It is convenient to express the effects of viscosity in terms of 

 r rather than by the more usual grain-size Reynolds number, UD/v, 

 equal to t<^^^'^, because r depends only on the materials used and 

 is typically a constant during a series of experiments. When the 

 materials involved are quartz sand and water and Pg/p and v can be 

 regarded as constant, r serves as a measure of D. Also, r^ is pro- 

 portional to a Reynolds number based on the Stokes fall velocity of the 

 grains and their mean diameter D. Not included in the above are param- 

 eters describing the effects of grain-size distribution and grain 

 shape and the degree of packing of the sand surface. Thus, the angle 

 of repose does not appear. The variable (j), except for the absence 

 of the drag coefficient, is a "Shields parameter" and provides a con- 

 venient, if approximate, measure of the intensity of sand motion. 

 Dingier (1975) refers to <)) as the "relative stress", and i^^'^ has been 

 named the "sediment number" by Carstens (1966) . 



In terms of the above variables, the criteria for the initiation 

 of sand motion and for the spontaneous appearance of ripples can be 

 expressed as 



) (^, r, £^) (sand motion) ^^'^ 



c U p 



b (^, r, ^) (ripple formation) (-4-) 



and, for example, the dependent equilibrium ripple length X can be 

 expressed as 



Thus, the criteria for the initiation of sand motion, and of ripple 

 formation, are described by dimensionless plots without distinguishing 

 whether the boundary layer, or stress, is laminar or turbulent. Such 

 distinction has been attempted by Komar and Miller (1973) and others. 

 The degree of turbulence can be defined by the character of the stress 

 coefficient which is itself some function of the above variables. 



3. Previous Observations . 



a. Facilities . Results of previous investigations may best be 

 introduced by a brief evaluation of facilities used in the investiga- 

 tions most often cited. Bagnold (1946) and Manohar (1955) used 

 sediment-filled trays oscillating in still water. This device 

 conveniently provided wide ranges of period and amplitude, but subjected 



14 



