Page:EB1911 - Volume 14.djvu/90

Rh will be represented by ac. In a deeper stream such as that in fig. 130, the average height to which particles are lifted, and, since the rate of vertical fall through the water may be assumed the same as before, the average distance a′c′ of transport will be greater. Consequently, although the scouring action may be identical in the two streams, the velocity of transport of material down stream is greater as the depth of the stream is greater. The effect is that the deep stream excavates its bed more rapidly than the shallow stream.

§ 126. Bottom Velocity at which Scour commences.—The following bottom velocities were determined by P. L. G. Dubuat to be the maximum velocities consistent with stability of the stream bed for different materials.

Darcy and Bazin give, for the relation of the mean velocity vm and bottom velocity vb.

vm = vb + 10.87 √ (mi).

But

√ mi = vm √ ( / 2g);

∴ vm = vb / (1 − 10.87 √ ( / 2g)).

Taking a mean value for, we get

vm = 1.312 vb,

and from this the following values of the mean velocity are obtained:—

The following table of velocities which should not be exceeded in channels is given in the Ingenieurs Taschenbuch of the Verein “Hütte”:—

§ 127. Regime of a River Channel.—A river channel is said to be in a state of regime, or stability, when it changes little in draught or form in a series of years. In some rivers the deepest part of the channel changes its position perpetually, and is seldom found in the same place in two successive years. The sinuousness of the river also changes by the erosion of the banks, so that in time the position of the river is completely altered. In other rivers the change from year to year is very small, but probably the regime is never perfectly stable except where the rivers flow over a rocky bed.

If a river had a constant discharge it would gradually modify its bed till a permanent regime was established. But as the volume discharged is constantly changing, and therefore the velocity, silt is deposited when the velocity decreases, and scour goes on when the velocity increases in the same place. When the scouring and silting are considerable, a perfect balance between the two is rarely established, and hence continual variations occur in the form of the river and the direction of its currents. In other cases, where the action is less violent, a tolerable balance may be established, and the deepening of the bed by scour at one time is compensated by the silting at another. In that case the general regime is permanent, though alteration is constantly going on. This is more likely to happen if by artificial means the erosion of the banks is prevented. If a river flows in soil incapable of resisting its tendency to scour it is necessarily sinuous (§ 107), for the slightest deflection of the current to either side begins an erosion which increases progressively till a considerable bend is formed. If such a river is straightened it becomes sinuous again unless its banks are protected from scour.

§ 128. Longitudinal Section of River Bed.—The declivity of rivers decreases from source to mouth. In their higher parts rapid and torrential, flowing over beds of gravel or boulders, they enlarge in volume by receiving affluent streams, their slope diminishes, their bed consists of smaller materials, and finally they reach the sea. Fig. 131 shows the length in miles, and the surface fall in feet per mile, of the Tyne and its tributaries.

The decrease of the slope is due to two causes. (1) The action of the transporting power of the water, carrying the smallest debris the greatest distance, causes the bed to be less stable near the mouth than in the higher parts of the river; and, as the river adjusts its slope to the stability of the bed by scouring or increasing its sinuousness when the slope is too great, and by silting or straightening its course if the slope is too small, the decreasing stability of the bed would coincide with a decreasing slope. (2) The increase of volume and section of the river leads to a decrease of slope; for the larger the section the less slope is necessary to ensure a given velocity.

The following investigation, though it relates to a purely arbitrary case, is not without interest. Let it be assumed, to make the conditions definite—(1) that a river flows over a bed of uniform resistance to scour, and let it be further assumed that to maintain stability the velocity of the river in these circumstances is constant from source to mouth; (2) suppose the sections of the river at all points are similar, so that, b being the breadth of the river at any point, its hydraulic mean depth is ab and its section is cb2, where a and c are constants applicable to all parts of the river; (3) let us further assume that the discharge increases uniformly in consequence of the supply from affluents, so that, if l is the length of the river from its source to any given point, the discharge there will be kl, where k is another constant applicable to all points in the course of the river.

Let AB (fig. 132) be the longitudinal section of the river, whose source is at A; and take A for the origin of vertical and horizontal coordinates. Let C be a point whose ordinates are x and y, and let the river at C have the breadth b, the slope i, and the velocity v.

Since velocity × area of section = discharge, vcb2 = kl, or b = √ (kl/cv).

Hydraulic mean depth = ab = a √ (kl/cv).

But, by the ordinary formula for the flow of rivers, mi = v2;

∴ i = v2 / m = (vundefined / a) √ (c / kl).

But i is the tangent of the angle which the curve at C makes with the axis of X, and is therefore = dy/dx. Also, as the slope is small, l = AC = AD = x nearly.

∴ dy/dx = (vundefined / a) √ (c / kx);

and, remembering that v is constant,

y = (2vundefined / a) √ (cx / k);

or

y2 = constant × x;

so that the curve is a common parabola, of which the axis is horizontal and the vertex at the source. This may be considered an ideal longitudinal section, to which actual rivers approximate more or less, with exceptions due to the varying hardness of their beds, and the irregular manner in which their volume increases.

§ 129. Surface Level of River.—The surface level of a river is a plane changing constantly in position from changes in the volume of water discharged, and more slowly from changes in the river bed, and the circumstances affecting the drainage into the river.

For the purposes of the engineer, it is important to determine (1) the extreme low water level, (2) the extreme high water or flood level, and (3) the highest navigable level.

1. Low Water Level cannot be absolutely known, because a river reaches its lowest level only at rare intervals, and because alterations in the cultivation of the land, the drainage, the removal of forests, the removal or erection of obstructions in the river bed, &c., gradually alter the conditions of discharge. The lowest level of which records can be found is taken as the conventional or approximate low water level, and allowance is made for possible changes.

2. High Water or Flood Level.—The engineer assumes as the highest flood level the highest level of which records can be obtained. In forming a judgment of the data available, it must be remembered that the highest level at one point of a river is not always simultaneous