H = [zeta]u²/2gi,
we fall upon the case of uniform motion. The results just stated may
be tabulated thus:--
For h = 0, H, > H, [oo],
the numerator has the value -[oo], 0, > 0, 1.
Next consider the denominator. If h becomes very small, in which case
u must be very large, the denominator tends to the limit -[oo]. As h
becomes very large and u consequently very small, the denominator
tends to the limit 1. For h = u²/g, or u = [root](gh), the denominator
becomes zero. Hence, tabulating these results as before:--
For h = 0, u²/g, > u²/g, [oo],
the denominator becomes
-[oo], 0, > 0, 1.
[Illustration: FIG. 122.]
§ 118. _Case_ 1.--Suppose h > u²/g, and also h > H, or the depth
greater than that corresponding to uniform motion. In this case dh/ds
is positive, and the stream increases in depth in the direction of
flow. In fig. 122 let B0B1 be the bed, C0C1 a line parallel to the bed
and at a height above it equal to H. By hypothesis, the surface A0A1
of the stream is above C0C1, and it has just been shown that the depth
of the stream increases from B0 towards B1. But going up stream h
approaches more and more nearly the value H, and therefore dh/ds
approaches the limit 0, or the surface of the stream is asymptotic to
C0C1. Going down stream h increases and u diminishes, the numerator
and denominator of the fraction (1 - [zeta]u²/2gih)/(1 -u²/gh) both
tend towards the limit 1, and dh/ds to the limit i. That is, the
surface of the stream tends to become asymptotic to a horizontal line
D0D1.
The form of water surface here discussed is produced when the flow of
a stream originally uniform is altered by the construction of a weir.
The raising of the water surface above the level C0C1 is termed the
backwater due to the weir.
§ 119. _Case_ 2.--Suppose h > u²/g, and also h < H. Then dh/ds is
negative, and the stream is diminishing in depth in the direction of
flow. In fig. 123 let B0B1 be the stream bed as before; C0C1 a line
drawn parallel to B0B1 at a height above it equal to H. By hypothesis
the surface A0A1 of the stream is below C0C1, and the depth has just
been shown to diminish from B0 towards B1. Going up stream h
approaches the limit H, and dh/ds tends to the limit zero. That is, up
stream A0A1 is asymptotic to C0C1. Going down stream h diminishes and
u increases; the inequality h>u²/g diminishes; the denominator of the
fraction (1 - [zeta]u²/2gih)/(1 - u²/gh) tends to the limit zero, and
consequently dh/ds tends to [infinity]. That is, down stream A0A1
tends to a direction perpendicular to the bed. Before, however, this
limit was reached the assumptions on which the general equation is
based would cease to be even approximately true, and the equation
would cease to be applicable. The filaments would have a relative
motion, which would make the influence of internal friction in the
fluid too important to be neglected. A stream surface of this form may
be produced if there is an abrupt fall in the bed of the stream (fig.
124).
[Illustration: FIG. 123.]
[Illustration: FIG. 124.]
[Illustration: FIG. 125.]
On the Ganges canal, as originally constructed, there were abrupt
falls precisely of this kind, and it appears that the lowering of the
water surface and increase of velocity which such falls occasion, for
a distance of some miles up stream, was not foreseen. The result was
that, the velocity above the falls being greater than was intended,
the bed was scoured and considerable damage was done to the works.
"When the canal was first opened the water was allowed to pass freely
over the crests of the overfalls, which were laid on the level of the
bed of the earthen channel; erosion of bed and sides for some miles up
rapidly followed, and it soon became apparent that means must be
adopted for raising the surface of the stream at those points (that
is, the crests of the falls). Planks were accordingly fixed in the
grooves above the bridge arches, or temporary weirs were formed over
which the water was allowed to fall; in some cases the surface of the
water was thus raised above its normal height, causing a backwater in
the channel above" (Crofton's _Report on the Ganges Canal_, p. 14).
Fig. 125 represents in an exaggerated form what probably occurred, the
diagram being intended to represent some miles' length of the canal
bed above the fall. AA parallel to the canal bed is the level
corresponding to uniform motion with the intended velocity of the
canal. In consequence of the presence of the ogee fall, however, the
water surface would take some such form as BB, corresponding to Case 2
above, and the velocity would be greater than the intended velocity,
nearly in the inverse ratio of the actual to the intended depth. By
constructing a weir on the crest of the fall, as shown by dotted
lines, a new water surface CC corresponding to Case 1 would be
produced, and by suitably choosing the height of the weir this might
be made to agree approximately with the intended level AA.
§ 120. _Case_ 3.--Suppose a stream flowing uniformly with a depth
h<u²/g. For a stream in uniform motion [zeta]u²/2g = mi, or if the
stream is of indefinitely great width, so that m = H, then
[zeta]u²/2g = iH, and H = [zeta]u²/2gi. Consequently the condition
stated above involves that
[zeta]u²/2gi < u²/g, or that i > [zeta]/2.
If such a stream is interfered with by the construction of a weir
which raises its level, so that its depth at the weir becomes h1 >
u²/g, then for a portion of the stream the depth h will satisfy the
conditions h < u²/g and h > H, which are not the same as those assumed in the two
previous cases. At some point of the stream above the weir the depth h
becomes equal to u²/g, and at that point dh/ds becomes infinite, or
the surface of the stream is normal to the bed. It is obvious that at
that point the influence of internal friction will be too great to be
neglected, and the general equation will cease to represent the true
conditions of the motion of the water. It is known that, in cases such
as this, there occurs an abrupt rise of the free surface of the
stream, or a standing wave is formed, the conditions of motion in
which will be examined presently.
It appears that the condition necessary to give rise to a standing
wave is that i > [zeta]/2. Now [zeta] depends for different channels
on the roughness of the channel and its hydraulic mean depth. Bazin
calculated the values of [zeta] for channels of different degrees of
roughness and different depths given in the following table, and the
corresponding minimum values of i for which the exceptional case of
the production of a standing wave may occur.
+-----------------------------+----------------+-------------------------+
| | Slope below | Standing Wave Formed. |
| |which a Standing| |
| Nature of Bed of Stream. | Wave is +-------------+-----------+
| | impossible in |Slope in feet|Least Depth|
| | feet peer foot.| per foot. | in feet. |
+-----------------------------+----------------+-------------+-----------+
| | | / 0.002 | 0.262 |
| Very smooth cemented surface| 0.00147 | < 0.003 | .098 |
| | | \ 0.004 | .065 |
| | | | |
| | | / 0.003 | .394 |
| Ashlar or brickwork | 0.00186 | < 0.004 | .197 |
| | | \ 0.006 | .098 |
| | | | |
| | | / 0.004 | 1.181 |
| Rubble masonry | 0.00235 | < 0.006 | .525 |
| | | \ 0.010 | .262 |
| | | | |
| | | / 0.006 | 3.478 |
| Earth | 0.00275 | < 0.010 | 1.542 |
| | | \ 0.015 | .919 |
+-----------------------------+----------------+-------------+-----------+