728 



[chap. 21 



Roll explains the initial formation of short ripples as the "bow waves" of 

 traveling air-pressure disturbances which are convected over the water surface 

 by the wind. Since the wind speed is always large compared to the phase 

 velocity of the ripple, the ripples' crests are formed almost parallel to the wind. 

 Phillips (1957) has constructed a theory of wave generation based on similar 

 ideas. It appears, however, from laboratory channel studies (Cox, 1958) that 

 the mechanism is not capable of explaining ripples in the second stages of 

 development. (We associate the transition between Roll's two stages with the 

 pronounced increase in mean square slope of waves observed in the laboratory 

 (Fig. 6). Presumably the ripples of the first stage are of such small slope that 

 they make an immeasurable contribution to the mean square slope.) Estimates 

 made of the angular distribution of ripples and wavelets in the laboratory 

 show that the waves travel much more nearly in the direction of the wind than 

 is consistent with Phillips' hypothesis. There is no reason why the limited 

 width of the laboratory channel should limit the angular distribution of waves 



0.04- 



o 0.02 - 



Fig. 6. Mean square slope Sx^ as a function of fetch for various wind speeds U measured 

 at an anemometer height of 6 cm. Laboratory measurements by Cox (1958). 



(except by controlling the direction of the wind) since waves at arbitrarily 

 wide angles to the channel axis can be propagated unimpeded by a series of 

 reflections at the channel walls. 



Another possible mechanism for supplying energy to ripples is by air-pressure 

 variations caused directly by flow over the ripples themselves. As contrasted 

 with Phillips' mechanism, the air-pressure variations are now locked in phase 

 with the waves rather than allowed to be at random. Calculations have been 

 made which show that pressure variations of proper phase to induce wave 

 growth are associated both with laminar (Lock, 1954) and turbulent (Miles, 

 1957, 1959) boundary -layer flow over infinitesimal gravity waves. To what 

 extent similar processes can account for the growth of ripples oi finite amplitude 

 is unknown. 



Finally, non-linear interactions between gravity waves certainly can con- 

 ceivably produce ripples (e.g. some ripples are produced by splashes from 

 white caps). The importance of this and other non-linear interactions is un- 

 known. 



