104 Prof. A. E. Richardson on 



The Francis empirical formula for such cases shows 3 3', 

 as the coefficient. 



(a) Extension to the case of depressed nappes. 



If the pressures over the upper and lower surfaces are 

 different, 



G,(fc + iQ)-G^)= [^(| +H-y)]* 



9 



m +s )-m- < u > 



In most cases G 2 { w ) is D °t the same as G, for the nearest 

 singularity to $ on the down-stream side occurs where the 

 nappe ceases to be a free stream-line, and this will be 

 different in the two cases. 



However, so long as this place is not close to the edge 

 c will not be altered very much, and the effect of a partial 

 vacuum behind the nappe will be to increase the flow. 



Experimental results show that the flow is approximately 

 given by Q = cK. 



Hence, if c' refers to a different shaped weir face and the 

 nappe springs clear in both cases, c'/c should be nearly inde- 

 pendent of the head. This agrees with Bazin's experiments*. 



(b) Case Q(w)~~B — e w {approximation to flow over a iveir). 



TakeB>l. 



dw 



dz 

 The singularities of -f- in the finite part of the w-plane 



are 



10 = 1717T. 



w = log B -f i 2W7T. 



Confine attention to the strip in the w-plane for which 

 0<^<7r. The singularities are then 



on i/r = 0, = logB and </> = 0, 



on yjr = 7r, <j) = 0. 



* Bovey, 'Hydraulics,' p. 101. 



