defined by Bagnold. However, the similarity o£ the photos in Figure 

 14 and in Figure 7 of Bagnold (1946) make the identity clear. 



4. Initial Ripple Length . 



The length of the vortex ripples at about the time they had 

 spread over most of the bed was measured from the photographic record. 

 Such ripples are shown in Appendix B which contains photos of the 

 initial and final ripples in all 24 experiments started with a leveled 

 bed. This initial ripple length is an approximate average value, 

 because the ripples leading the advance are smaller than the older 

 ripples nearer the point of origin of the train, and because three- 

 dimensional areas were excluded from the averaging. The initial ripple 

 length (Aj^) is shown in Figure 15 in the form of Xi/D plotted against 

 a/D. For each sand, values of Aj^ are remarkably independent of a. 

 Average values of X^ for the 0.18-, 0.21-, and 0.55-millimeter sands are 

 4,5, 5.4, and 8.5 centimeters respectively. 



V. GROWTH AND VARIABILITY OF EQUILIBRIUM RIPPLES 



1 . Two- and Three-Dimensional Forms . 



As mentioned earlier, in the experiments with previously leveled 

 beds, ripple development was left to proceed at U = U^. or, in a few 

 cases, at U = U-p. Although the small initial ripples were always two 

 dimensional, further growth involved the suppression and coalescing of 

 crests which often occurred first near a channel wall. That is, 

 during growth the bed form became to some degree three dimensional . 

 With the 0.55-millimeter sand this degree was relatively small and as 

 the ripples approached equilibrium dimensions they became and remained 

 two dimensional. This pattern of growth, with a transitory three- 

 dimensional phase, is illustrated in Figure 16 by a sequence of photos 

 from experiment 51. Two-dimensional growth involves a reduction in the 

 number of crests per unit length of the profile. This can result from 

 a spreading of the profile into unrippled areas or, in these experi- 

 ments, into the spools. When further spreading is blocked, some crests 

 shrink and disappear. Typically, as a crest disappears, its neighbors 

 move toward it and it is replaced by a new trough. Less often, two 

 neighboring crests approach each other and merge into a single crest. 

 Growth need not begin from a leveled bed, and occurs more generally in 

 response to an increase in a. Also, when a is reduced, the process 

 just described is reversed, new crests arise, and X shrinks. Two- 

 dimensional processes of shrinkage and growth are shown in Figures 17 and 

 18 by sequences of photos from experiments 48 and 49. These record the 

 successive responses of a profile when, first, the amplitude was reduced 

 by one-half and then doubled to its original value. In these two 

 experiments, the profiles were contained between rigid end crests, as 

 described in Section V,5. 



47 



