Page:EB1911 - Volume 14.djvu/88

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 − ζu²/2gih) / (1 − u²/gh) tends to the limit zero, and consequently dh/ds tends to ∞. 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).

Fig. 124.

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.

Fig. 125.

§ 120. Case 3.—Suppose a stream flowing uniformly with a depth h < u²/g. For a stream in uniform motion ζu²/2g = mi, or if the stream is of indefinitely great width, so that m = H, then ζu²/2g = iH, and H = ζu²/2gi. Consequently the condition stated above involves that

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 (h ₀Ω₀ − h₁Ω₁) pounds, and the impulse in t seconds is G (h ₀Ω₀ − 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.