20 BERNOULLI'S THEOREM APPLIED TO AN OPEN CHANNEL 



be consumed in lifting the water. At first, when the water has a high 

 velocity, sufficient energy can be supplied with but a slight decrease in 

 velocity. Hence, on the lower part of the slope the depth will increase 

 very slowly; but as greater heights are reached, it requires increasingly 

 greater changes in velocity to supply the required energy, and therefore 

 the depth, which varies inversely as the velocity, changes more rapidly. 



Fig. 202. Diagram Illustrating Flow in an Open Conduit of Constant Width. 



No friction or impact. The profile of the bottom is that necessary to raise the surface of the water 

 on the uniform slope EO; that is, the decrease in velocity head accompanying the increase in depth is 

 just sufficient to raise the water along the line EO. The critical conditions of flow exist at the section 

 NP. 



When the water reaches the section NP, the depth must increase at 

 the same rate as the surface rises. Hence, at this section, the bottom 

 must be level and the depth is the critical depth previously described 

 and shown at NP in Fig. 201. As stated before, at this critical point, 

 the velocity head is one-half of Dc. 



In order that the water surface may rise as the water flows beyond 

 this section, the bottom must descend as indicated in Fig. 202. If the 

 depth becomes infinite, so that the entire kinetic energy may be con- 

 verted into static head, the surface may rise to as a maximum limit. 



The bottom need not follow exactly the curve shown. If it follows 

 any smooth curve rising to the same maximum height P, the water sur- 

 face will follow a corresponding curve having for each elevation the 

 proper height above the bottom. 



A third method might be used to raise the water by converting 

 velocity head into static head. If the bottom of the conduit is kept 

 level as in Fig. 201, but if the sides, instead of remaining parallel to give 

 a uniform width to the channel, as in that diagram, are made to approach 

 each other and then diverge, any desired rate of change of the velocity 

 head may be secured. Figure 203 shows the variation in width neces- 

 sary to secure a uniform slope to the water surface. The curves show- 

 ing the variable width, if prolonged, would be asymptotic at the two 

 ends to the straight lines AG and OK. Any other curves having the 



