FLOW IN OPEN CHANNELS 



303 



the necessary flushing. On the assumption that ?; = C Vm i, where (7 = 

 constant, the only essential condition to be satisfied for v to be inde- 

 pendent of the depth is that the hydraulic mean depth shall also be 

 independent of the depth of water. 



Thus the required channel must have sides formed by a continuous 

 curve such that the area bounded by the sides, and any two horizontals 

 varies as the length of the arcs intercepted between these horizontals. 

 No curve can be found to satisfy these conditions, though close 

 approximations may be obtained. 



Obviously, a rectangular section of great depth compared with its 

 width would satisfy the conditions approximately, and especially if its 



k/77 



FIG. 181. 



bottom were constructed so as to offer less resistance per unit area than 

 its sides. 



A construction which gives accurate results between certain limits 

 may, however, be obtained as follows. 



In Fig. 131 let x be the half breadth of the section at a height y above 

 M' M where the half breadth is b, and let s be the length of the arc M N 



The position of the axis M' M and the breadth 6 are usually fixed 

 from a consideration of the minimum discharge to be expected through 

 the channel, a trapezoidal channel having an upper breadth M 1 M(= 2 b) 

 being designed to take this minimum discharge when running full. 

 Let m be the hydraulic mean depth of this small channel, let p be its 

 half perimeter, and a its half area. It is required to continue the sides 

 of this channel so as to give a section for which the hydraulic mean 

 depth A -v- P shall be equal to m for all depths of water. 



