310 HYDRAULICS AND ITS APPLICATIONS 



fall in the level of the bed and in the depth of the water, is equal to the 

 increase of kinetic energy together with the loss in friction per unit 

 length of channel. 



For uniform flow such as occurs in a culvert with slope, velocity, and 



depth of water constant, we have = 0, ' ' = 



il I il I 



. _/V /> 

 = 20' A 



or the total potential energy is absorbed in overcoming frictional resist- 

 ances. This gives the relation between the slope and the velocity, or the 



A /2 (i i A 

 discharge Q, tor since v = - V ' . ., 



so that for a given slope we can find the section for v to be a constant, 

 and to give any required discharge. 



With a rectangular section, breadth /;, we get for either uniform or 

 non-uniform flow, if b is constant and if Q is constant 



Q v b U = const. 

 /. v h = const. 

 . d v . d h 



h -<ri + v in= 



d v _ _v d h 

 ' ~dl~ h' cTl' 

 Substituting this value, equation (2) becomes 



' _ C U l = Jii d JL -L/''' 2 P 



dl g h d I """ 2 g ' A 



._* P 



dji _ <2g 'A 



'' dl :.= \-2 (3) 



ffh 



n -j -I 



Here 7 still represents the rate of increase of kinetic energy 



fj It Cl i 



with length, and shows that the K. E. increases when the depth 

 diminishes, i.e., when - is negative. 



If b be great in comparison with //, we may write = 



'* 1 b It h 



(approximately), especially if, as is very usual in open channels, the 

 bottom is rougher than the sides. 



