this phase of the problem is beyond the scope of the present study. In 

 a practical application it is not as important to know the exact value of 

 the critical Reynolds number as to be able to predict with sufficient con- 

 fidence that under the existing conditions the flow regime in the boundary 

 layer is not laminar and consequently that the theoretical laminar solu- 

 tion is no more applicable. This type of information can be obtained by 

 experimental methods. The studies of Li (1954) and Manohar (1955) are 

 two outstanding sources of such information. The procedure used by these 

 two investigators as well as their results will be discussed briefly in 

 Appendix A. 



(ii) The Turbulent Case. After we have established the fact 

 that the flow in the boundary layer is unstable we proceed with our main 

 task which is the description of the velocity distribution within this 

 layer. We define the instantaneous velocity components in the two di- 

 rections as 



(3-18) 



and 



where u is a simple harmonic function of time and space while u' and v' 

 are the turbulence components. We claim that the amplitude of u is a 

 function of y only and that its frequency is the same as the frequency 

 of the wave. A reasonable expression for u will be then of the form (a 

 form similar to equation (3-15) of the laminar case) 



u (y,t) = u q jsincut - f 1 (y) sin Lt -*" 2 (y)] } (3-19) 



where f (y) and f (y) are functions of y only. These functions can be 

 determined by experimental methods. 



This, of course, implies actual measurement of the velocity at points 

 very close to the boundary. The effective thickness, however, of the 

 boundary layer in the actual case does not exceed a few millimeters and 

 even if it were possible to make velocity measurement within such a thin 

 layer the experimental flume must have practically prototype dimensions. 

 The experimental work can be simplified considerably by making use of the 

 principle of superposition mentioned above. Therefore, setting 



(3-20) 



where u = U = u sintut 

 1 °° o w 



and 



f^y) sin ^t-f 2 (y)J (3-21) 



the velocity component u in the boundary layer for given values of a and 

 cu can be obtained from measurement of f (y) and f~(y) in a flume in which 

 the water surface is maintained at rest while the bottom, is oscillating 



