18 BERNOULLI'S THEOREM APPLIED TO AN OPEN CHANNEL 



the gradual conversion of velocity head into pressure is not sufficient to 

 raise the water, and hence the previous accumulation of pressure is 

 gradually drawn upon in raising the surface to higher and higher 

 elevations. 



If the whole apparatus is open to the air, the water at H would no 

 longer follow the upper slanting surface, but would break loose and 

 flow away with level surface from this point. If the air were excluded 

 by suitable means beyond the point H, it might be possible to have the 

 upper water surface continue to cling to the slanting face, in which case 

 the water beyond H would be under a vacuum, increasing in intensity 

 the farther the expansion of cross section is continued. 



The pattern of the variation of pressure along the top EH is obviously 

 reasonable from the following considerations. For a given small change 

 in cross section, the change in velocity near E will be much greater than 

 near H. Also, since the velocity head is proportional to the square of 

 the velocity, for a given change in velocity, the change in velocity head 

 will be greater as the velocity itself is greater. To illustrate, the differ- 

 ence between the squares of 101 and 100 is 201, though the difference 

 between the squares of 11 and 10 is only 21. That is, a change of 1 in a 

 velocity of 100 would have about 10 times as much effect on the velocity 

 head as a change of 1 in a velocity of 10. For both of these reasons, then, 

 the pressure changes most rapidly at E. 



The cross section at which the upward pressure on the surface EH is a 

 maximum, shown on Fig. 201 at NP, is of particular interest. At this 

 place a tangent to the curve at G is parallel to the water surface at N, 

 and the change of velocity head is just sufficient to produce the change 

 in elevation of water surface, so that there is neither accumulation nor 

 reduction of pressure. On account of this peculiar balance between 

 velocity head and depth, this particular section may be said to mark a 

 critical point or a condition of critical flow. We shall find hereafter that 

 this critical point appears in numerous important relations. Let the 

 subscript c denote values at the critical point. From the value of the 

 static pressure on the bottom given by equation (203), it is easily proved 

 by the differential calculus that 



Vc = V^ = </'^ [204] 



and 



Dc = — = ^1^ [205] 



From equation (205) it is evident that at the critical point the velocity 

 head is one-half the depth. This leads to a convenient definition or 



