76 BACKWATER CURVES IN UNIFORM CHANNELS 



is given by either 



/ 7 \2 q2 



depending upon whether the Manning or Kutter formula is to be used. 

 By the Manning formula, 5/ is a function of the depth alone for a given 

 discharge in a given channel. This is also true when the Kutter 

 formula is used, for though C is a function of the slope, the correct slope 

 to use is not the variable water-surface slope, nor the slope of the 

 channel bottom, but the friction slope Sf. 

 The rate of change of velocity head is 



d (V\ d Q^ -Q^dA -Q^dAdy -Q^Tdy 



© 



dx\2g/ dx2gA^ gA'^ dx gA"^ dy dx gA"^ dx 



[702] 



In this expression T is the surface width, A is the area of flow, dy is an 

 increment of the variable depth of flow corresponding to dx, an ele- 

 mentary distance along the stream, and the other factors have their cus- 

 tomary significance, so that the multiplier of dy/dx is seen to be a 

 function of the depth alone for a given discharge in a given channel. 



Equations (701) and (702) may now be substituted in the general 

 differential equation developed in Chapter VI, 



dx dx \2g/ 

 giving, after separation of the variables, 



1 _^ 



/I 3 



dx = ^ dy [703a] 



So — Sf 



This equation is of the form 



dx = f(y) dy [703b] 



The function f{y) is usually of a form that is difficult if not impossible 

 to integrate by the methods of ordinary calculus, so that graphical 

 integration is to be preferred. Values of f{y) are computed corre- 

 sponding to values of y covering the range of depths of the backwater 

 curve. Values of f(y) are then plotted against y on Cartesian coor- 

 dinates, and a smooth curve drawn through the points. The area 

 under the curve /(>») between yi and y2 is the distance along the stream 

 from where the depth is yi to where it is y2' 



