This equation is satisfied if H0 = H1, which corresponds to the case
of uniform motion. Dividing by H0 - H1, the equation becomes
(H1/H0)(H0 + H1) = 2u0²/g; (3)
.: H1 = [root](2u0²H0/g + ¼H0²) - ½H0. (4)
In Bidone's experiment u0 = 5.54, and H0 = 0.2. Hence H1 = 0.52, which
agrees very well with the observed height.
[Illustration: FIG. 127.]
§ 122. A standing wave is frequently produced at the foot of a weir.
Thus in the ogee falls originally constructed on the Ganges canal a
standing wave was observed as shown in fig. 127. The water falling
over the weir crest A acquired a very high velocity on the steep slope
AB, and the section of the stream at B became very small. It easily
happened, therefore, that at B the depth h < u²/g. In flowing along
the rough apron of the weir the velocity u diminished and the depth h
increased. At a point C, where h became equal to u²/g, the conditions
for producing the standing wave occurred. Beyond C the free surface
abruptly rose to the level corresponding to uniform motion with the
assigned slope of the lower reach of the canal.
[Illustration: FIG. 128.]
A standing wave is sometimes formed on the down stream side of bridges
the piers of which obstruct the flow of the water. Some interesting
cases of this kind are described in a paper on the "Floods in the
Nerbudda Valley" in the _Proc. Inst. Civ. Eng._ vol. xxvii. p. 222, by
A. C. Howden. Fig. 128 is compiled from the data given in that paper.
It represents the section of the stream at pier 8 of the Towah
Viaduct, during the flood of 1865. The ground level is not exactly
given by Howden, but has been inferred from data given on another
drawing. The velocity of the stream was not observed, but the author
states it was probably the same as at the Gunjal river during a
similar flood, that is 16.58 ft. per second. Now, taking the depth on
the down stream face of the pier at 26 ft., the velocity necessary for
the production of a standing wave would be u = [root](gh) =
[root](32.2 × 26) = 29 ft. per second nearly. But the velocity at this
point was probably from Howden's statements 16.58 × {40/26} = 25.5 ft.
per second, an agreement as close as the approximate character of the
data would lead us to expect.