their histories. However, in experiments 37 to 49 with fixed end 

 crests (Fig. 25), there was a variety o£ ripple forms, corresponding 

 to different values of a with a fixed value of X, within the 

 limits of equations (24) and (25). In addition, when the profiles 

 were strained beyond these limits, a hysteresis resulted (discussion of 

 Figs. 17 and 18 in Sec. V,5). On the unbounded seabed, ripples might 

 be expected to show strain while still adjusting to previous changes 

 in a, though, presumably, the time of adjustment, and with it the 

 maximum strain encountered, would be reduced by the freedom of the 

 profile to extend and contract longitudinally. A quicker adjustment 

 might result from the opportunity for lateral motions combined with devi- 

 ations of the bed forms from two dimensionality. On the seabed examples 

 where a crestline stops or branches into two are common. The lateral 

 movement of such features into or out of an area could change the 

 average ripple length almost continuously. This mechanism was alluded 

 to in Section VII, 2 in regard to the growth of ripples. 



4. Two- and Three-Dimensional Bed Forms . 



Equilibrium bed forms were always two dimensional with the 0.55- 

 millimeter sand but both two and three dimensional with the two finer 

 sands. Attempts to further define the conditions for two- and three- 

 dimensional forms are inconclusive. With the 0.1 8 -millimeter sand, 



two- or three-dimensional forms were observed, with fair consistency, 

 as (j> was less than or greater than 21.3 (U = 25.2 centimeters per 

 second; Fig. 27). With the 0.21-millimeter sand the separation of two- 

 and three-dimensional forms was not so simple. A suggestion for this 

 difference in behavior between the two finer sands, based on grain-size 

 distribution, was offered in Section VI, 2. Generally, irregularities 

 in the occurrence of two- and three-dimensional forms might stem from 

 limitations of time and space. It was noted in Section V,l that pro- 

 files are prone to become three dimensional during periods of growth 

 and that, with the finer sands, during slow or imperceptible growth, 

 recurrence of two and three dimensionality was not uncommon. There 

 remains then a possibility that some of the observed final bed forms 

 might have modified or changed character if the experiments had been 

 prolonged. Although the channel walls clearly suppress large three- 

 dimensional motions, they also initiate small ones, since at a wall 

 the vorticity vector of the flow must be parallel to the wall, rather 

 than normal to it as in strictly two-dimensional flow. Once established, 

 symmetrical slightly three-dimensional flows and associated bed forms 

 might become unstable, lose symmetry and become increasingly three 

 dimensional. Since such wall effects on the transition from two- to 

 three-dimensional bed forms in these experiments are not known to be 

 negligible, the curves in Figures 27 and 28 must be used with caution. 

 After transition, the effects of the walls on the further development of 

 three-dimensional bed forms must be presumed significant, and for this 

 reason their final characteristics have not been described in detail . 



77 



