The existing wind wave growth formulas (WWGFs) in form of power function are analyzed. It is shown that these WWGFs after eliminating the fetch are uniformly consistent with the 3/2 power law, although originally there are considerable discrepancy among them. It is found that the similarity of integrals of the three wind input parameterizations exists by analogy. Likewise it is found that the similarity of integrals of the three dissipation parameterizations exists by analogy. The fractional fetch power law for wind wave growth in deep water is presented by combining the energy balance equation for significant wave with the similarities of wind input and dissipation parameterizations, after invoking the 3/2 power law. The semi-empirical WWGFs have been proposed by fitting the existing WWGFs with the derived fetch law. The proposed formulas are agreeably consistent with the available observational data.
So far a great deal of effort has been paid to investigate the fetch-limited growth of wind wave with field and laboratory measurement s. By fitting the observational data many wind wave growth formulas (WWGFs) have been proposed, most of which are in form of power function (Mitsuyasu et al., 1971; Hasselmann et al., 1973; Davidan, 1980; Kahma, 1981; Donelan et al., 1985; Dobson et al., 1989; Evans and Kibblewhite, 1990; Babanin and Soloviev, 1998).
Nevertheless these formulas are almost uniformly consistent with an alternative equivalent form of the 3/2 power law proposed by Toba (1972) by eliminating the dimensionless fetch from each pair of formulas (Guan and Sun, 2001). How to understand the contradictory phenomenon above? A plausible interpretation is given as follow. It is known that a pair of WWGFs consists of wind speed at 10m-height above sea surface, significant wave height (or variance of surface elevation), period (or peak frequency), and fetch.