Q = 4⁄15 cB √(2g) H1/2,

Fig. 46.

When a notch is used to gauge a stream of varying flow, the ratio B/H varies if the notch is rectangular, but is constant if the notch is triangular. This led Professor James Thomson to suspect that the coefficient of discharge, c, would be much more constant with different values of H in a triangular than in a rectangular notch, and this has been experimentally shown to be the case. Hence a triangular notch is more suitable for accurate gaugings than a rectangular notch. For a sharp-edged triangular notch Professor J. Thomson found c = 0.617. It will be seen, as in § 41, that since 1⁄2BH is the area of section of the stream through the notch, the formula is again of the form

Q = c × 1⁄2BH × k √(2gH),

where k = 8⁄15 is the ratio of the mean velocity in the notch to the velocity at the depth H. It may easily be shown that for all notches the discharge can be expressed in this form.

Coefficients for the Discharge over Weirs, derived from the Experiments of T. E. Blackwell. When more than one experiment was made with the same head, and the results were pretty uniform, the resulting coefficients are marked with an (*). The effect of the converging wing-boards is very strongly marked.

Heads in
inches
measured
from still
Water in
Reservoir.		 Sharp Edge. Planks 2 in. thick,
square on Crest.		 Crests 3 ft. wide.
3 ft. long. 10 ft. long. 3 ft. long. 6 ft. long. 10 ft. long.10 ft. long,
wing-boards
making an
angle of 60°.		3 ft. long.
level.		 3 ft. long,
fall 1 in 18.		3 ft. long,
fall 1 in 12.		 6 ft. long.
level.		10 ft. long.
level.		 10 ft. long,
fall 1 in 18.
1 .677 .809  .467  .459 .435[4] .754 .452 .545 .467 .. .381 .467
2 .675 .803  .509* .561  .585* .675 .482 .546 .533 .. .479* .495*
3 .630 .642* .563* .597* .569* .. .441 .537 .539 .492* .. ..
4 .617 .656  .549  .575  .602* .656 .419 .431 .455 .497* .. .515
5 .602 .650* .588  .601* .609* .671 .479 .516 .. .. .518 ..
6 .593 .. .593* .608* .576* .. .501* .. .531 .507  .513 .543
7 .. .. .617* .608* .576* .. .488 .513 .527 .497  .. ..
8 .. .581  .606* .590* .548* .. .470 .491 .. .. .468 .507
9 .. .530  .600  .569* .558* .. .476 .492* .498 .480* .486 ..
10 .. .. .614* .539  .534* .. .. .. .. .465* .455 ..
12 .. .. .. .525  .534* .. .. .. .. .467* .. ..
14 .. .. .. .549* .. .. .. .. .. .. .. ..
Fig. 47.

§ 44. Weir with a Broad Sloping Crest.—Suppose a weir formed with a broad crest so sloped that the streams flowing over it have a movement sensibly rectilinear and uniform (fig. 47). Let the inner edge be so rounded as to prevent a crest contraction. Consider a filament aa′, the point a being so far back from the weir that the velocity of approach is negligible. Let OO be the surface level in the reservoir, and let a be at a height h″ below OO, and h′ above a′. Let h be the distance from OO to the weir crest and e the thickness of the stream upon it. Neglecting atmospheric pressure, which has no influence, the pressure at a is Gh″; at a′ it is Gz. If v be the velocity at a′,

v2/2g = h′ + h″ − z = h − e;