as given by equation (9) ; the curve for the finer sands has a somewhat 

 lower asymptote around A/a = 1.25. 



The curves show that the growth of both n and X is fairly 

 complete by the time (D/a)n equals 1. This suggests the simple 

 criterion that, with U/UcWl, the ripples are mature if n > a/D. Other 

 stages of ripple growth can be defined in similar terms. Such criteria, 

 which make the time of development proportional to Ca/D)T, appear to be 

 in conflict with Inman's (1957) comment that ripples in fine sand are 

 more rapidly modified than ripples in coarse sand. It is not clear under 

 what conditions or restrictions, if any, this comparison a.pplies. In 

 any case, U need not approximate Uc as in Figures 19 and 20. Only 

 three experiments with the finer sands are included in these figures and 

 some further dependence upon grain size (r) may well exist. 



Along the asymptotes in Figures 19 and 20, the data show considerable 

 scatter which is largely due to end effects (discussed in Sec. V,4) . 

 Sudden and substantial increases in X and n result when a shrinking 

 crest finally ceases to be counted. These increases result from 

 averaging over the limited number of ripples in the test section and do 

 not imply actual abrupt changes in the profile. 



3. Equilibrium Two-Dimensional Ripple Profiles . 



Photos of the final equilibrium profiles for all the experiments 

 started from a leveled bed are contained in Appendix B. The two- 

 dimensional profiles differ widely in scale, but are rather similar in 

 shape, with variations of types discussed below. The photos reveal a 

 variety of configurations at the ends of the profiles. The scour some- 

 tim.es appearing at the curved ramp acting as a persistent trough, may- 

 serve to "anchor" the end of the profile. 



Final equilibrium values of X/D are plotted against 2a/D in 

 Figure 21. Included in the data are all the experiments with the 0.55- 

 millimeter sand, and 33 experiments with the 0.18- and 0.21-millimeter 

 sands which were sufficiently two dimensional to permit analysis. In 

 this form, the data can be compared directly with the previous observa- 

 tions in Figure 5, by their relation to equation (10) which is the 

 straight line comm.on to both Figures 5 and 21. The data of this study 

 (Fig. 21) follow this line reasonably well, though their trend shows 

 X/D to increase, nearly, with the three-fourths power of 2a/D, ratlier 

 than v;ith the first power. Deviations of the data from equation (10) 

 are not accounted for by the regular and relatively minor effect of <(> . 

 However, the deviations are within the range of inherent scatter and 

 end effects discussed in Section V,4. Also, values of X/D falling 

 below the line for the four highest values of 2a/D may be the beginning 

 of the trend for X/a to decline with 2a/D, which was found in several 

 of the plots in Figure 5, and discussed in Section I,3,d. Here, as 



55 



