502 HYDROMECHANICS [HYDRAULICS. the part unaffected by the obstruction was 5 54 feet per second. Above the point where the abrupt change of depth occurred u~ = 5 51 2 = 307, and #/i = 32 2x 0-2 = 6-44; hence if was >g>/i. Justbe- 0"2 low the abrupt change of depth u = 5 54 x = 1 97 ; w 2 = 3 8S; and yli 32 - 2 x 56 = 18 03 ; hence at this point u-<^gh. Between these two points, therefore, u = gh ; and the condition for the pro duction of a standing wave occurred. The change of level at a standing wave may be found thus. Let fig. 127 represent the longitudinal section of a stream and /< a I y>^x j m * HO U 2 b T> i t d d Fig. 127. ab, ed cross sections normal to the bed, which for the short distance considered may be assumed horizontal. Suppose the mass of water abed to come to a b c d in a short time t ; and let v , ij be the veloci ties at ab and cd, QO,^ the areas of the cross sections. The force causing change of momentum in the mass abed estimated horizon tally is simply the difference of the pressures on ab and cd. Putting h , h-L for the depths of the centres of gravity of ab and cd measured down from the free water surface, the force is G(7i Q - h^) pounds, and the impulse in t seconds is G (A ft - A, 1 fi 1 ) t second pounds. The horizontal change of momentum is the difference of the momenta of cdc d and aba b ; that is, Hence, equating impulse and change of momentum, G 00 1 "" g For simplicity let the section be rectangular, of breadth B and depths H and H 1} at the two cross sections considered; then h = - H , m But, since QM l) =QjU ll we have This equation is satisfied if H = H 1; which corresponds to the case of uniform motion. Dividing by H - H lf the equation becomes In Bidone s experiment = 5 54, and H =0 2. Hence H 1 = 52, which agrees very well with the observed height. Fig. 128. 111. 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. 128. The water falling over the weir crest A acquired a very high velo city 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 < . 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 , the conditions for producing the stand ing 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. 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. of Civil Engineers, vol. xxvii. p. 222, by Mr A. C. How- den. Fig. 129 is compiled from the data given in that paper. It repre sents the section of the stream at pier 8 of the Towali Viaduct, during the flood of 1865. Tin- ground level is m>t exactly given by Mr 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 feet per second. Now, taking the depth on the down stream face of the pier at 26 feet, the velocity necessary for the production of a standing wave would be u = V</7t= V(32 2 x 26) = 29 feet per second nearly. But the velo city at this point was probably from Mr Howden s statements 16 58 x |^ = 25 5 feet, an agreement as close as the approximate character of the data would lead us to expect. XL ON STREAMS AND RIVERS. 112. 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 fVom 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 vege tation, part may escape by percolation into neighbouring districts. The following table gives the relation of the average stream dis charge to the average rainfall on the catchment basin (Tiefenbacher). Ratio of average Discharge to average Rainfall. Loss by Evapora tion, <fcc., in pel- 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 untissured mountains . 55 to -60 40 to 45 113. 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 water* of the most distant affluents reach the point considered, simultane ously with those from nearer points. The larger the catchment basin the less probable is it that all the conditions tending to pro duce 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 Itosin. In flat country 87 to 12 5 cub. ft. In hilly districts 17 5 to 22 5 ,, In moderately mountainous districts 36 2 to 45 ,,
In very mountainous districts 50 to 75 - , ,
