into ripples characteristic of steady flow. Such ripples are of 

 limited length. Yalin (1977) cites the approximation. 



^« 1,000 [steady flow) (26) 



(and replaces the constant, 1,000, with a function of Reynolds number. 

 Yalin's data have an average X/D of around 800). A relation of the 

 form of equation (26), with the constant lowered, moderately, to between 

 500 and 600, is consistent with the trends of the observations of Inman 

 (1957) and Dingier (1975) for large a/D (see Fig. 5,e and f ) . Then, for 

 oscillatory ripples, if <^ is held constant and both a and T 

 become large, X/a must ultimately become small. Such behavior is not 

 found in the plots of Mogridge and Kamphuis (1972) which, as T becomes 

 infinite and X2 (eq. 7) approaches zero, require X/a to remain, approxi- 

 mately, constant, as in equation (8). 



The transition from oscillatory to steady-flow ripples suggests an 

 analogy with the transition from laminar to turbulent flow in a rough 

 pipe. This analogy is illustrated by schem.atic plots in Figure 31, For 

 pipe flow (Fig. 31a), as the Reynolds number, R, increases, a friction 

 coefficient, Cf, breaks away from its laminar asymptote at the onset of 

 turbulence ultimately to approach a constant value depending on the 

 wall roughness defined by a parameter, k. For ripples (Fig. 31b), as 

 a/D increases, A/D diverges from its oscillatory asymptote, ultimately 

 to approach the constant value for steady flow. The transition has 

 been presumed to depend on some function K(((),r) which combines the 

 effects of (|) and r (possibly like U/w as discussed in Section I,3,d) 

 and is analogous to the parameter k in pipe flow. 



In the case of pipe flow, transition to turbulent flow is enhanced 

 and shifted to lower values of R by disturbances in the incoming flow 

 such as slight turbulence or swirl. Extending the analogy, transition 

 from oscillatory to steady-flow ripples might be enhanced and shifted 

 to lower values of a/D by deviations from pure sinusoidal two-dimensional 

 flow over the bed. The analogy then suggests that, for given a/D, ()), r, 

 values of A/D would tend to be larger (closer to eq. 10) in a water 

 tunnel, which provides a sinusoidal two-dimensional flow, and smaller 

 (closer to eq. 26) on the sea bed where the flow is perturbed, tendencies 

 which were observed in Figure 5 and discussed in Section I,3,d. Effects 

 of perturbations in the flow might also account for some of the scatter 

 found in plots of A/D and A/a, particularly from field observations. 



Results of this study (summarized in App. A) show that two- 

 dimensional profiles, developing under the same given values of a 

 and T and with minimum end constraints, attain nearly identical final 

 forms, even when the initial bed forms were very different. This 

 indicates that unconstrained equilibrium profiles are independent of 



75 



