- ^xlO 3 

 (ii) Two-dimensional roughness f (y) = e " 



f 2 (y) = 0.5 (gy) 273 



-133 y 

 (iii) Three-dimensional roughness f (y) = 0.5 e a^D 



f 2 (y) = 0.5 (py) 2/3 



We observe that these equations do not satisfy the boundary condition at 

 the wall which is not a very serious limitation since it can be easily 

 circumvented by assuming the formation of a laminar sublayer as in a 

 steady mean flow. 



We want to emphasize, however, that the above expressions are nothing 

 more than a convenient representation of actual measurements. It may be 

 that the true distribution follows some other law which conceivably will 

 be described by an equation derived by a more rigorous analytical process. 

 Nevertheless, one will expect that close to the boundary, which is the 

 region we are mostly concerned with, the results by the two methods will 

 be closely the same since the experimental work was conducted within the 

 range of conditions which most likely will materialize in an actual case. 

 We claim therefore in concluding this phase of the study that by means of 

 the expressions given above it is possible to describe the mean velocity 

 distribution near the ocean bed in terms of the surface wave character- 

 istics, the depth of the water and the grain size of the uniform material 

 forming the bed. 



4. HYDRODYNAMIC EFFECTS OF THE FLOW ON THE BOUNDARY 



The strong interaction between the flow and the loose material form- 

 ing the bed of a stream has been studied at great length by numerous 

 investigators. The results of these studies led to the development of 

 theories describing the phenomenon of sediment transport in rivers. It 

 has been reported some time ago, (Einstein, 1948), that a basic similarity 

 exists between the motion of sediment in a river and in a large body of 

 water in which the main motion is oscillatory. This, of course, does not 

 imply that the laws governing the two phenomena are exactly the same , nor 

 that the theories describing the motion in one case are directly applic- 

 able to the other. It is, however, reasonable to assume that some funda- 

 mental concepts used in the derivation of one theory may be applied in 

 the derivation of the other. Since we intend to use in our study some of 

 the basic principles associated with the motion of sediment by steady 

 streams of water we consider it appropriate to give a brief outline of 

 the historical development of theories related to this problem. 



a. Theories of Sediment Transport in Rivers 



The older equations which were used to determine rates of sediment 

 transport in rivers have been derived mostly by means of empirical methods. 



