawn from the dispersion relation of the coupled Green–Naghdi model with the atmospheric dynamics. Different values of the dimensionless water depth parameter $\delta = \frac{gh}{U_1}$, with g the gravity and $U_1$ a characteristic wind velocity, produce two families of growth rate γ in function of the dimensionless theoretical wave-age $c_0$: a family of γ with h constant and $U_1$ variable and another family of $\gamma$ with $U_1$ constant and $h$ variable. The allowed minimum and maximum values of $\gamma$ in this model are exhibited.