6. Occurrence of Ripples . 



With the criteria of Manohar (1955) and Dingier (1975) providing a 

 maximum (J) above which ripples disappear, and with the "composite" 

 curve for ()>(, in Figure 2, together with the remarks in Section VII, 1 re- 

 garding a minimum (|) below which ripples cannot be sustained in equilib- 

 rium, an attempt may be made to map out the domain where ripples occur. 

 Observations of ripples and of sheet flow in the field by Inman (1957) 

 and Dingier (1975) are plotted in Figure 32 according to their values of 

 a/D and (^ . The criteria for sheet flow have been drawn and the composite 

 curve for (j)^ as function of a/D has been transcribed from Figure 2. A 

 lower bound for equilibrium ripples, which is poorly defined, is here 

 suggested as (}> = (()c/5, which lies along the lower edge of the field 

 observations just below the (f)c/4 suggested by Lofquist's (1975) laboratory 

 study, and somewhat farther below the point from Carstens, Neilson, and 

 Altinbilek's (1969) experiment 62 which is marked by an arrow. A few 

 observations that fall below the curve are to be expected as representing 

 cases of relict ripples formed by previous and stronger flows. It is 

 interesting that more than half of the field observations of ripples are 

 for flow conditions too weak to move sand on a flat bed according to the 

 composite curve for laboratory data in Figure 2. Also, for about half of 

 the experiments in this study values of ()) were below <^^, and the ob- 

 served sand motion showed that the profiles were not relict but were in 

 equilibrium with the flow. The upper-bound criterion of Dingier (1975) 

 is, of course, in accord with his own and Inman' s (1957) observations of 

 sheet flow; the criteria of Manohar (1955) and of Chan, Baird and 

 Round (1972), represented by two selected values of r, are also in 

 fair agreement. 



Although (|) and a/D are independent variables, the distribution 

 of the points of ripple observations in Figure 32 shows a trend of ()) 

 with a/D with hardly more scatter than is often tolerated in interpret- 

 ing functional relationships. In this case, ({) increases roughly as 

 the square of a/D. This trend reflects a limited range of T in the 

 field observations of Inman (1957) and Dingier (1975) . For two-thirds 

 of these observations, T fell in the range from 8 to 12 seconds. 

 This range is characteristic of the California coast and may not appear 

 elsewhere. The observed trend of (j) with a/D is then not a functional 

 relationship but a circumstantial correlation. It makes separation of 

 the effects of ^ and a/D upon X/D or n/D very difficult. 



With pg/p regarded as constant for natural sands, the occurrence 

 of ripples, as well as other of their characteristics, depends on the 

 three independent variables a/D, (f; , and r. The choice of a/D, 

 rather than r, as abscissa in Figure 32 owes much to the fact that 

 the trend of ^^^ with a/D is fairly clear while the effects of r on 

 ([)(; remain contradictory and obscure. (In field data, r depends 

 primarily on D.) Similarly, X/D and n/D have been found to depend more 



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