ζu2 / 2gi < u2 / g, or that i > ζ/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 > ζ/2. Now ζ depends for different channels on the roughness of the channel and its hydraulic mean depth. Bazin calculated the values of ζ 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.

Nature of Bed of Stream. Slope below
which a Standing
Wave is
impossible in
feet peer foot.		 Standing Wave Formed.
Slope in feet
per foot.		 Least Depth
in feet.
Very smooth cemented surface 0.00147 0.002 0.262
0.003  .098
0.004  .065
Ashlar or brickwork 0.00186 0.003  .394
0.004  .197
0.006  .098
Rubble masonry 0.00235 0.004 1.181
0.006  .525
0.010  .262
Earth 0.00275 0.006 3.478
0.010 1.542
0.015  .919

Standing Waves

§ 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.

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, Ω0, Ω1 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 (h0Ω0 − h1Ω1) pounds, and the impulse in t seconds is G (h0Ω0 − h1Ω1) t second pounds. The horizontal change of momentum is the difference of the momenta of cdc′d′ and aba′b′; that is,

(G/g) (Ω1u12 − Ω0u02) t.