The movement of loose bed material is governed by the inertial forces 

 of the particles and of the water against them, by the weight of the par- 

 ticles, and by the viscous forces acting between the water and the parti- 

 cles. Three physical laws have evolved from an analysis of these forces: 

 Newton's law of inertia, the law of gravitation, and the viscous friction 

 law of Newtonian fluids. These laws have provided two well-known dimen- 

 sionless terms which must be equated between the model and the prototype 

 for kinematic and dynamic similarity to prevail; i.e., the Reynolds number 

 and the Froude number. 



and 



R„ = IT ('-« 



^n=(^. '"' 



where the symbols have been previously defined. 



The simultaneous conformation of the model and the prototype to both 

 the Reynolds number and Froude number yields the familiar problem that 

 the length-scale factor becomes a function of the scale factor of the 

 kinematic viscosity. This determines that no readily available fluid 

 possesses the kinematic viscosity to make a useful model fluid. Schuring 

 (1977) reasons that, since the same fluid for model and prototype re- 

 quires less than perfect similarity but probably must be used, design 

 requirements can be relaxed if the inertial forces of the sediment are 

 much smaller than the rest of the forces and therefore can be neglected. 

 Then, Newton's law of inertia must only be applied to the fluid. A fur- 

 ther sinqjlification, without loss of generality, is achieved by restrict- 

 ing the law of gravitation to the weight difference of water and sediment. 

 With these two modifications, a qualified Froude number evolves often re- 

 ferred to as a densimetric Froude number) , and the length-scale factor is 

 freed from its dependence on kinematic viscosity; 



V 

 F = Y? V-^i — . C7-8) 



[(y.w - 'M' 



The penalty for this simplification is a restriction of the particles 

 to a state of rolling or sliding with small or no inertial forces acting 

 upon them. The model becomes invalid when the particles begin to leave 

 the bed and are carried upward, such as in the surf zone or in relatively 

 shallow water affected by surface gravity waves. Very good correlation 

 between variables was achieved in flume experiments with unidirectional 

 flow (Schuring, 1977). 



A different approach, advanced by Gessler (1971), assumes that both 

 the prototype sediment and the material ;;sed as model sediment are given, 

 and the model geometric scales are determined to fit the requirements of 

 these materials. In this approach, supplemental information should be 

 used in the form of the Shields parameter regarding the critical tractive 



473 



