GΩi dl − χ dl ƒ(v) = 0,
ƒ(v) / G = Ωi / χ = mi.
(1)
But it has been already shown (§ 66) that ƒ(v) = ζGv2/2g,
∴ ζv2 / 2g = mi.
(2)
This may be put in the form
v = √ (2g/ζ) √ (mi) = c √ (mi);
(2a)
where c is a coefficient depending on the roughness and form of the channel.
The coefficient of friction ζ varies greatly with the degree of roughness of the channel sides, and somewhat also with the velocity. It must also be made to depend on the absolute dimensions of the section, to eliminate the error of neglecting the variations of velocity in the cross section. A common mean value assumed for ζ is 0.00757. The range of values will be discussed presently.