In cases where u can be directly determined, these formulae give the
discharge quite simply. When, however, u is only known as a function
of the section of the stream in the channel of approach, they become
complicated. Let [Omega] be the sectional area of the channel where h1
and h2 are measured. Then u = Q/[Omega] and [h] = Q²/2g [Omega]².
This value introduced in the equations above would render them
excessively cumbrous. In cases therefore where [Omega] only is known,
it is best to proceed by approximation. Calculate an approximate value
Q´ of Q by the equation
Q´ = (2/3)cb [root](2g) {h2^(3/2) - h1^(3/2)}.
Then [h] = Q´²/2g[Omega]² nearly. This value of [h] introduced in the
equations above will give a second and much more approximate value of
Q.
[Illustration: FIG. 49.]
§ 46. _Partially Submerged Rectangular Orifices and Notches._--When
the tail water is above the lower but below the upper edge of the
orifice, the flow in the two parts of the orifice, into which it is
divided by the surface of the tail water, takes place under different
conditions. A filament M1m1 (fig. 49) in the upper part of the orifice
issues with a head h´ which may have any value between h1 and h. But a
filament M2m2 issuing in the lower part of the orifice has a velocity
due to h´´ - h´´´, or h, simply. In the upper part of the orifice the
head is variable, in the lower constant. If Q1, Q2 are the discharges
from the upper and lower parts of the orifice, b the width of the
orifice, then
Q1 = (2/3)cb [root](2g) {h^(3/2) - h1^(3/2)}
(3)
Q1 = cb (h2 - h) [root](2gh).
In the case of a rectangular notch or weir, h1 = 0. Inserting this
value, and adding the two portions of the discharge together, we get
for a drowned weir