FLOW IN OPEN CHANNELS 309 



will be greater in a shallow channel having a rough bed, but in general 

 the results calculated on this assumption may be taken as substantially 

 correct. 



If p = wetted perimeter of section, and if (v) 2 = mean square of the 

 velocity from A to B 



If A B = S I we have r = ^ . B I, while if V A = r, V R = r -f ' '. 8 / 



and (i>) 2 = (v + m ~5~T ^ > where m is less than unity. 

 So that, neglecting small quantities of the second order 



o . & v x ] 



drj V dl .fv* P*, 

 dl Bl = -*T + J *lj-A* 1 

 dr_v civ fv* P 

 21 ' 31^ 2> ' A' 



This is the general equation of flow in an open channel, v being the 

 mean velocity at a cross section, and though the assumptions made in its 

 conception are not altogether justified by the result of experiment, yet it 

 forms a useful guide and is capable of a wide range of application in the 

 general problems of channel flow. 



If h is the depth of water at A (measured vertically from the surface), 

 the depth at B is given by 



Again the depth at B is given by 



- _ 



dl~ dl' 

 Substituting this value in (1) we get 



. dh_v dv , fv* P , . 



~Ti-~ 9 J7 H "~27/'Z' 

 giving the general equation in terms of the slope of the bed. 



The physical interpretation of this equation is that the total loss of 

 (potential and pressure) energy per unit length of the channel, due to the 



