On the growth and decay of a deep water wave component

A wave component grows exponentially in time owing to some feedback interaction with the mean wind above the sea. Initially, this growth is opposed primarily by wave breaking. By assuming that the rate of loss of energy at a given wave frequency is proportional to the mean square slope of waves at lower frequencies, an equation is derived for the temporal behaviour of a wave. One prediction of this equation is that a growing wave overshoots its final equilibrium energy level: a phenomenon observed to occur in sea waves. 1