It appears reasonable that some further modification of the 

 Shields criterion might make it more pre cise i n oscillatory flow. 

 Shields (1936) interpreted the factor vy/p/t^ occurring in the abscissa 

 of his plot of "y as a measure of the thickness of the boundary layer. 

 In oscillatory flow, the boundary layer thickness is dependent, in 

 part, on the period. This dependence has not been taken into account 

 by either of the criteria discussed here. 



IV. EARLY STAGES OF RIPPLE FORMATION 



1. Induced and Spontaneous Ripples . 



As has been observed by Bagnold (1946) and by Carstens, Neilson, 

 and Altinbilek (1969), a foreign object or a slight local elevation of 

 the sand surface on an otherwise flat bed enhances the formation of 

 ripples in that locality and, once begun, the ripples spread out over 

 the surface while they grow in size. Also, along the borders of a 

 sand bed there is a tendency to early grain motion and scour which 

 then becomes the origin of a spreading ripple pattern. In these ways 

 ripples can be started and spread at values of U below U,,, the value 

 needed to initiate sand motion on an ideal flat bed. How much below 

 depends on the local perturbation of the flow which the foreign object 

 or surface irregularity causes. (It is recalled that the "scow" con- 

 striction in these experiments was designed to offset the tendency to 

 premature scour at the ends of the bed) . Ripples generated by such 

 local disturbances are of the vortex type, and even when small they 

 have sharp crests. 



The concept of "spontaneous ripple formation" is that of ripples 

 arising suddenly and everywhere at the same rate over an infinite 

 flat bed. The value of U for this condition, Uj., cannot be less 

 than Uq, since ripples cannot arise unless grains are in motion. 

 Ripples can be observed to form spontaneously in the laboratory if the 

 rate of increase of U is so large that U is reached before all of 

 the bed is overrun by ripples generated by local perturbations. As U 

 approaches U„, the rate of spread of rippled areas increases rapidly, 

 and "spontaneous generation" can be regarded as equivalent to an 

 "infinite rate of spreading." Ripples may be said to arise "quasi- 

 spontaneous ly" when, though first observed in separate areas, their 

 subsequent spread over the entire area is accomplished within the time 

 of a few periods. 



2 . Observations with Coarse and Finer Sands . 



In all the experiments, with all three sands, vortex ripples were 

 found to begin, either quasi-spontaneously or at some point on the 

 boundary, at values of U near to or somewhat below U^. Thus, in most 

 experiments with a previously leveled bed, ripple development was 



44 



