■However, for 2a/D > 2x10^, the data in Figure 5 show a tendency 

 for X/D and X/a to fall below values according to equations (10) and 

 (9) . This tendency appears in the laboratory observations in Figure 

 5(b and d) and, more strongly, in the field observations in Figure 

 5(e and f ) . Circumstances, especially in the field, appear to favor 

 a positive correlation between the "independent" variables 2a/D and (j), 

 so that their effects are not readily separated. This problem is dis- 

 cussed further in Section VII, 6. The data falling below the envelope 

 curve of Mogridge and Kamphuis (1972) in Figure 5 include large values 

 of (}), between 50 and 200, which can account for some of this deviation. 

 However, the reductions in X/D thus accounted for are a rather small 

 part of those actually found, and substantial reductions are found also 

 with small values of <t) . Thus, the data, with their scatter, reveal no 

 clear effect of <j) upon X/D. Dingier 's (1975) data, unlike those of 

 Inman (1957), form two separate clusters, despite fairly continuous 

 distributions of 2a/D and (j) . Small values of X/a appear to be 

 characteristic of small sand grains, and values of X/a < 1/2 are most 

 often associated with values of D < 0.2 millimeters. This would suggest 

 an effect of grain size, or of r, not included in the curves of 

 Mogridge and Kamphuis (1972). In their experiments, r remained within 

 the rather narrow range 28 < r < 43, so that its possible effects were 

 hardly explored. 



Nielsen (1977) has suggested that X/a should be a function of U/w, 

 where w is the fall velocity of the sand grains, and has plotted data 

 from various sources in this form. His curve drawn through the data 

 shows an accelerating decline in X/a with increasing U/w similar to 

 that with (f) in equation (8). U/w can be written as (jj-'-'^f (r) , where 

 the function f(r) depends on the variation of the drag coefficient of 

 the grains with Reynolds number. Over the range of practical interest, 

 f (r) decreases with r almost as rapidly as 1/r. Then, a reduction 

 in r, as well as an increase in (j), increases U/w and so, by 

 Nielsen's curve, reduces X/a. Such a behavior seems appropriate for a 

 modification of the curves of Mogridge and Kamphuis (1972) with (U/w)^ 

 replacing <)) . A rather large degree of scatter in the data of Nielsen's 

 plot obscures to what extent separate effects of (^ and r combine 

 into the single effect of U/w. However, Nielsen (1977) has identified 

 U/w as a variable likely to be useful in analyses of ripple forms. 



Several studies (not included in Fig. 5) report behavior of X 

 quite different from the results of Mogridge and Kamphuis (1972) . 

 Manohar (1955), using an oscillating tray in still water, found X/D to 

 be almost independent of a and to be approximately proportional to U. 

 Guided by the process of vortex shedding by cylinders, Homma and 

 Horikawa (1962) cite the field data of Inman (1957) and their observations 

 and those by Scott (1954) in wave tanks to show that X/a is primarily 

 a function of the Reynolds number Ua/v . Horikawa and Watanabe (1967) 

 obtained a similar result when sand was replaced with plastic pellets. 



23 



