FLOW IN OPEN CHANNELS 



317 



dh 



(equation 6) is now positive and the depth increases down-stream. 

 cl I 



Since, as h increases, the nu- R 

 merator vanishes before the de- 

 nominator -j: = in the limit, 



i.e., the surface curve tends to 

 become asymptotic to the line 

 R R' (Fig. 139). 



This state of affairs is at- 



FIG. 139. 



tained at a sluice in a stream having a slope greater than ^. 

 Case 2 (6). 

 Let r > 1, and let h be greater than H and less than /JV - . H. 



J 



dh 



f 



Here -=j (equation 6) is negative and the depth diminishes down- 



Cl> L 



stream. As h diminishes the numerator vanishes before the denominator 

 and in the limit -j-% = 0, or the curve becomes asymptotic to the 



line R R', and the stream settles 

 down to the uniform depth H. 

 This state of affairs is realised 

 where an obstacle in a river 

 bed may have caused the level 

 to rise to within the required 

 limits. The surface curve is 

 then as shown in Fig. 140. Up- 

 stream, as h increases it finally 



FIG. 140." 



reaches the value 



d h 



J 



Here ~ is oo , and the curve becomes 



perpendicular to the bed of the stream. As h increases still further the state 

 of affairs considered in Case 2 (c) is attained. This vertical front is seen 

 when a sudden rush of water, such as may be produced by the bursting of an 

 embankment, is caused in a channel of fairly rapid slope. It is also seen in 

 the bores which occur at certain states of the tide in the Seine between 

 Havre and Kouen, and in various other rivers and contracted channels. 

 Case 2 (c). f 2 



> 1 





Let 



