ALTERNATE DEPTHS 19 



physical picture of the condition of critical flow. In any rectangular 

 channel critical flow exists when the velocity head is one-half the depth 

 of the moving stream. Thus in Fig. 201 at P the velocity head, repre- 

 sented by the vertical distance from G to the horizontal axis OL, is 

 equal to one-half the depth, NP. 



But a little further discussion of Fig. 201 is necessary. The curve 

 EFGH, if extended according to its mathematical equation, has as 

 asymptotes the straight lines OL and OK. The cover EH need not 

 follow a straight line. Any smooth curve would do. The curve EFGH 

 would change correspondingly in shape, but would have the same 

 properties as enumerated above, and would, for each depth of flow, be 

 at the same distance above the top. 



The depths at E and H always have a definite relation to each other 

 because at these two points the sum of the depth and the velocity head 

 is the same. Depths so related are called alternate depths. 



Let Di = depth at E 



D2 = depth at H 



Then 



V 2 y 2 



17 + ^' = 17 + ^^ [206] 



Substituting for velocities their values in terms of Q, and transposing, 





= D2- Di 



= D2-D1 



2g D2^D,^ 

 Dividing through by D2 — D\, and substituting D^ for Q^/g, 



If the values of any two of the depths are given, the value of the third 

 can readily be obtained. The equation is symmetrical in Di and Z>2- 

 When Di equals Dc, D2 will also equal Dc- When Dy is less than Dc, 

 D2 must be greater than Dc, and vice versa. 



The method described above is not the only device by which theoreti- 

 cally the depth of water flowing in the rectangular channel may be 

 varied. Suppose that instead of the rigid top applied to the conduit as 

 shown in Fig. 201, the bottom be raised on a gradual ascent as shown in 

 Fig. 202. The water will then flow up the incline as indicated in the 

 figure. As the water rises from £, some of the initial kinetic energy will 



