a a ee 
 eross sections of the issuing streams will be 
similar, and the coefficient of contraction there- 
_ fore constant. 
SES i ee 
GENERAL PRINCIPLES OF HYDRAULICS 453 
‘the area of the cross section of the stream above the weir. Calculate 
Q, the discharge over the weir, neglecting velocity of approach, then 
2 
v= Q’/A approximately, and h)= e ; 
392. Triangular Notches.—A triangular or V notch has one great 
advantage over the rectangular notch. In the former the linear 
dimensions are in a fixed ratio to one another, whatever be the depth of 
water in the notch, and it follows that the 
Let the edges of the notch (Fig. 745) have 
equal inclinations to the vertical, and let the angle 
between them be 20. Neglecting in the first 
instance the contraction of the jet and the effect of friction, consider a 
strip of the notch at a depth y and of width dy. The length of this 
strip is 2(h —y) tan 0, its area is 2(h-—y) tan @- dy, and the velocity of 
the water through it is ./2gy. Hence 
dQ=2%(h-y) tan 0+ dy /2gy=2 tan 0 /2g(hy! — y!)dy, 
h 
and Q=2 tan 6 75) (hy — y? dy = 2 tan 6 ,/29(Zh! — Zh!4) 
0 
=; tan 0 /2g - hi. 
If & is the coefficient of discharge, then the actual discharge is 
Q =, tan 6 /2y-hi. 
The late Professor James Thomson found & to be 0°617 ; taking this 
value of &, and making 20=90°, which is the usual angle of the notch, 
Q =2°64h! = 2-641? /h. 
393. Partially Submerged Orifices.—When a rectangular orifice is 
partially submerged, as shown in Fig. 746, the orifice may 
be considered as made up of two parts, the upper of depth - 
h, —h, —h’, and breadth b discharging into the atmosphere 
under a head varying from h, —h’ to h,, and the lower of 
depth h’ and breadth } fully submerged and discharging 
under a constant effective head h,—h’. Let Q, and Q, 
denote the discharges from these upper and lower parts 
respectively, then, by Articles 389 and 383, 
Q, = $b J2g{(hg—h’)!—h3}, and. Q. = kbh’ ./2g(hy — h’). 
The total discharge is Q, + Q,. 
394. Drowned Weirs.—A weir is said to be drowned or submerged 
when the tail water level is above the crest of the weir, as shown in 
Fig. 747. The formule of the preceding Article .— 
Fia. 745. 
Fia. 746. 
may be applied to a drowned weir by putting }, “\’ 
h,=0 and changing h, to h, then the discharge he ert ! 
over the weir is D 
Q=Q) + Q. = kd J 2h —h’)i + koh’ J2gh — h’) Se 
= 2kb /2g(h -W)(h+ 4h’) 
