appear with plastic (opalon) beads. They also report that "the 

 equilibrium ripple wavelength was observed to form at an early stage 

 of ripple growth for all sediment materials used" and that the ripple 

 length increased with both a and oj. Carstens, Neilson, and 

 Altinbilek (1969) describe an (apparently) unstable progression from 

 rolling grains through a stage of rolling-grain ripples and into the 

 stage of vortex ripples, which attain an equilibrium length, approxi- 

 mately proportional to a. They find a "small lag" between incipient 

 motion and the spontaneous formation of ripples which, by an average 

 of their three observations, occurs around U = 1.16 Uc . 



d. Equilibrium Two-Dimensional Ripples . The preceding descriptions 

 include references to the stable forms which ripple profiles attain. 

 The primary source of information on two-dimensional equilibrium ripple 

 forms is the study of Mogridge and Kamphuis (1972), obtained with a 

 large oscillatory-flow water tunnel and using three types of plastic 

 grains and a single quartz sand (with D = 0.36 millimeters). Observa- 

 tions of the equilibrium ripple length and height are summarized by 

 curves in Figures 17 and 18 of their report. These figures give X/D 

 and n/D as functions of 2a/D for selected values of the parameter. 



:C^-l)T^ 



%C2.r. t') 



Figure 17 of Mogridge and Kamphuis (1972) is reproduced in Figure 3 with 

 the addition of lines of constant <}) drawn according to equation (7). 

 The figure shows that, over wide ranges of a/D and ij), lines of constant 

 (\> are approximately straight with a slope of plus one. Thus, along each 

 such line, X/a is approximately constant and independent of a/D. That is. 



^ ~ f(<}>) . (8) 



As <}) increases, X/a ultimately decreases while its dependence upon a/D 

 increases gradually. In particular. 



i ~ i 

 a ~ 3 



(9) 

 < (f> < 30 . 



Equation (9), remarkably, states that X/a is nearly constant and inde- 

 pendent of U or T over a wide range of 4" > which (as this study confirms) 



