CHANNEL IN WHICH HYDRAULIC RADIUS REMAINS CONSTANT 51 



where p represents dy/dx. Clearing the equation of fractions and 

 radicals 



p^{x^ -F?) -B? = 



Differentiating with respect to p, 



p{x^ - R^) = 



Now eliminating p between these two equations we obtain the singular 

 solution of the differential equation, 



X = zizR 



which is the equation of the pair of tangents common to the family of 

 curves y = R cosh~^x/R + C for different values of C. This solu- 

 tion satisfies the differential equation as well as the physical require- 

 ments of the problem. It, too, requires a bottom section. The two 

 solutions may be used in combination, as shown in Fig. 407. 



The channel of constant hydraulic radius is clearly impractical for 

 earthen canals, for the only shapes in which the bottom portion would 

 be stable allow but little variation of the water surface. It was once 

 thought to be practicable, owing to the error of an early writer in fit- 

 ting a semicircular bottom to the catenary curve at the point where 

 the sides are vertical. The channel does, however, offer an experi- 

 mental approach for the determination of the adequacy of the hydraulic 

 radius as the sole shape parameter in the friction velocity formulas. 



