Analytical formulas representing the idealized growth of wind-waves, duration-limited and fetch-limited

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The SMB method is an historical wave-forecasting method, which is based on fetch-limited and duration-limited growth of wind-waves. However, wind-wave growth envisaged by the words fetch-limited and duration-limited has been ambiguous, and clear explanation seems to have been absent. A simplest conceptual model is proposed here for the idealized growth of wind-waves, where the model equation is based on several assumptions: (1) a well-established fetch relation for small to moderate fetches, (2) ubiquitous equilibrium governed by the 3/2-power law of Toba, (3) propagation of wind-waves as a whole at the group velocity of waves with the significant wave period, (4) presence of the upper bound of the significant wave period, and (5) tanh-type fetch dependence of wave energy. In particular, the fetch relation of (5) is a modified Wislon's formula I, where variables are normalized by a single scale of fectch and upper bounds of wind-waves. The model yields analytical formulas that clearly illustrate the growth of wind-waves, both fetch-limited and duration-limited. The result suggests that wave growth is better described by substantial duration t_<sd> than the conventional discrimination of duration-limited and fetch-limited; t_<sd> allows a unified description of wave-growth. Related problems are examined and discussed as well: fetch relations at large fetches, normalized and approximate formulas of fetch relations, momentum and energy input, and so on.

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