BACKWATER FUNCTION 323 



Then as before H = 2 ; 1 - ^- = 1. 



J 6 

 Also in equation (10), we have 



hi = 1-5 ; li = ; <f> (wn) = < ( ^ ) = 1-159 (from table). 



To determine the depth at a point 50 feet up stream we have 

 Z 2 = 50 feet, so that the equation becomes 



1-50 - ;/ 9 = -002 (50) + g 1-159 - <j> ( -~ 



\ 



^ _ ^ ( ^ > _ ^o _ - 10 _ -773 



' ' 2 3 9 V 2 ) 



= -627. 



Putting fea " g * ( 2" ) = * 627 = 2/ we get 



if /? 2 = 1-8, y = 1-8 - 1-0146 - -627 = + '1584 

 fca' = 1-9, y = 1-9 - 1-1793 - '627 = + '0937 

 /i 2 = 1-95, y = 1-95 - 1-3393 - -627 = - '0163 

 On plotting values of /? 2 and ?/, the curve shows that y 2 is zero when 

 // 2 = 1-946 (approximately), and this, therefore, gives the depth at the 

 given point. 



At a point 20 feet up stream we have 



**\ = -667 



On solving this equation in the same way, we find that the depth here 

 is 1*932 feet (approximately). Since the breadth is constant, the mean 

 velocity at any point is inversely proportional to the depth, so that at the 

 two points 20 and 50 feet up stream the velocities are increased by the 



2 2 



fall in the ratios and T respectively. 



This action becomes increasingly important as the slope is diminished. 

 For example, in the previous case, if the slope were diminished to "001, 

 the velocities at the same two points would be increased by the fall in the 



ratios ^^ and ^=^ respectively. 



I'OO X* | a 



As a further example of the use of these formulae, consider the case of 

 a flume of rectangular section, feeding a forebay from which a turbine 

 is to be supplied. 



The breadth of flume is 20 feet, the slope 1 in 1,000, the length 1,000 

 yards, and the value of /== "003. The discharge required is 500 cubic 

 feet per second. 



Y 2 



