62 



so that 



By geometry, 

 so that 



BACKWATER CURVES — INTRODUCTORY 



d_ 

 dx 



Ou, — Oo 



dx 



©=-(?) 



^di 

 dx 



[601] 

 [602] 



5o = 





-(?) 



3n 



5o c^x = 1 — ( — J dy 



Integrating, and letting re = when y = jc 



[603] 



The graph of this equation is shown in Fig. 601. It is evident that the 

 two branches of the curve shown are different backwater curves, that is, 

 a water-surface profile could not possibly follow continuously through 



Fig. 601. Backwater Curves in a Frictionless Rectangular Channel. 



both. Note that one branch is asymptotic to the bottom, while the 

 other is asymptotic to a horizontal line at an elevation yc/2 above the 

 critical depth. This corresponds to a complete regain of the velocity 

 head. Both curves cross the critical depth vertically. Except in the 

 range near the critical depth, these curves would provide a close approxi- 

 mation to the surface curves in a steep rectangular channel, over dis- 



