NOTCHES AND WEIES 



139 



between the limits // and 0. It might also be deduced from equation (2) 

 of p. 129 by writing H = ; d = H. 



The value of (7, in addition to varying with the breadth and depth of 

 the weir, varies largely with its position with respect to the sides and 

 bottom of the approach channel. Where the sides of this channel are so 

 far removed as not to affect the contraction of section at either side of the 

 weir, this is said to have two end contractions, while where the channel 

 is of the same breadth as the weir, the latter has no end contractions, and 

 is termed a Suppressed Weir. The bottom contraction is also affected by the 

 nearness of the sill of the weir to the bottom of the channel of approach 

 and by the inclination of the weir face. 



Where the " nappe," or falling sheet 

 of water, comes into contact with the 

 top or down-stream face of the weir, 

 as shown in Figs. 80 and 88, a further 

 and in general irregular variation 

 takes place in C. In what immedi- 

 ately follows, only those weirs will be 

 considered in which the nappe springs 

 clear of the crest and discharges freely 

 into the air, the crest being vertical, 

 narrow in comparison with the head, 

 and having a sharp upstream edge. 



Boussinesq's Theory. A theoretical treatment of the flow over a 

 suppressed weir, which is of much interest, is due to Boussinesq. 



Let H = depth above sill. 



,, a be the amount by which the arched under side of the vein rises 

 above the sill. (Assumed, and verified by Bazin, to be a definite fraction 

 (13) of H) (Fig. 76). 



At the vertical section, at which a is a maximum, let d be the vertical 

 thickness of the vein. 



If we assume that, over this section, the stream lines are portions of 

 concentric circles and are normal to the section, and also assume that 

 there is no loss of energy up to this section we have the conditions 

 obtaining in free vortex flow and as in that case, if r is the radius of 

 curvature of any stream line, v r is constant. 



Consider any filament, x above the sill, of velocity v, and radius r. 



Let v a = vel. at A ; v b = vel. at B. 



FIG. 76. 



Then v a = * 2 g (H - a) ; v = * 2 g (H a} . 



