Hence, equating impulse and change of momentum,
For simplicity let the section be rectangular, of breadth B and depths H0 and H1, at the two cross sections considered; then h0 = 12H0, and h1 = 12H1. Hence
But, since Ω0u0 = Ω1u1, we have
This equation is satisfied if H0 = H1, which corresponds to the case of uniform motion. Dividing by H0 − H1, the equation becomes
In Bidone’s experiment u0 = 5.54, and H0 = 0.2. Hence H1 = 0.52, which agrees very well with the observed height.
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.
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| 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 = √ (gh) = √ (32.2 × 26) = 29 ft. per second nearly. But the velocity at this point was probably from Howden’s statements 16.58 × 4026 = 25.5 ft. per second, an agreement as close as the approximate character of the data would lead us to expect.
XI. ON STREAMS AND RIVERS
§ 123. Catchment Basin.—A stream or river is the channel for the discharge of the available rainfall of a district, termed its catchment basin. The catchment basin is surrounded by a ridge or watershed line, continuous except at the point where the river finds an outlet. The area of the catchment basin may be determined from a suitable contoured map on a scale of at least 1 in 100,000. Of the whole rainfall on the catchment basin, a part only finds its way to the stream. Part is directly re-evaporated, part is absorbed by vegetation, part may escape by percolation into neighbouring districts. The following table gives the relation of the average stream discharge to the average rainfall on the catchment basin (Tiefenbacher).
| Ratio of average Discharge to average Rainfall. | Loss by Evaporation, &c., in per cent of total Rainfall. | |
| Cultivated land and spring-forming declivities. | 3 to .33 | 67 to 70 |
| Wooded hilly slopes. | .35 to .45 | 55 to 65 |
| Naked unfissured mountains | .55 to .60 | 40 to 45 |
§ 124. Flood Discharge.—The flood discharge can generally only be determined by examining the greatest height to which floods have been known to rise. To produce a flood the rainfall must be heavy and widely distributed, and to produce a flood of exceptional height the duration of the rainfall must be so great that the flood waters of the most distant affluents reach the point considered, simultaneously with those from nearer points. The larger the catchment basin the less probable is it that all the conditions tending to produce a maximum discharge should simultaneously occur. Further, lakes and the river bed itself act as storage reservoirs during the rise of water level and diminish the rate of discharge, or serve as flood moderators. The influence of these is often important, because very heavy rain storms are in most countries of comparatively short duration. Tiefenbacher gives the following estimate of the flood discharge of streams in Europe:—
| Flood discharge of Streams per Second per Square Mile of Catchment Basin. | ||
| In flat country | 8.7 to 12.5 | cub. ft. |
| In hilly districts | 17.5 to 22.5 | ” |
| In moderately mountainous districts | 36.2 to 45.0 | ” |
| In very mountainous districts | 50.0 to 75.0 | ” |
It has been attempted to express the decrease of the rate of flood discharge with the increase of extent of the catchment basin by empirical formulae. Thus Colonel P. P. L. O’Connell proposed the formula y = M √ x, where M is a constant called the modulus of the river, the value of which depends on the amount of rainfall, the physical characters of the basin, and the extent to which the floods are moderated by storage of the water. If M is small for any given river, it shows that the rainfall is small, or that the permeability or slope of the sides of the valley is such that the water does not drain rapidly to the river, or that lakes and river bed moderate the rise of the floods. If values of M are known for a number of rivers, they may be used in inferring the probable discharge of other similar rivers. For British rivers M varies from 0.43 for a small stream draining meadow land to 37 for the Tyne. Generally it is about 15 or 20. For large European rivers M varies from 16 for the Seine to 67.5 for the Danube. For the Nile M = 11, a low value which results from the immense length of the Nile throughout which it receives no affluent, and probably also from the influence of lakes. For different tributaries of the Mississippi M varies from 13 to 56. For various Indian rivers it varies from 40 to 303, this variation being due to the great variations of rainfall, slope and character of Indian rivers.
In some of the tank projects in India, the flood discharge has been calculated from the formula D = C3√ n2, where D is the discharge in cubic yards per hour from n square miles of basin. The constant C was taken = 61,523 in the designs for the Ekrooka tank, = 75,000 on Ganges and Godavery works, and = 10,000 on Madras works.
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| Fig. 129. |
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| Fig. 130. |
§ 125. Action of a Stream on its Bed.—If the velocity of a stream exceeds a certain limit, depending on its size, and on the size, heaviness, form and coherence of the material of which its bed is composed, it scours its bed and carries forward the materials. The quantity of material which a given stream can carry in suspension depends on the size and density of the particles in suspension, and is greater as the velocity of the stream is greater. If in one part of its course the velocity of a stream is great enough to scour the bed and the water becomes loaded with silt, and in a subsequent part of the river’s course the velocity is diminished, then part of the transported material must be deposited. Probably deposit and scour go on simultaneously over the whole river bed, but in some parts the rate of scour is in excess of the rate of deposit, and in other parts the rate of deposit is in excess of the rate of scour. Deep streams appear to have the greatest scouring power at any given velocity. It is possible that the difference is strictly a difference of transporting, not of scouring action. Let fig. 129 represent a section of a stream. The material lifted at a will be diffused through the mass of the stream and deposited at different distances down stream. The average path of a particle lifted at a will be some such curve as abc, and the average distance of transport each time a particle is lifted
