FLOW IN OPEN CHANNELS 315 



i 

 >- 



2 i 3 / 2 i 



Case 1 (b). Here ->- < 1, while /i is greater than V - H and is 



}ess than H. 



In equation (6) the numerator is now negative, while the denominator 



is positive, so that ^ j is negative, or the depth h diminishes down- 



stream. 



As the velocity increases and h diminishes, the denominator of the 



fraction - vanishes before the numerator, so that -= tends to 



,,3 _ J H s 



:{ / 2l 

 a limiting value oo where h = V y H. 



At this point the surface curve becomes vertical as shown in Fig. 137. 

 If produced by a sudden 'drop in the bed of a stream, as shown in this 

 figure, h increases up-stream, and 

 approaches more nearly to H as this 

 distance increases, the surface curve 

 being asymptotic to the line R R r . 



Such a drop in the bed may cause 

 an appreciable increase in the velo- 



city of flow for a considerable dis- FlG 137 



tance up-stream and may thus affect 



the foundations of structures (bridges, etc.) which may be placed 

 up-stream, besides causing serious erosion of the bed. 



In the case of the sluice (Fig. 135), the water after rising to the 

 height h 2 is governed by this second set of conditions, so that the level 



again falls until h = \/ r- . H. Inertia then causes the level to fall 



below this, when we have the conditions of Case 1 (a) repeated. Thus a 

 series of stationary waves are produced, the level alternately rising and 



/TTt 



falling above and below that given by h = \/ ^ . H. At each 



successive jump a loss of energy occurs, and the velocity energy after 

 the jump is therefore diminished. It follows that the value of // 2 must 

 be greater after each successive jump, and ultimately will become equal 

 to H, after which steady flow occurs. 



The same reasoning applies to the stream after passing the drop in the 

 bed (Fig. 137) the depth ultimately settling down to H. 



The state of affairs outlined in this second case may be met with where 



