﻿Problem of the Weir. 785 



Hence, h < h' < /i + d ; 



so that the stream-surface is always depressed on passing 

 from the upper to the lower regions, but the lower stream is 

 deeper than the upper. 



§ 12. Putting <£>(£)= sin 2 ^ f in (11) the pressure HP may 



be found: 



^=7T^+U 2 /t[^(^^ 



where v 2 =l + /e(S — vT). 



Thus £P can be determined for any particular values of U, 

 h, and d, when in has been found for that case from (20). 



§ 13. The following Table (p. 786) of values of «r, v, and 

 &/U 2 h has been calculated for various values of k and 6 in 

 order to institute a comparison between the three types of 

 motion which have been worked out. 



From the Table it appears that irrotational motion differs 

 in character from the cases of rotational motion considered. 

 In the former case the depth of the lower stream decreases 

 as the height of the weir increases, tc being kept constant, 

 while in rotational motion the reverse is the case. It follows 

 that calculations based on the assumption of irrotational 

 flow will differ widely in this respect from those based on 

 the more physical assumption of rotational flow. 



As a particular case, illustrating the use of the Table, 

 consider a stream whose depth in the upper reaches is 4 ft. 

 Let the height of the weir be 1 ft. and the surface velocitv 

 of the upper stream 16 ft. per sec. These values give 



S=l--r-rf/A = l-25, fc = 2gh/U 2 = l. 



From column I. we find *r= '8431, v= 1*1861, £Pj U 2 h = '1138, 

 „ II. „ ^=1-2125, v=l-0186^/U 2 /< = -1344, 

 „ HI. „ cr=: 1-1721, v = l-0383,^/U 2 / i = -1280. 



Hence, for 



Case I. A r = 3'3724 ft, I.J' = 18-978 fp.s.,^= 116-53 lb. wt. 

 „ II. h' = 4-8500 ft., U' = 16-298 f p.s., ^= 137*63 lb. wt. 

 „ III.A'=:4-6884ft., U' = 16-614f.p.s.^.= 131-071b.wt. 



Here h' is the depth of the lower stream, U' its surface- 

 velocity, and HP the horizontal pressure on the weir per foot 

 breadth. If the water were at rest with its surface 4 it. 



Phil. Mag. S. 6. Vol. 23. No. 137. May 1912. 3 F 



