7.2.4 The Transiticxi between Linear and Exponential Growth 



Of considerable interest is the transition distance at which the processes 

 of linear growth and exponential growth are equal (cc = ^ F). As postualted by 

 Phillips and Katz (1961), this is the fetch distance required for a particular spectral 

 component to occupy the steep forward face of the energy spectrum. The point has 

 arisen as to how good an assumption this is. To answer this question we have used 

 Figure 16 and the closed form solution of (4) for the downwind component to arrive 

 at estimates of the transition distance. These were then compared with the fetch 

 distances shown on Figure 17 (to be discussed in the next paragraph). The result was 

 that in 1 1 out of 13 estimates the fetch distance was roughly 2 to 4 Hmes greater than 

 the transition distance. Snyder and Cox found a typical factor of 7 although their 

 data was not particularly suited to making the estimate. At any rate, it would appear 

 that the transition distance is not equivalent to the fetch distance. 



The estimates of fetch distances obtained from the present data are shown in 

 Figure 17, a representation which is similar to that used by Phillips and Katz. Also 

 shown on this Figure are the measurements of Burling (1955), Kinsman (1960), and 

 Snyder and Cox (1966). The distances are given in numbers of wave lengths (fetch 

 (x)/ wavelength (L)) versus wave speed (c) over wind speed (W)the latter measured at a 

 height of one wave length. The wind speed actually used in Figure 17 is an average 

 of values observed by the aircraft, for there is no desire on the part of the authors to 

 extend the logrithmic profile to the heights required. The uncertainty in the wind 

 speed is thought to be of order 10 percent, which does not materially affect the results. 



It will be seen that most of the data fits well together. The data are generally 

 not in agreement with the theoretical curves of Phillips and Katz (not shown) which 

 were derived from the theory of Miles (1957). However, the 1961 work of Phillips 

 and Katz has been amended by Phillips (personal communication) so that there is now, 

 at least, a qualitative agreement between theory and observation.. It will be inter- 

 esting to compare the quantitative details of Phillips' forthcoming work with the data 

 of Figure 17. Finally, the data shown on Figure 17 lend support to the contention of 

 Snyder and Cox that there is a universal relation between the fetch distance (measured 

 in wavelengths) at which a component occupied the "steep" forward face of the 

 spectrum and the wind speed measured in units of phase velocity. 



7.3 Wave Dissipation 



A detailed attempt to account for the limitations of wave growth as observed 

 in this experiment will not be undertaken. Instead, only the end result, the equil- 

 ibrium range of the wave spectrum, will be discussed. As mentioned in Section 7.1, 

 (iii), spectral components with frequencies somewhat higher than the wind frequency 

 (fw) seem to grow past or "overshoot" their eventual equilibrium value. This fact is 



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