short curved ones actually used CFig. 6). In present circumstances, 

 not only the ramps with scoured troughs and crestlike simmiits, but the 

 screens in the spools act to position and hold crests, since the sand 

 which cannot pass through a screen tends to accumulate a short distance 

 in front of the screen, in the form of a crest (Fig. 6). Such crests 

 are liable to move, and cases of cyclic instabilities of sand crests 

 within the spools, with an otherwise stable profile, were observed. 

 Finally, it seems that a stable profile cannot end abruptly, but 

 requires the accumulation of a certain amount of sand in the spools 

 to provide a gradual transition to the unrippled region beyond. This 

 accumulation usually occurs by migration of a ripple crest into the 

 spool where the crest is rearranged into separate peaks on the bare 

 floor of the spool. The migration of the ripple crest into the spool 

 may extend the profile and make X > Aj^. This effect was tested in 

 experiment 57 by interrupting the experiment before the ripples reached 

 equilibrium and adding a quantity of sand, by hand, into each spool. 

 The subsequent equilibrium \ was found to be reduced by 20 percent 

 from that in experiment 29 with the same a and T. In experiments 30 

 to 50, pieces of brass bent into a crest shape, 2.8 centimeters high and 

 with a vertex angle of 120°, were mounted in the spools at various posi- 

 tions between the screens and the ends of the sand bed. The record for 

 experiments 29 to 36 in Table A-1 shows that X was reduced by moving 

 the brass crests forward in the spools, while the redistribution of 

 sand in experiment 31 had a relatively small effect. Thus, the brass 

 crests provide a relatively definite, if extreme, end effect. With 

 the brass crests removed, the end effects remain unclear in nature and 

 detail but appear to have an order of magnitude quite able to account 

 for the scatter in Figures 20, 21, and 22. 



5. Variability and Hysteresis of Stable Profiles . 



Although the mechanism of the end effect could not be analyzed, it 

 was possible to measure its limits. A series of experiments (37 to 47) 

 with the 0.55 -millimeter sand was performed to determine over what 

 range of A/a a profile could remain stable and how the profile shape 

 varied with A/a. The brass crests were mounted atop the ramps 254 centi- 

 meters apart with seven sand crests between. Since the brass crests 

 fixed the ends of the profile, the average ripple length. A, remained 

 constant in this series (equal to 254/8 = 31.8 centimeters) as long as 

 the number of sand crests remained unchanged. As in equation (9), Aj^ is 

 approximately proportional to a, with A/Aj^ proportional to A/a. Each 

 experiment of this series had a different value of a (and hence A/a) 

 and T was adjusted to keep U approximately constant at 34.5 centimeters 

 per second — U^^. The variable n was observed. Photos of the center 

 third of the final profiles for these experiments are shown in Figure 24. 

 Figure 25 is a plot of final n/A as function of A/a including the numbers 

 of the consecutive experiments. The series begins at experiment 37 with 

 a low A/a; that is, a compressed profile. In experiment 38, with A/a 

 further reduced to about 1.1, signs of instability appeared (note the 

 incipient shrinking of the center crest in Fig. 24) and the experiment 

 was quickly stopped. Previous experiments had shown that, if the 



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