§ 121. The formation of a standing wave was first observed by Bidone.
Into a small rectangular masonry channel, having a slope of 0.023 ft.
per foot, he admitted water till it flowed uniformly with a depth of
0.2 ft. He then placed a plank across the stream which raised the
level just above the obstruction to 0.95 ft. He found that the stream
above the obstruction was sensibly unaffected up to a point 15 ft.
from it. At that point the depth suddenly increased from 0.2 ft. to
0.56 ft. The velocity of the stream in the part unaffected by the
obstruction was 5.54 ft. per second. Above the point where the abrupt
change of depth occurred u² = 5.54² = 30.7, and gh = 32.2 × 0.2 =
6.44; hence u² was > gh. Just below the abrupt change of depth u =
5.54 × 0.2/0.56 = 1.97; u² = 3.88; and gh = 32.2 × 0.56 = 18.03; hence
at this point u² < gh. Between these two points, therefore, u² = gh;
and the condition for the production of a standing wave occurred.
[Illustration: FIG. 126.]
The change of level at a standing wave may be found thus. Let fig. 126
represent the longitudinal section of a stream and ab, cd cross
sections normal to the bed, which for the short distance considered
may be assumed horizontal. Suppose the mass of water abcd to come to
a´b´c´d´ in a short time t; and let u0, u1 be the velocities at ab and
cd, [Omega]0, [Omega]1 the areas of the cross sections. The force
causing change of momentum in the mass abcd estimated horizontally is
simply the difference of the pressures on ab and cd. Putting h0, h1
for the depths of the centres of gravity of ab and cd measured down
from the free water surface, the force is G(h0[Omega]0 - h1[Omega]1)
pounds, and the impulse in t seconds is G (h0[Omega]0 - h1[Omega]1) t
second pounds. The horizontal change of momentum is the difference of
the momenta of cdc´d´ and aba´b´; that is,
(G/g)([Omega]1u1² - [Omega]0u0²)t.
Hence, equating impulse and change of momentum,
G(h0[Omega]0 - h1[Omega]1)t = (G/g)([Omega]1u1² - [Omega]0u0²)t;
.: h0[Omega]0 - h1[Omega]1 = ([Omega]1u1² - [Omega]0u0²)/g. (1)
For simplicity let the section be rectangular, of breadth B and depths
H0 and H1, at the two cross sections considered; then h0 = ½H0, and h1
= ½H1. Hence
H0² - H1² = (2/g)(H1u1² - H0u0²).