hysteresis in the differences between the profiles at the start of experi- 

 ment 48 (following experiment 47} where X = 31.8 centimeters, and at the 

 end of experiment 49, where X = 25.4 centimeters. As the record in 

 Table A-1 shows, both profiles are in equilibrium with the same combina- 

 tion of a and T, but in experiment 48, a was half the value of 

 experiment 47, and in experiment 49 a was returned to the original 

 value of experiment 47. The first profile in Figure 17 (resulting from 

 experiment 47) is unstrained with seven crests on the bed. The last 

 profile in Figure 18 (resulting from experiments 48 and 49 in sequence) 

 is compressed with nine crests. For the same variation in a, the 

 hysteresis, or net change in A, would have been more had the first 

 profile of Figure 17 been extended with only six sand crests on the 

 bed. In the few possible cases where hysteresis might have occurred 

 with the brass end crests removed, its observation is obscured by 

 scatter in the data in Table A-1. 



VI. CONDITIONS FOR TWO- AND THREE-DIMENSIONAL BED FORMS 



1. Approach . 



As described in Section V,l, with the 0.55-millimeter sand the 

 final bed forms were always two dimensional, but with the 0.18- and 

 0.21-millimeter sands the final bed forms were either two or three 

 dimensional or even some combination of both. Although experiments 

 with the finer sands yielded data for the preceding descriptions of 

 two- and quasi two-dimensional profiles, they were also intended to 

 identify the conditions for two- and three-dimensional final bed forms. 

 To this end, the experiments were located again on plots of N and T, 

 as in Figures 10 and 11, with symbols denoting the two- or three- 

 dimensional character of the final bed forms. As before, N is a 

 measure of amplitude a (a = 0.4576 N centimeters). The symbols and 

 definitions are listed in Table 2. Although the definitions are sub- 

 jective, it is expected that, after the symbols are plotted, a curve 

 can be drawn separating two-dimensional from three-dimensional bed 

 forms . 



2. Results . 



Symbols showing the character of the final bed forms of the experi- 

 ments with the 0.18- and 0.21-millimeter sands have been plotted in 

 Figures 27 and 28 at the points (T,N) which define the experiments. 

 These points are the same as in Figures 10 and 11 from which the number 

 of each experiment can easily be obtained. 



In Figure 27, regions of predominantly- two- and three-dimensional 

 final bed forms can be separated by the straight line U = 25.2 centi- 

 meters per second, with two- and three-dimensional forms on the sides of 



65 



