FLOW IN OPEN CHANNELS 311 



Substituting this value in equation (3) we have, for such a channel, 



d h % Q h . ! % f) hi \ //n 



TT? = "72" * 1 'a { W 



l L. i 



h (I h j 



This gives the slope of the water surface, and from a solution of this 

 quation, the profile of this surface may be determined. 



Thus for steady uniform flow 1 -^- = 0, i.e., the depth is constant 



throughout. 



fv 9 



~fgTi = 

 h = (5) 



ART. 89. NON-UNIFORM FLOW. 



If, as is usual, the motion while being constant at a point, varies from 

 point to point in the length of the stream, so that -j and therefore t 



Clr i Cl V 



is not zero, let H be the equivalent depth of an uniform stream of 

 breadth b which would give the same discharge as the variable stream in 

 unit lime. 



Then // = {-- from (5) 



where Q = V b H or V = -~- 



. (p_ _ 2 fi i H or ^_ __ ^ _ \giJP_ 



Substituting - * . for v 2 in (4), this becomes 



f l HI 1 



' d <rr= { \ ~ -jr^n (6) 



*~W- T 



J -J 



This is the differential equation to the curve forming the longitudinal 

 profile of the surface. 



Since the value of -j \ depends both on the ratio which may be 

 d L n. 



2 i 



artificially adjusted in any stream, and on the ratio r which is fixed 



