With the 0.18- and 0. 21-inillimeter sands, the degree o£ three 

 dimensionality during growth was relatively large and in many cases, 

 the bed forms remained more or less three dimensional even when fully 

 developed. Sometimes with these finer sands a profile, or a part of 

 it, would evolve through two- and three-dimensional phases and on 

 occasion this evolution seemed a type of cyclic instability. In a 

 few cases the attainment of a steady state and its character remained 

 in doubt. Although attaining an equilibrium as to average character- 

 istics and an unchanging overall appearance, three-dimensional bed 

 forms, unlike two-dimensional ripples, never became static. When 

 compared with two-dimensional forms which developed with the coarser 

 0.55 -millimeter sand under otherwise similar conditions or, indeed, 

 with the same finer sand under conditions which may have been only 

 slightly different, the three-dimensional form.s were sharper, more com- 

 plex and smaller (it is recognized that these terms are subjective and 

 ill defined) . Characteristic three-dimensional bed foims are showm in 

 most of the photos of final bed forms for the finer sands in Appendix B. 



Only three experiments (72, 90, and 96) with the 0. 18-. and 0.21- 

 millimeter sands could be included in an analysis of the rates of 

 growth of two-dimensional ripple profiles. This analysis CSec. V,2) 

 thus becomes almost limited to the coarser 0.55-millimeter sand. How- 

 ever, in about two-thirds of the experiments with the finer sands, the 

 final bed forms were sufficiently two dimensional to yield equilibrium 

 values of X and t\, which are included in a subsequent analysis 

 (Sec. V,3), 



2. Growth of Two-Dimensional Ripples . 



For U/UqWI, n/a and A/a are shown as functions of time in Figures 

 19 and 20. The time is expressed in the form. (D/a)n, where n is the 

 number of cycles since the time of origin of the ripples, t^; that is, 

 n = (t-tQ)/T. For each experiment, n, X, and t were taken from the 

 photographic record, n was then plotted against t, and t was read 

 at the intersection of the extrapolated curve and the axis n = 0. 

 During conditions of rapid growth, measurements of n and A are made 

 less precise by the partially three-dimensional character of the profiles. 

 However, all the data from the experiments with the 0.55-millimeter sand 

 with untamped beds lie near a single curve, showing that n/a is approxi- 

 mately a function of (D/a)n, and that any further dependence upon a/D 

 is relatively minor. Points from the experiments with tamped beds plot 

 somewhat above this curve, and points for the finer sands lie along a 

 separate (dotted) curve. As time increases, these curves approach 

 asymptotes of about n/a = 0.24 for the 0.55 -millimeter sand and 

 n/a = 0.18 for the finer sands. The difference between these curves 

 indicates a slight effect of grain size or of r. Using the values of 

 to and n already determined. A/a has been plotted as function of (D/a)n 

 in Figure 20. In this case reduction to a single curve is not possible 

 since, unlike n, A does not become zero at t = t (as shown in Fig. 15). 

 Even so, the separate curves for different a/D with the coarse 0.55- 

 millimeter sand appear to approach a single asymptote, of about A/a = 4/3 



52 



