We study numerically solutions of Hasselmann equation for different wind input terms, including those used in operational wave prediction models, for limited fetch growth statement. Some of the wind input terms are well physically justified and some are not, but simulation results with first ones exhibit remarkable universal properties -- self-similar behavior of total energy and frequency as a functions of the fetch distance, described by power laws obtained from self-similarity analysis of Hasselmann equation.
It is more remarkable that even for not so well physically justified wave input terms, we do observe absence of good self-similar behavior of the total energy and spectral peak frequency, but still can construct specific "magic" combinations of specific powers of total energy and peak frequency, obtained from self-similar analysis of Hasselmann equation for limited fetch waves growth, which are preserved with good accuracy along the fetch. We explain this "strange" coincidence as locally realized self-similar regimes in individual locations of the fetch.
In summary, we observe the universality of the wind wave growth for wide spectrum of the wind input terms historically developed for Hasselmann equation, exhibiting itself in either direct manifestation in the form of self-similar laws of the total energy and peak frequency as function of fetch, or preservation of the specific combinations of the powers of theoretical self-similar solutions.
In our opinion, the observed properties emphasize the key role of nonlinearity in the evolution of wave ocean surface.