in those cases, sands with small grain sizes are involved. Further, 

 these four experiments were among those with profiles showing some 

 three dimensionality, which is characterized by smaller bed forms, 

 and in two of them (experiments 64 and 82 with the 0.18-millimeter 

 sand) an occasional recurrence of three-dimensional bed forms may 

 have prevented attainment of two-dimensional equilibrium. With these 

 considerations and the previous observations (Fig. 5), the data in this 

 study are regarded as agreeing with equation (10) reasonably well, 

 rather than as indicating a three- fourths power law. 



To observe the effects of <p , which were not revealed in Figure 21, 

 the same data have been replotted in Figure 22 to give X/a as function 

 of (f) . Flags have been added to the plotted symbols to give values of 

 N (or a/D) as shown in Table 1. The curve in Figure 22 is another replot 

 of Figure 17 in Mogridge and Kamphuis (1972). This curve indicates that 

 in the experiments of this study the effect of <J> upon X/a remains 

 slight but still dominates that of a/D which equations (8) and (9) show 

 to be negligible. 



Final equilibrium values of n/a are plotted against (j) in Figure 

 23. Included experiments (with values of (J) and symbols) are the same 

 as in Figure 22. The curves for constant values of a/D in Figure 23 

 are a replot of Figure 18 in Mogridge and Kamphuis (1972) . In this 

 case, over the range of the data, the effects of varying a/D are 

 noticeable, and separate curves have been drawn. However, the curves 

 form a rather narrow band, and do not serve to reduce the scatter. The 

 data do not follow the curves as closely as in Figure 22, with points 

 for the coarse sand falling above and points for the finer sands falling 

 below. These moderately separate trends indicate a noticeable dependence 

 on r not contained in Figure 18 of Mogridge and Kamphuis (1972). 



4. End Effects. 



Figures 19 to 23 show a considerable degree of scatter, attributable 

 to variations in the effects of the ends of the channel on the average 

 ripple length. Assuming that the particular geometry of the ends some- 

 how tends to hold crests in certain fixed positions near the ends (as 

 does the local scour before a ramp), two such crests, one at either end 

 of the channel, will be "fixed" a distance S apart. To be stable, the 

 profile between must have an average ripple length A = S/m where m is 

 some integer. There is usually no integer which makes X equal to X^, 

 the "natural" ripple length which, for the given a and T, would form 

 on an unbounded bed. If X^ > X the profile will tend to press the end 

 crests outward or, if Xj^ < X, to pull them inward, increasing or decreas- 

 ing S, and in either case acting against the end effect tending to hold 

 the crests in place. Thus, the resulting displacement of the crests is 

 limited and leaves X still unequal to X^^. Preferably then, the ends 

 should be indifferent to the position of the end crests and, if space 

 had allowed, long gently sloping ramps might have been preferable to the 



57 



