SECT. 5] RIPPLES 723 



The constant j8 = 0.0015 has been evaluated from Burling's observations on a 

 reservoir and from wave-pole measurements at sea. It is independent of wind 

 speed. 



The upper frequency limit of validity for this spectrum is not precisely 

 known. Burling's observations extend to a frequency of 1.9 c/s. 



The spectrum of wave slopes can now be calculated provided one knows the 

 relation between wave number k and frequency. Since the waves in the equih- 

 brium range are limited by non-linear processes and are somewhat influenced 

 by surface tension, one supposes that the formula for infinitesimal gravity 

 waves, 



4772/2 = g]c, (5) 



will be only approximately correct. Nevertheless we shall use it. The slope 

 spectrum (regardless of the direction of steepest ascent) is 



S^+8y = k^A, (6) 



Combining (4), (5) and (6) yields 



S^+Sy = iS/-i. (7) 



The slope spectrum times angular frequency is, therefore, a constant, j8, in 

 the equilibrium range, as shown in Fig. 2. The tank measurements of the 

 spectrum of a single component of slope, Sx, will necessarily give a lower limit 

 to an estimate oiSx + Sy. (Cox's measurements of the ratio SyjSx show that this 

 ratio is about 0.2 but varies with frequency.) 



The low frequency peaks on Cox's spectra are seen to be too high to fit on 

 smoothly with equation (7): This behavior is, however, consistent with the 

 limitation of gravity waves by non-linear processes. Thus the waves in the 

 laboratory channel tend to grow until the slopes reach limiting values. Since 

 only short waves have time to grow appreciably, the spectrum of gravity waves 

 covers only a narrow range ; therefore a higher spectral density is required on 

 the short laboratory fetch to reach the same slope as found after a long fetch 

 on the sea. 



These considerations lead to the prediction that the low frequency peak 

 shown by spectra offSx for short fetches will not exist for long fetches. What 

 then of the high frequency peak? If ripples are produced solely as a by-product 

 of non-linear interactions of gravity waves, then one would expect the ripple 

 spectrum to decrease with increasing fetch because of the increasing average 

 separation between sharp crests of gravity waves. On the other hand, if ripples 

 are produced directly by the wind then one would expect the ripple spectrum 

 to be only slightly changed with fetch. 



Comparison of mean square slopes measured at sea and in the laboratory 

 favors the second interpretation. 



The mean square total slope, Sx^ + Sy^, has been measured for a 300-m fetch 

 on a river (Schooley, 1954) and on the open sea with large fetch (Cox and 

 Munk, 1954). Both measurements are consistent with the mean square slope 



