ALTERNATE DEPTHS 



39 



taneously. For both, the derivative of the discharge with respect to 

 the velocity should be zero. In the first definition the discharge is to 

 be a maximum, and in the second it is to be constant. 



d d dA —A 



The derivative of the total head above the bottom of the channel, with 

 respect to the velocity, is likewise equal to zero. In the first case it is 

 to be constant, and in the second it is to be a minimum. 



dV 



\2g J dV 



-V 



The values of dA/dV and dD/dV are the same as in the proof of the 

 preceding definition. Substitution and elimination of dD/dV will there- 

 fore give equation (401), as before, so that the flow must be critical. 



Alternate depths. For a given discharge and total head only two 

 different depths of flow are possible, except when the flow is critical, 

 when the depth may vary continuously over a limited range. The 



.1-4 



*3 



a2 



012345678 

 D+ -r— = Total Head Above Bottom of Channel, in Feet 



Fig. 401. Total Head as a Function of Discharge and Depth of Flow, in a Typical 

 Trapezoidal Channel. 



relation between these alternate depths is easily obtained for channels 

 with cross sections in which the area is a simple function of the depth. 

 Equation (207) gives the relation for rectangular channels, and similar 

 expressions may be derived for triangular and trapezoidal cross sections. 

 Alternate depths in channels having more complicated cross sections, 

 however, are best obtained by a semi-graphical method. The procedure 

 is indirect. A number of depths are first selected, covering the range 



