504 HYDROMECHANICS [HYDRAULICS. 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 sinous- ness 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 ease, is not without interest. Let it be assumed, to make the con ditions 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 cb^ r where a and c are constants appli cable to all parts of the river; (3) let us further assume that the discharge in- A D X ^-^ C Y Fig. 133. B creases uniformly in consequence of the supply from affluents, so that, if I 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. 133) 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 x area of section = discharge, vcb^ = kl, or b = / cv Hydraulic mean depth = a6-=rtA/ . But, by the ordinary formula for the flow of rivers, mi=& ; -_C^ 2 _C^ /~ m a kl But i is the tangent of the angle which the curve at C makes with the axis of X, and is therefore = -J- . I = AC = AD = x nearly. Also, as the slope is small, dx = >3 l~c_ . a fcx and, remembering that v is constant, ? /~ v or y 2 = constant x x ; so that the curve is a common parabola, of which the axis is hori zontal 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. 118. Surface Level of Jiiver.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 approxi mate low water level, and allowance is made for possible changes. (2) Hir/h Water or Flood Level. The engineer assumes as the highest flood level the highest level of which records can be ob tained. 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 with the attainment of the highest level at other points, and that the rise of a river in flood is very different in different parts of its course. In temperate regions, the floods of rivers seldom rise more than 20 feet above low water level, but in the tropics the rise of floods is greater. (3) Highest Navigable Level. When the river rises above a certain level, navigation becomes difficult from the increase of the Weight of a Cubic Foot in tt>. In Air. In Water. Basalt 187-3 130-0 112-0 170-0 144-0 116-144 124-9 67-6 49-6 107-6 81-6 53-6-81-6 Brick Brickwork Granite and limestone Sandstone Masonry velocity of the current, or from submersion of the tow paths, or from the headway under bridges becoming insufficient. Ordinarily the highest navigable level may be taken to be that at which the river begins to overflow its banks. 119. Relative Value of Different Materials for Submerged Works. That the power of water to remove and transport different materials lepends on their density has an important bearing on the selection of materials for submerged works. In many cases, as in the aprons or floorings beneath bridges, or in front of locks or falls, and in the formation of training walls and breakwaters by pierres pcrdus, which have to resist a violent current, the materials of which the structures are composed should be of such a size and weight as to be able individually to resist the scouring action of the water. The heaviest materials will therefore be the best ; and the different value of materials in this respect will appear much more striking, if it is remembered that all materials lose part of their weight in water. A block whose volume is V cubic feet, and whose density in air is w lb per cubic foot, weighs in air vu~V lb, but in water only (w-62 4) V lb. 120. Inundation Deposits from a River. When a river carrying silt periodically overflows its banks, it deposits silt over the area flooded, and gradually raises the surface of the country. The silt is deposited in greatest abundance where the water first leaves the river. It hence results that the section of the country assumes a peculiar form, the river flowing in a trough along the crest of a ridge, from which the land slopes downwards on both sides. The silt deposited from the water forms two wedges, having their thick ends towards the river (fig. 134). Fig. 134. This is strikingly the case with the Mississippi, and that river is now kept from flooding immense areas by artificial embankments or levees. In India, the term deltaic segment is sometimes applied to that portion of a river running through deposits formed by inunda tion, and having this characteristic section. The irrigation of the country in this case is very easy, a comparatively slight raising of the river surface by a weir or annicut gives a command of level which permits the water to be conveyed to any part of the district. 121. Deltas. The name delta was originally given to the A-shaped portion of Lower Egypt, included between seven branches of the Nile. It is now given to the whole of the alluvial tracts round river mouths formed by deposition of sediment from the river, where its velocity is checked on its entrance to the sea. The characteristic feature of these alluvial deltas is that the river traverses them, not in a single channel, but in two or many bifurcating branches. Each branch l. ShdalJ V_x Fig. 135. has a tract of the delta under its influence, and gradually raises the surface of that tract, and extends it seaward. As the delta extends itself seaward, the conditions of discharge through the different branches change. The water finds the passage through one of the branches less obstructed than through the others ; the velocity and
scouring action in that branch are increased ; in the others they
