with A 5 a constant of proportionality and u B the velocity from equation 

 (3-24) at an arbitrary distance, say y = D, from the theoretical bed. 

 A now can be calculated from the equation 

 5 2tt n 2D 



u(y , t)dydcot 

 — (6-7) 



4 TT D j U | y = D 



A large number of values could be obtained from (6-7) for a wide range 

 of variation of the independent variables, and then averaged out. This 

 average value may be considered as a universal constant and be used to 



calculate u,,, from (6-6) with u fi = |ujy_ D . 



The work could be substantially reduced by the use of a computer; 

 otherwise the operation is not much easier than the more accurate one 

 mentioned previously. Its main advantage over the latter is that it has 

 to be performed only once. Because computer time was not available when 

 the study reached this point, a simpler but less accurate method was used 

 to calculate u m . This method is adequate at least in establishing an • 

 order of magnitude. 



The method consisted of a numerical integration of equation (3-19) 

 in which the two functions f^ (y) and f 2 (y), as we have seen in section 



3 , were of the form 



133y 



f 1 (y) = 



.5e a p D 



f 2 (y) = 



2/3 

 .5 (Gy) 



One may observe that for given values of the parameters a$ and gD both 

 f 1 (y) and f£(y) are functions of the ratio y/D only. As a characteristic 

 set of these parameters, the mean values from the twenty-seven runs men- 

 tioned in a previous section were used. The ratio y/D was made to vary 

 in increments of .4, beginning with y/D = .2 and ending with y/D = 1.8. 

 The angle (jut was varying between and 2rr in increments of tt/12 beginning 

 with cut = TT/24 and ending with yjt = 47 rr/24. Typical velocity profiles 

 constructed this way and corresponding to different values of the angle 

 mt are shown of Figure 10. The value of the constant A5 obtained from 

 the integration of these profiles was found to be equal to .618. Equation 

 (6-4) can be written now as 



q B 

 c = .618 (6-8) 



2D I u I y=D 



This is the final result of our study. When c Q from (6-8) is intro- 

 duced in equation (6-2) the desired value of Q B is obtained provided, of 



22 



