HYDRAULICS.] HYDROMECHANICS 501 approaches the limit 0, or the surface of the stream is asymptotic to CoCj. Going down stream h increases and u diminishes, the Zgih ~ numerator and denominator of the fraction both tend towards the limit 1, and -j- to the limit i. That is, the surface of the stream tends to become asymptotic to a horizontal line D,^. The form of water surface here discussed is produced when the flow of a stream originally uniform is altered by the construction of a weir. The raising of the water surface above the level CgCj is termed the backwater due to the weir. 108. Case 2. Suppose /C> , and also h<^R. Then -= is nega- 9 da tive, and the stream is diminishing in depth in the direction of flow. In fig. 124 let r B B X be the s?= stream bed as be- A. fore ; C^^ a line drawn parallel to BjjBj at a height above it equal to H. By hypo thesis the surface AgAjof the stream is below C C 1; and the depth has just been shown to diminish from B towards B. Going up stream h approaches the limit H, and - tends to the limit zero. That is, up cts stream A Aj is asymptotic to CoCj. Going down stream h diminishes and u increases ; the inequality 7i> diminishes; the denominator of the fraction SI- tends to the limit zero, and dh ( jh consequently -r- tends to <x> . That is, down stream A,^ 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. 125). Fig. 125. 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 accord ingly 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 " (Orofton s Report on the Ganges Canal, p. 14). Fig. 126 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 con sequence of the presence of the ojjee 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 con structing 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. A Fig. 126. 109. Case 3. Suppose a stream flowing uniformly with a depth tt 2 U^ . For a stream in uniform motion = mi, or if the stream v? is of indefinitely great width, so that ??i=H, then = iH, and tt j H = CTT^ Consequently the condition stated above involves that igi , 2gi g 2 If such a stream is interfered with by the construction of a weir v? which raises its level, so that its depth at the weir becomes h^ > , then for a portion of the stream the depth /;. will satisfy the con- Zt 2 ditions &< - and /C>H, which are not the same as those assumed 9 in the two previous cases. At some point of the stream above the 11? dh weir the depth h becomes equal to , and at that point becomes g ds infinite, or the surface of the stream is normal to the bed. It is ob vious that at that point the influence of internal friction will be too great to be neglected, and the general equation will cease to repre sent 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 O*^- Now depends for different channels on the roughness of the channel and its hydraulic mean depth. M. Bazin has calculated the values of 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 Stand ing Wave is impossible in feet per foot. Standing Wave Formed. Slope in feut per foot. Least Depth in feet. Very smooth cemented ) surface ) 0-00147 0-00186 0-00235 ] 0-00275 I 0-002 0-003 0-004 0-003 0-004 0-006 0-004 0-006 o-oio 0-006 010 0-015 0-262 098 065 394 197 098 1-181 525 262 3-478 1-542 919 Ashlar or brickwork Rubble masonry Earth STANDING WAVES. 110. The formation of a standing wave was first observed by Bidone. Into a small rectangular masonry channel, having a slope of 023 feet per foot, he admitted water till it flowed uniformly with a depth of 2 feet. He then placed a plunk across the stream which raised the level just above the obstruction to 95 feet. He found that the, stream above the obstruction was sensibly unaffected up to a point 15 feet from it. At that point the depth suddenly
increased from 2 feet to 56 feet. The velocity of the stream in
