with boundary conditions 



u (0, t) = 



S | (3-14) 



u (», t) = U = u sinuit 



The solution of (3-13) satisfying (3-14) is of the form 



u (y, t) = u q | sincut - e~ 0y sin (u>t-py)~L (3-15) 



where R = / T~ 



We recognize the second term in the parentheses of (3-15) as the 

 solution of the diffusion equation 



(3-16) 



(3-17) 



which describes the flow near an oscillating flat plate (Schlichting, 1955). 

 The structure of the solution (3-15) suggests that it is possible to deter- 

 mine the velocity component in the x direction within the boundary layer 

 of an oscillating body of water by means of simple superposition of the 

 irrotational component at the bottom and of the solution of the oscillating 

 flat plate. This property has significant importance especially in con- 

 nection with the experiment. The solution thus obtained is a good ap- 

 proximation of the actual case provided that the flow within the boundary 

 layer is laminar. 



However, even when the surface of the wall is hydraulically smooth 

 one would expect that at a certain value of the Reynolds number defined 

 as 



u 6 

 "R = — 



at 



S 2 u 



- v — p 



Zy z 



ith boundary 



conditions 





u (o,t) = u sinuut 

 o 





U (co,t) = 



Nn 



the flow will become unstable. The knowledge of the critical value of 

 Reynolds number is very important to our problem, because the dynamic 

 effects on the particles under laminar flow conditions are quite different 

 than in turbulent flow. It would be very helpful, therefore, to establish 

 criteria of flow stability covering a wide range of wave characteristics 

 and bed roughness. This is rather an involved case in the class of prob- 

 lems of hydrodynamic stability. If we recall the lengthy calculations 

 required to solve the relatively simple case of the Blasius profile on a 

 smooth flat plate we may conclude without much deliberation that the 

 attempt to seek a theoretical solution for our case, of a mean unsteady 

 flow and a rough plate, does not offer much hope for success. Besides 



