experiment were continued, one crest of the compressed profile would 

 shrink and disappear (and thereby increase X). In following experi- 

 ments, A/a was successively increased, and the observed n/A successively 

 decreased. The troughs became longer and shallower, and each eventually 

 developed a gentle "rise", characteristic of an extended profile. In 

 experiment 44 one of the trough rises began to grow. The experiment 

 was stopped but apparently not soon enough; in the following experiment 



45, even with A/a reduced to 4/3, this growth, though slower, continued. 

 Not until experiment 46, with A/a in the compressed range, did the new 

 crest shrink and disappear. The number of sand crests was still counted 

 as seven and A unchanged in determining A/a and n/A for experiments 



45 and 46. The appearance and disappearance of this incipient crest 

 (Fig. 24) give the curves in Figure 25 a hysteresis which would not have 

 occurred had A/a been kept below some value between that for experiments 

 43 and 44 and the new crest not allowed to start. Following experiment 



46, A/a was again increased, and the point for experiment 47 falls on 

 the former descending curve. On the basis of these observations, the 

 stable ranges of A/a and n/A are taken to be, approximately. 



1.1 < A/a < 1.6 (24) 



(stable) 



0.22 > n/A > 0.125 (25) 



The middle of the range of A/a in equation (24) is near 4/3, so that 

 Aji can be identified as the ripple length predicted by equation (9) . 

 In Figure 25 the profile has been described as "compressed" or "extended" 

 as A/a is less than or greater than 4/3. The points from this series of 

 experiments with their variations in A/a and n/a, are identified by the 

 diamond symbol in Figures 22 and 23. It is seen that their ranges are 

 about the same as the ranges of the scatter. Therefore, the scatter 

 is probably associated with various degrees of profile strain. 



Characteristics of "strained" profiles are shown in Figure 26. As 

 in experiments 37 to 47, A is constant and, from top to bottom, A/a 

 increases as a decreases. 



Based on the information summarized in Figures 25 and 26, the 

 following implica-tions for equilibrium ripple shape are drawn. When 

 the end crests of a two-dimensional profile are fixed, the conditions 

 of strain imply that changes in A responding to variations in a must 

 show hysteresis. As a increases, such a profile becomes compressed 

 and, as crests disappear, A increases in steps such that A > 1.1a 

 (eq. 24). As a decreases, the profile becomes extended and as new 

 crests appear, A decreases in steps such that A < 1.6a. Thus, A 

 depends, in part, on whether a has been increasing or decreasing and 

 must show hysteresis as a is varied from and returned to its original 

 value, provided that the variation in a was large enough to eliminate 

 an old crest or create a new one. The photos in Figures 17 and 18 show a 



63 



