50 HYDRAULIC JUMP IN NON-RECTANGULAR CHANNELS 



radius by R and the area and wetted perimeter by A and P, 

 jf_'^_-^+dA__dA_ xdy 

 ~P~lP+~dP^dP^VWTW 

 The general solution of this differential equation is 



y = R cosh-i ^ + ^ 



Axes may be chosen so as to eliminate the constant, so that 



y = R cosh ^ — X = R cosh -^ 



R 



R 



Inasmuch as the curve is not closed at the bottom, even at infinity, 

 the " channel " will not hold water, and we may conclude that a channel 

 for which the hydraulic radius is a constant for any depth does not 



Fig. 407. Channels in which the Hydraulic Radius is Constant for all Depths of 

 Flow Above the Bottom Sections of Elementary Shape. 



exist. If we allow the restriction that the depth of flow be always 

 above a certain level, a bottom may be fitted to the curve. It is nec- 

 essary for the bottom to have a hydraulic radius equal to R, but it may 

 join the upper curve at any horizontal line. Figure 407 shows circular 

 and rectangular elements of the correct dimensions fitted to the curve 

 to form channels which have constant hydraulic radius as long as the 

 water level is above the top of the elementary section. 



This, however, is not the only possible solution. Returning to the 

 differential equation 



dy _ R _ 



dx ~ Vx^ - i?2 ~ ^ 



