FLOW IN OPEN CHANNELS 313 



get the water surface below this level, then on passing the sluice the 

 depth of water increases as shown. Finally, when h = ^J 1 . H, the 



curve should be vertical where it intersects this line. Before this limit 

 is reached, however, the hypothesis that the stream lines are sensibly 

 parallel ceases to be even approximately true, and the curve becomes 

 modified as shown in the dotted lines. This is exemplified in the case 

 of a sluice fitted in a channel having 

 a very small slope. 



If hi and /? 2 be the depths of the 

 stream before and after its sudden 

 change of level, the value of h% may 

 be calculated. 



Let vi and v% be the velocities at 

 sections lii and A 2 . 



It is not now legitimate to assume that the loss due to shock at the 



/ \2 



sudden change of section is - - as in the case of pipe flow, since 



c/ 



the pressure over the area E' E (Fig. 136) is no longer uniform and 

 equal to that from E to F but varies with the depth, and hence one 

 of the fundamental assumptions made in deducing this formula is 

 unjustified. 



On applying the equation of momentum, however, to this particular 

 case, we have 

 Difference of forces acting in the direction of motion,] _ 



on the faces C D and G H \ ~ Pl AI ~ V* A *> 



where p\ and pz are the mean pressures over the areas AI and A 2 . 



Then 



The change of momentum per second, in) W ( . 2 A' 2! 



passing the sections C D and G H ~ $~ ( 



Also AI i'i = A 2 t'2, and if the section is rectangular, lii v\ = h% r 2 , so 

 that on equating the momentum per second to the force producing it, 

 we have 





2 i- 2 2 ~ 



" i' r 1 ~^ = v j'' 1 ~ 



fl _ 7 



-h, ~ 2 Aj ** 



