Based on a previous linear theory, turbulent air flow over a surface gravity wave of finite amplitude is studied analytically by the methods of matched asymptotic expansions and multiple-scale analysis. In particular, an initial-value problem for weakly nonlinear waves is solved, where the initial conditions are prescribed by a single Stokes wave, displacing the water surface. The water is inviscid and incompressible, and there is no mean shear current. Wave-wave interactions are not taken into account. The validity of the theory is restricted to slow waves and small drag coefficient.