342 HYDRAULICS AND ITS APPLICATIONS 



Bazin, experimenting on a stream having the maximum velocity in the 

 surface, obtained 



_ aw = 25-4 V m am 



- v b = 10-87 V m sin~tf 



- v b = 36*27 V m sin 

 the general equation being 



v = (v s } max - 36-27 V m sin ( |) ^ 



Here v mmM , is the mean velocity over the whole section, and m is the 

 hydraulic mean depth, the dimensions being taken in feet and the velocities 

 in feet per second. He also states that wherever the position of maximum 

 velocity, the relation 



v max v b = 36-27 V m sin 



holds true. In the vertical plane containing the filament of maximum 

 velocity, we have from equation (7) 



W sin 

 36-27 v m sin = ^ {h yi} 2 



Substituting the value of thus found, in equation (5), when the 



2i //. 



maximum velocity is below the surface 



a* - v = 36-27 



a formula which gives fairly accurate results in practice. 



Rankine states that the maximum, mean, and bottom velocities may be 

 taken as being in the ratio 5 : 4 : 3 in ordinary cases, and in the ratio 

 4 : 3 : 2 in very slow currents, and these ratios may be taken as being 

 approximately correct for streams and rivers of moderate size. 



Velocity at Mid-depth. 



From equation (5) of this article, we may obtain the mean velocity 

 over any vertical by integrating the sum of such terms as v 8 y over the 

 vertical, and by dividing this sum by its length h. Thus 



rh W sin B, xo) , 





(9) 



while from (5) we may obtain the velocity at mid-depth, i.e., where 



h 



