assumed equal to the momentum exchange coefficient. Assuming that shear 
due to viscosity may be neglected, compared with that due to momentum 
transport, the depth, d, may be introduced: 
t= t) [(d-Y)/d] = Sv 0 {[u - $16 (au/av)| i lu tole (au/ayy|} , (26) 
where 
To = shear stress at the bed; equal to goRS 
t = shear stress at elevation, Y 
R = hydraulic radius (in feet) 
S = energy slope 
g = acceleration due to gravity 
o = density of the water 
u = horizontal flow velocity 
Using the logarithmic formula based on von Karman's (1934) similarity law 
for the distribution of flow velocity, du/dY may be calculated: 
du/dY = (1.0/0.4) (u,/Y) , (27) 
where u, is the shear velocity and equal to Gone. Substituting this 
value into equation (26) and solving for 1/2 1, v yield: 
Fle v = (-0.4) Yu, (d-Y)/d . (28) 
Using this value in equation (25), separating variables and introducing 
the abbreviation: 
Zia VE CORA uy) (29) 
the result can be integrated from a to Y. The solution is: 
(C/o Ned Wi nayy duane! 2 (30) 
It has been found that equation (30) gives the correct form of the 
distribution function, but the value of the exponent Z _ given by equa- 
tion (29) does not always agree with the exponent that fits the measured 
data. Let Z' be the exponent which best fits the data. It was found 
that torshagh) values o£ Z)/CZ>10)),) 2" was) stenifacantliy, less. Asi iZ) jas 
65 
