Ranee and Warren (1968) who, using a large water tunnel and limestone 

 chips, attained values of r > 10^; and from Chan, Baird, and Round 

 (1972) who, using a small pipe but with various viscous liquids, 

 attained values of r < 0.2. The various trends reflect a considerable 

 scatter in the data, on the order of ± 40 percent. Some of this 

 scatter may derive from the neglected effects of grain-size distribution 

 and grain shape which, as Collins and Chesnutt (1976) describe, can be 

 significant in the evolution of beach profiles or from the degree of 

 compaction of the sand surface. However, most of the scatter can be 

 attributed to the subjective element in determining the condition of 

 incipient grain motion. The definition and determination of U^ and 

 (j)^, are discussed in Section 111,1. Generally, the data represented in 

 Figure 2 show a weak correlation of larger values of <\>q with smaller 

 values of r. For convenient reference, a single composite curve, for 

 all values of r, has been drawn through the trends shown in the 

 figure. For small a/D, this curve becomes parallel to the trend of 

 Ranee and Warren (1968), with ^^ proportional to (a/D)-^^; for large 

 a/D, the curve makes (f)^ proportional to (a/D)^'^, a behavior suggested 

 by the known stress on a smooth bed combined with a Shields criterion, 

 as defined in Section 111,4. 



c. Early Stages of Ripple Development . Observations of ripple 

 development following the initiation of sand motion as reported by 

 various investigators are summarized below. Bagnold (1946) identified 

 a "rolling-grain" type of ripple stable over the range U,, < U < 2U^ 

 with ripple length and height, for constant a, increasing with co. 

 At Uw2Uc these ripples rather suddenly change character and become 

 "vortex-type" ripples, with vortices developing behind the ripple 

 crests and over the troughs. The vortex ripple length is independent 

 of 0) and is proportional to a, for small a, and constant for 

 larger a. Bagnold states that these ripples can form "from any 

 sufficiently large surface feature" at values of U even below U^. He 

 describes artificial generation of ripples for U < U^. (also described 

 in more detail by Carstens, Neilson, and Altinbilek, 1969). Manohar 

 (1955) describes stable regimes of rolling-grain motion and "first-stage" 

 ripples very similar to the rolling-grain ripples of Bagnold. He also 

 describes "second-stage" ripples which appear to be the vortex ripples 

 of Bagnold. Throughout the process of ripple formation, Manohar finds, 

 for given a, stable ripples at each value of gj, and ripple length 

 increasing with oj. He provides data for the "initiation of ripples" 

 which occurs, on the average, at a value of U which exceeds the 

 values for "initial" and "general" motion of "sediment in the turbulent 

 boundary layer" by 24 and 13 percent. Presumably, this refers to the 

 initiation of second-stage or vortex ripples. Kennedy and Falcon (1965) 

 provide no data on the inception of either sand motion or ripple forma- 

 tion, but observe that, when there is "general motion of the upper 

 layers" of the bed particles, ripples appear simultaneously and "once 

 they grow to a certain height, vortices are shed from the ripple crests." 

 They state that, in their experiments, the rolling-grain ripples 

 described by Bagnold (1946) did not appear with quartz sand but did 



