The linear Miles-Phillips model of the coupling between water waves and a turbulent wind field is generalized to include fluctuations in the air-sea coupling parameter in the dynamic equations. The model is solved exaclty, assuming the fluctuations to be delta correlated in time. The energy spectrum of the water wave field is shown to grow exponentially at rates higher than those predicted by the linear Miles-Phillips model. These increased initial growth rates are consistent with those observed in open ocean measurements.