The growth of duration-limited wind waves

Laboratory measurements have been made of the one-dimensional spectra of the duration-limited wind waves which are generated when a wind abruptly begins to blow over a water surface, maintaining a constant speed during the succeeding period of time. The duration dependences of the wave energy E and the spectral peak frequency f m determined from the measured spectra are slightly different from those inferred from the fetch dependences of these quantities. The normalized spectra of the duration-limited wind waves are also slightly different from those of fetch-limited wind waves: the concentration of the normalized spectral energy near the spectral peak frequency is smaller, in many cases, for the duration-limited wind waves than for fetch-limited wind waves. The exponential growth rates β of the duration-limited wind-wave spectra are generally larger than those of fetch-limited wind-wave spectra. Furthermore, both for the duration-limited wind waves and for fetch-limited wind waves the exponential growth rate has a behaviour which is different from the empirical formula of Snyder & Cox (1966). A new empirical formula for the growth rate of the wave spectrum is proposed, from which the empirical formula of Snyder & Cox (1966) can be derived as a special case. Agreement between the new empirical formula and the experimental results is satisfactory for fetch-limited wave spectra, but is confined to the qualitative features for the duration-limited wave spectra.