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 δ = gh/U1, with g the gravity and U1 a characteristic wind velocity, produce two families of growth rate γ in function of the dimensionless theoretical wave-age c0: a family of γ with h constant and U1 variable and another family of γ with U1 constant and h variable. The allowed minimum and maximum values of γ in this model are exhibited.
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