g. Limitations of Time and Space . Except for Carstens, Neilson, 

 and Altinbilek C1969) discussion of the evolution of a duned bed and 

 for occasional descriptions of the early spread of ripples over a flat 

 bed, the bed forms described in the references are presumed to be in 

 equilibrium with their environment, not changing with time and requir- 

 ing increments in a or U to induce further change. The presumption 

 of stability is sometimes stated and sometimes implied. Inman (1957) 

 remarks that for ripples in coarse sand the condition of equilibrium is 

 made less certain by their slow response to changes in the environment, 

 but that ripples in fine sand are more rapidly modified. Dingier (1957) 

 observes that following sheet-flow ripples can reform within a few 

 cycles. In any case, references to duration or rate of processes are 

 rare. 



Some of the discrepancies in the preceding survey invite specula- 

 tion that stable states had not in fact been established, particularly 

 in cases where the growth of X and n stopped short of the values 

 predicted by equation (9) and related curves of Mogridge and Kamphuis 

 (1972). For example, Bagnold (1946) took his equilibrium ripple length 

 to be that of three adjacent ripples which had grown to equal size from 

 an initial local disturbance, not discussing the possibility that they 

 might continue to grow, perhaps at a reduced rate. Again, the unusual 

 growth of X with increasing w (rather than a) observed by Manohar 

 (1955) suggests a continuing response to an earlier selection of a. 

 If the increases in w were fairly continuous, an effect on the con- 

 current growth of X might be unduly inferred. In general, rates of 

 change might be so small as to remain undetected and yet persist to 

 produce significant effects. 



These discrepancies also suggest a closer examination of limita- 

 tions in the experimental facilities and of their possible effects; in 

 particular, end effects combined with the shortness of the sand bed. At 

 the ends of the sand beds or trays there is a tendency for local scour, 

 which is sensitive to the local geometry of the rigid container. The 

 local scour tends to define a ripple trough and so to inhibit motions 

 which may accompany the normal development of the ripple profile. The 

 effect of both ends of the bed acting as permanent ripple troughs is 

 to permit only those ripples for which X is equal to the length of 

 the bed divided by some integer. It would appear that end effects 

 become more serious as the bed becomes shorter. Surprisingly, end 

 geometries and effects are seldom mentioned in the studies referred to 

 above . 



h. Emerging Questions . The preceding discussion shows that more 

 information is needed to provide a satisfactory description of ripples 

 and their development. Areas where new observations may be most useful 

 include: 



(a) Whether and to what extent, on a flat bed, the 

 criteria for the initiation of sand motion and for the spon- 

 taneous generation of ripples are distinct; 



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