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STANDING WAVES § 121. The formation of a standing See also:wave was first observed by Bidone. Into a small rectangular See also:masonry channel, having a slope of 0.023 ft. per See also:foot, he admitted See also:water till it flowed uniformly with a See also:depth of 0.2 ft. He then placed a See also:plank across the stream which raised the level just above the obstruction to 0.95 ft. He found that the stream above the obstruction was sensibly unaffected up to a point 15 ft. from it. At that point the depth suddenly increased from o•2 ft. to o•56 ft. The velocity of the stream in the See also:part unaffected by the obstruction was 5.54 ft. per second. Above the point where the abrupt See also:change of depth occurred 142=5.542=30.7, and gh =32.2 X0.2 =6.44; hence u2 was>gh. Just below the abrupt change of depth u=5.54X0.2/0.56=1.97; u2=3.88; and gh= 32'2 X0-56=-18.03; hence at this point u2 <gh. Between these two points, therefore, u2=gh; and the See also:condition for the See also:production of a standing wave occurred. The change of level at a standing wave may be found thus. Let fig. 126 represent the See also:longitudinal See also:section of a stream and ab, cd c c See also:cross sections normal to the See also:bed, which for the See also:short distance considered may be assumed See also:horizontal. Suppose the See also:mass of water abed to come to a'b'c'd' in a short See also:time t ; and let uo, ui be the velocities at ab and cd, 5le, 52i the areas of the cross sections. The force causing change of momentum in the mass abed estimated horizont-' ally is simply the difference of the pressures on ab and cd. Putting ho, hi for the depths of the centres of gravity of ab and cd measured down from the See also:free water See also:surface, the force is G(ho52o—hi52i) pounds, and the impulse in t seconds is G (ho52o—hi52i) t second pounds. The horizontal change of momentum is the difference of the momenta of cdc'd' and See also:aba'b'; that is, (See also:Gig)(521ui2 S2ou 2)t. Co a a' uo b 16' ' di ii FIG. 126. Xi ON STREAMS AND See also:RIVERS] Hence, equating impulse and change of momentum, G(hot).—hit2,)t = (G/g)(t21u12—Slouo2)t; .'. hotto —h1t21=(t21u12—t2ouo2)/g• (I) For simplicity let the section be rectangular, of breadth B and depths Ho and H1, at the two cross sections considered; then ho = ZHo, and h1= 2H1. Hence Hot—H12 = (2/g)(Hlu12—Houo2). But, since t2euo=t21ui, we have u12 = uo2Ho2/H 12, See also:Hoe—Hi2 = (2uo2/g)(H02/H1—Ho). (2) This See also:equation is satisfied if Ho=H1, which corresponds to the See also:case of See also:uniform See also:motion. Dividing by Ho—H1, the equation becomes (H1/Ho)(Ho+H1) =2uo2/g; (3) H1=J (2uo2Ho/g+ a IIo2)—ZHo. (4) In Bidone's experiment uo=5.54, and Ho=0.2. Hence H1=o.52, which agrees very well with the observed height. § 122. A standing wave is frequently produced at the foot of a See also:weir. Thus in the See also:ogee falls originally constructed on the See also:Ganges See also:canal a standing wave was observed as shown in fig. 127. The water falling over the weir See also:crest A acquired a very high velocity on the77 Ratio of See also:average Loss by Evaporation, See also:Discharge to &c., in per cent of average Rainfall. See also:total Rainfall. Cultivated See also:land and See also:spring- 1 3 to 33 67 to 70 forming declivities . 35 to `45 55 to 65 Wooded hilly slopes . Naked unfissured mountains •55 to •6o 40 to 45 § 124. See also:Flood Discharge.--The flood discharge can generally only be determined by examining the greatest height to which Hoods 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 See also:great that the flood See also:waters of the most distant affluents reach the point considered, simultaneously with those from nearer points. The larger the catchment See also:basin the less probable is it that all the conditions tending to See also:pro-duce a maximum discharge should simultaneously occur. Further, lakes and the See also:river bed itself See also:act as storage reservoirs during the rise of water level and diminish the See also:rate of discharge, or serve as flood moderators. The See also:influence of these is often important, because very heavy See also:rain storms are in most countries of comparatively short duration. Tiefenbacher gives the following estimate of the flood ///"A (/,ZZ/07-4:'t'r/Z.A. 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 See also: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 See also:rose to the level corresponding to uniform motion with the assigned slope of the See also:lower reach of the canal. A standing wave is sometimes formed on the down stream See also:side of See also:bridges the piers of which obstruct the flow of the water. Some interesting cases of this See also:kind are described in a See also:paper on the " Floods in the See also:Nerbudda Valley " in the Proc. Inst. Civ. Eng. vol. See also: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 See also:pier 8 of the Towah Viaduct, during the flood of 1865. The ground level is not exactly given by How- den, but has been in- ferred from data given on another See also: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 See also:face of the pier at 26 ft., the velocity necessary for the production of a st=z9Jft) andi(gng // wave would be u h =J (3z•2 X26) = per second nearly. But the velocity at this point was probably from Howden's statements 16.58 =25.5 ft., an agreement as See also:close as the approximate See also:character of the data would See also: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 See also:district, termed its catchment basin. The catchment basin is surrounded by a See also:ridge or See also:watershed See also:line, continuous except at the point where the river finds an outlet. The See also:area of the catchment basin may be determined from a suitable contoured See also:map on a See also:scale of at least I in 100,000. Of the whole rain-fall 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 See also: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). discharge of streams in See also:Europe: Flood discharge of Streams per Second per Square Mile of Catchment Basin. In See also:flat See also: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•o to 75.0 It has been attempted to See also:express the decrease of the rate of flood discharge with the increase of extent of the catchment basin by empirical formulae. Thus See also:Colonel P. P. L. O'Connell proposed the See also:formula y=MJx, where M is a See also:constant called the modulus of the river, the value of which depends on the amount of rainfall, the See also: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 See also: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 See also:British rivers M varies from 0.43 for a small stream draining meadow land to 37 for the See also:Tyne. Generally it is about 15 or 20. For large See also:European rivers M varies from 16 for the See also:Seine to 67.5 for the See also:Danube. For the See also:Nile M = I I, a See also: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 See also:Mississippi M varies from 13 to 56. For various See also:Indian rivers it varies from 40 to 303, this variation being due to the great See also:variations of rainfall, slope and character of Indian rivers. In some of the tank projects in See also:India, the flood discharge has been calculated from the formula D = C l n2, where D is the discharge in cubic yards per See also:hour from n square See also:miles of basin. The constant C was taken =61,523 in the designs for the Ekrooka tank, =75,000 on Ganges and Godavery See also:works, and =lo,000 on See also:Madras works. § 125. See also:Action of a Stream on its Bed.—If the velocity of a stream exceeds a certain limit, depending on its See also:size, and on the size, heavi- suspension depends on the size a c and See also:density of the particles in FIG. 129. 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 See also:deposit and scour go on simultaneously over the whole river bed, but in some parts the rate 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 See also:curve as See also:abc, and the average distance of transport each time a particle is lifted ness, See also: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 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 See also:power at any given velocity. It is possible that the difference is strictly a difference of se' eezimm/wait/oi'e7esee se/a/,ioe/e/i aio/% /,isss s w' c" [ON STREAMS 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 See also: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. See also:Darcy and See also:Bazin give, for the relation of the mean velocity vm and bottom velocity vb. vm =1'b+to•87ei (mi). But 11 mi =em11 (r/2g) ; See also:Vie =vb/(I-IO.8711 (l'/2g)). Taking a mean value for we get vm= 1.3122)4, and from this the following values of the mean velocity are obtained :- " See also:Witte ' See also:HYDRAULICS r -- Bottom Velocity Mean Velocity =vb. =v.. 1. Soft See also:earth 0.25 •33 2. See also:Loam 0.50 •65 3. See also:Sand I.00 1.30 4. See also:Gravel . . . 2.00 2.62
5. Pebbles 3.40 4.46
6. Broken See also: . . 5.00 4.03 3.08 See also:Conglomerate of slaty fragments 7.28 6•io 4.90 Stratified rocks 8•oo 7.45 6•oo Hard rocks 14.00 12.15 10.36 happen if by artificial means the erosion of the See also:banks is prevented. If a river flows in See also: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 progres• sively till a considerable See also: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 See also:volume by receiving affluent streams, their slope diminishes, their bed consists of smaller materials, and finally they reach the See also: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. (I) The action of the transporting power of the water, carrying the smallest debris the greatest distance, causes the bed to be less See also: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 See also: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 See also: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 See also:supply from affluents, so that, if 1 is the length of the river from its source to any given point, the discharge there will be A 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 X area of section = discharge, vcb2 = kl, or b = - (kl/cv). Hydraulic mean depth es ab =tie/ (kl/ce). But, by the See also:ordinary formula for the flow of rivers, mi-i-v2; .'. i = i-v2/m = (i-el/a)11 (See also:Oki). But i is the tangent of the See also:angle which the curve at C makes with the See also:axis of X, and is therefore =dy/dx. Also, as the slope is small, l =AC =AD =x nearly. :. dy/dx = (i'v1/a)V (c/kx) ; and, remembering that v is constant, y=(2fvi/a)1I (ex/k); or y2 constant X x; so that the curve is a See also:common See also:parabola, of which the axis is hori- x D X § 127. Regime of a River Channel.-A river channel is said to be in a See also:state of regime, or stability, when it changes little in See also:draught or form in a See also:series of years. In some rivers the deepest part of the channel changes its position perpetually, and is seldom found in the same See also: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 See also: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 - 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 vary-mg hardness of their beds, and the irregular manner in 010' which their volume increases. c ' § 129. Surface Level of River.-The surface level of a • R.7yne See also:fix° sy; river is a See also:plane changing constantly in position from {4 changes in the volume of water discharged, and more 7}m,.---n ---- 9101: +e--._ ra, slowly from changes in the river bed, and the circum- 3.8fi 7f! r~ stances affecting the drainage into the river. »e. For the purposes of the engineer, it is important to 13f! 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, discharged is constantly changing, and therefore vei because a river reaches its lowest level only at rare inter- the velocity, silt is deposited when the velocity c5 vale, and because alterations in the cultivation of the decreases, and scour goes on when the velocity • • land, the drainage, the removal of forests, the removal increases in the same place. When the scouring a' • or erection of obstructions in the river bed, &c., gradu and silting are considerable, a perfect See also:balance `~~ e''z. -1 ally alter the conditions of discharge. The lowest level between the two is rarely established, and hence of which records can be found is taken as the conven- continual variations occur in the form of the river • tional or approximate low water level, and See also:allowance is and the direction of its currents. In other cases, made for possible changes. where the action is less violent, a tolerable balance maybe established, 2. High Water or Flood Level.-The engineer assumes as the highest and the deepening of the bed by scour at one time is compensated by flood level the highest level of which records can be obtained. In the silting at another. In that case the See also:general regime is permanent, forming a See also:judgment of the data available, it must be remembered that though alteration is constantly going on. This is more likely to 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 ft. 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, See also:navigation becomes difficult from the increase of the velocity of the current, or from submersion of the See also:tow paths, or from the See also:head-way under bridges becoming insufficient. Ordinarily the highest navigable level may be taken to be that at which the river begins to overflow its banks. § 130. Relative Value of Different Materials for Submerged Works.—That the power of water to remove and transport different materials depends 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 perdus, which have to resist a violent current, the materials of which the structures are composed should be of such a size and See also: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 See also:block whose volume is V cubic feet, and whose density in See also:air is w lb per cubic foot, weighs in air wV lb, but in water only (w-62.a) V lb. In Air. In Water. See also:Basalt 187.3 124.9 See also:Brick . . I 130.0 67.6 See also:Brickwork . 112.0 49'6 See also:Granite and See also:limestone 170.0 Io7.6 See also:Sandstone 144.0 81.6 Masonry 116-144 53.6-81.6 possible at uniform distances) in a line starting from the stake and perpendicular to the See also:thread of the stream. To obtain these, a See also:wire may be stretched across with equal distances marked on it by hang- See also:ing tags. The depth at each of these tags may be obtained by a See also:light wooden See also:staff, with a disk-shaped See also:shoe 4 to 6 in. in See also:diameter. If the depth is great, soundings may be taken by a See also:chain and weight. To ensure the wire being perpendicular to the thread of the stream, s it is desirable to stretch two other wires similarly graduated, one above and the other below, at a distance of 20 to 40 yds. A number of floats being then thrown in, it is observed whether they For large and rapid rivers the cross section is obtained by See also:sounding f in the following way. Let AC (fig. 135) be the line on which soundings are required. A See also:base line AB is measured out at right angles toward to AC, and ranging staves are set up at AB and at D in line with AC. s the river (fig. 133). A See also:boat is allowed to drop down stream, and, at the moment it comes in line with AD, the lead is dropped, and an observer in the boat takes, with a See also:box See also:sextant, See also:eA~ the angle AEB subtended by have a weight of 14 lb of lead, This is strikingly the case with the Mississippi, and that river is and, if the boat drops down now kept from flooding immense areas by artificial embankments or stream slowly, it may hang near levees. In India, the See also:term deltaic segment is sometimes applied to the bottom, so that the observathat portion of a river See also:running through deposits formed by inunda- f tion is made instantly. In ex-See also:Lion, and having this characteristic section. The See also:irrigation of the tensive surveys of the Missiscountry in this case is very easy; a comparatively slight raising of sippi observers with theodolites the river surface by a weir or annicut gives a command of level . were stationed at A and B. The i t which permits the water to be conveyed to any part of the district. See also:theodolite at A was directed towards C, that at B was kept § 132. Deltas.—The name See also:delta was originally given to the ~- , shaped portion of Lower See also:Egypt, included between seven branches of on the boat. When the boat g the Nile. It is now given to the whole of the alluvial tracts See also:round came on the line AC, the ob- river mouths formed by deposition of sediment from the river, where server at A signalled. the See also:sound- its velocity is checked on its entrance to the sea. The characteristic ing line was dropped, and the feature of these alluvial deltas is that the river traverses them, not observer at B read off the angle in a single channel, but in two or many bifurcating branches. Each ABE. By repeating observations a number of soundings are obbranch has a See also:tract of the delta under its influence, and gradually tained, which can be plotted in their proper position, and the form raises the surface of that tract, and extends it seaward. As the delta of the river bed See also:drawn by connecting the extremities of the lines. extends itself seaward, the conditions of discharge through the From the section can be measured the sectional area of the stream different branches change. The water finds the passage through tl and its wetted perimeter x; and from these the hydraulic mean one of the branches less obstructed than through the others; the depth m can be calculated. velocity and scouring action in that See also:branch are increased; in the § 135. Measurement of the Discharge of Rivers.—The area of cross others they diminish. The one channel gradually absorbs the whole section multiplied by the mean velocity gives the discharge of the of the water supply, while the other branches silt up. But as the stream. The height of the river with reference to some fixed See also:mark mouth of the new See also:main channel extends seaward the resistance in- should be noted whenever the velocity is observed, as the velocity creases both from the greater length of the channel and the formation and area of cross section are different in differest states of the river. of shoals at its mouth, and the river tends to form new bifurcations To determine the mean velocity various methods may be adopted; AC or AD (fig. 134), and one of these may in time become the main and, since no method is free from liability to See also:error, either from the
channel of the river. difficulty of the observations or from uncertainty as to the ratio of
§ 133. See also: Ft. in lb. river, and chained and levelled. The cross sections are referred to the line of stakes, both as to position and direction. The determination of the surface slope is very difficult, partly from its extreme smallness, partly from oscillation of the water. See also:Cunningham recommends that the slope be taken in a length of 2000 ft. by four simultaneous observations, two on each side of the river. § 134. Cross Sections —A stake is planted flush with the water, and its level relatively to some point on the line of levels is determined. Then the depth of the water is determined at a series of points (if silt periodically overflows its banks, it deposits silt over the area river. It hence results that the section of the country assumes a § 131. Inundation Deposits from a River.—When a river carrying flooded, and gradually raises the surface of the country. The silt is deposited in greatest abundance where the water first leaves the r pass the same See also:graduation on each wire. See also:peculiar form, the river flowing in a trough along the crest of a ridge, deposited from the water forms two wedges, having their thick ends them visible they may have a vertical painted See also:stem. In experi- by Cunningham. It consists of a hollow metal See also:ball connected to a mints on the Seine, See also:cork balls It in. diameter were used, loaded to float flush with the water, and provided with a stem. In A. J. C. Cunningham's observations at See also:Roorkee, the floats were thin circular disks of See also:English See also:deal, 3 in. diameter and 4 in. thick. For observations near the banks, floats 1 in. diameter ands in. thick were used. To render them visible a tuft of See also:cotton See also:wool was used loosely fixed in a hole at the centre. The velocity is obtained by allowing the float to be carried down, and noting the time of passage over a measured length of the stream. If v is the velocity of any float, t the time of passing over a length 1, then v=l,/t. To mark out distinctly the length of stream over which the floats pass, two See also:ropes may be stretched across the stream at a distance apart, which varies usually from 50 to250ft., according to the size and rapidity of the river. In the Roorkee experiments a length of run of 5o ft. was found best for the central two-fifths of the width, and 25 ft. for the See also:remainder, except very close to the banks, where the run was made 122 ft. only. The longer the run the less is the proportionate error of the time observations, but on the other See also:hand the greater the deviation of the floats from a straight course parallel to the axis of the stream. To mark the precise position at which the floats cross the ropes, Cunningham used short See also: In Cunningham's observations two chronometers were sometimes used, the time of passing one end of the run being noted on one, and that of passing the other end of the run being noted on the other. The chronometers were compared immediately before the observations. In other cases a single chronometer was used placed midway of the run. The moment of the floats passing the ends of the run was signalled to a time-keeper at the chronometer by shouting. It was found quite possible to See also:count the chronometer beats to the nearest See also:half second, and in some cases to the nearest See also:quarter second. § 137. Sub-surface Floats.—The velocity at different depths below the surface of a stream may be obtained by sub-surface floats, used precisely in the same way as surface floats. The most usual arrange- ment is to have a large float, of slightly greater density than water, connected with a small and very light surface float. The motion of the combined arrangement is not sensibly different from that of the large float, and the small surface float enables — an observer to See also:note the path and velo- See also:city of the sub-surface float. The in- strument is, however, not free from objection. If the large submerged float is made of very nearly the same density as water, then it is liable to be thrown upwards by very slight eddies in the water, and it does not maintain its position at the depth at which it is intended to float. On the other hand, if the large float is made sensibly heavier than water, the indicating or surface float must be made rather large, and then it to scme extent influences the motion of the submerged float. Fig. 137 shows one form of sub- surface float. It consists of a couple of See also:tin plates See also:bent at a right angle and soldered together at the angle. This is connected with a wooden ball at the surface by a very thin wire or See also:cord. As the tin alone makes a heavy submerged float, it is better to attach to the tin float some pieces of wood to diminish its weight in water. Fig. 138 shows the form of submerged float used slice of cork, which serves as the surface float. § 138. Twin Floats.—Suppose two equal and similar floats (fig. 139) connected by a wire. Let one float be a little lighter and the other a little heavier than water. Then the velocity of the comuil,ed 1 i D – 3"dia:- w-3"diem,+! FIG. 138. FIG. 139. floats will be the mean of the surface velocity and the velocity at the depth at which the heavier float swims, which is determined by the length of the connecting wire. Thus if v, is the surface velocity and vs the velocity at the depth to which the lower float is sunk, the velocity of the combined floats will be v = 2 (ve-}-vd). Consequently, if v is observed, and v., determined by an experiment with a single float, Vd=2V–vs. According to Cunningham, the twin float gives better results than the sub-surface float. § 139. Velocity Rods.—Another form of float is shown in fig. 140. This consists of a cylindrical See also:rod loaded at the lower end so as to float nearly vertical in water. A wooden rod, with a metal cap at the bottom in which shot can be placed, answers better than anything else, and sometimes the wooden rod is made in lengths, which can be screwed together so as to suit streams of different depths. A tuft of cotton wool at the See also:top serves to make the float more easily visible. Such a rod, so adjusted in length that it sinks nearly to the bed of the stream, gives directly the mean velocity of the whole vertical section in which it floats. § 140. Revy's Current See also:Meter.—No instrument has been so much used in directly determining the velocity of a stream at a given point as the See also:screw current meter. Of this there are a dozen varieties at least. As an example of the instrument in its simplest form, Revy's meter may be selected. This is an ordinary screw meter of a larger size than usual, more carefully made, and with its details carefully studied (See also:figs. 141, 142). It was designed after experience in gauging the great See also:South See also:American rivers. The screw, which is actuated by the water, is 6 in. in diameter, and is of the type of the Griffiths screw used in See also:ships. The hollow spherical See also:boss serves to make the weight of the screw sensibly equal to its displacement, so that See also:friction is much reduced. On the axis as of the screw is a See also:worm which drives the See also:counter. This consists of two worm wheels g and h fixed on a common axis. The worm wheels are carried on a See also:frame attached to the See also:pin 1. By means of a See also:string attached to l they can be pulled into See also:gear with the worm, or dropped out of gear and stopped at any instant. A See also:nut m can be screwed up, if necessary, to keep the counter permanently in gear. The worm is two-threaded, and the worm See also:wheel g has 200 See also:teeth. Consequently it makes one rotation for too rotations of the screw, and the number of rotations up to too is marked by the passage of the graduations on its edge in front of a fixed See also:index. The second worm wheel has 196 teeth, and its edge is divided into 49 divisions. Hence it falls behind the first wheel one See also:division for a See also:complete rotation of the latter. The number of hundreds of rotations of the screw are therefore shown by the number of divisions on h passed over by an index fixed to g. One difficulty in the use of the ordinary screw meter is that particles of grit, getting into the working parts, very sensibly alter the friction, and therefore the See also:speed of the meter. Revy obviates this by enclosing the counter in a See also:brass box with a See also:glass face. This box is filled with pure water, which ensures a constant coefficient of friction for the rubbing parts, and prevents any mud or grit finding its way in. In See also:order that the meter may place itself with the axis parallel to the current, it is pivoted on a vertical axis and directed by a large See also:vane shown in fig. 142. To give the vane more directing power the vertical axis is nearer the screw than in ordinary meters, and the vane is larger. A second horizontal vane is attached by the screws x, x, the See also:object of which is to allow the meter to See also:rest on the ground without the motion of the screw being interfered with. The string or wire for starting and stopping the meter is carried through the centre of the vertical axis, so that the See also:strain on it may not tend to pull the meter oblique to the current. The See also:pitch of the screw is about 9 in. The screws at x serve for filling the meter with water. The whole apparatus is fixed to a rod (fig. 142), of a length proportionate to the depth, or for very great depths it is fixed to a weighted See also:bar lowered by ropes, a plan invented by Revy. The instrument is generally used thus. The See also:reading of the counter is noted, and it is put out of gear. The meter is then lowered into the water to the required position from a See also:platform between two boats, or better from a temporary See also:bridge. Then the counter is put into gear for one, two or five minutes. Lastly, the instrument is raised and the counter again read. The velocity is deduced from the number of rotations in unit time by the formulae given below. For surface velocities the counter may be kept permanently in gear, the screw being started and stopped by hand. § 141. The Harlacher Current Meter.—In this the ordinary counting apparatus is abandoned. A worm drives a worm wheel, which makes an See also:electrical contact once for each too rotations of the worm. This contact gives a See also:signal above water. With this arrangement, a series of velocity observations can be made, without removing the instrument from the water, and a number of See also:practical difficulties attending the accurate starting and stopping of the ordinary counter are entirely got rid of. Fig. 143 shows the meter. The worm wheel z makes one rotation for too of the screw. A pin moving the See also:lever x makes the electrical contact. The wires b, c are led through a See also:gas See also:pipe B ; this also serves to adjust the meter to any required position on the wooden rod dd. The See also:rudder or vane is shown at WH. The galvanic current acts on the electromagnet m, which is fixed in a small metal box containing also the See also:battery. The magnet exposes and withdraws a coloured disk at an opening in the See also:cover of the box. § 142. See also:Amsler Laffon Current Meter.—A very convenient and accurate current meter is constructed by Amsler Laffon of See also:Schaffhausen. This can be used on a rod, and put into and out of gear by a ratchet. The peculiarity in this case is that there is a See also:double ratchet, so that one pull on the string puts the counter into gear and a second puts it out of gear. The string may be slack during the action of the meter, and there is less uncertainty than when thecounter has to be held in gear. For deep streams the meter A is suspended by a wire with a heavy lenticular weight below (fig. 144). The wire is payed out from a small winch D, with an index showing the depth of the meter, and passes over a See also:pulley B. The meter is in gimbals and is directed by a conical rudder which keeps it facing the stream with its axis horizontal. There is an electric See also:circuit from a battery C through the meter, and a contact is made closing the circuit every too revolutions. The moment the circuit closes a See also:bell rings. By a subsidiary arrangement, when the foot of the instrument, 0.3 metres below the axis of the meter, touches the ground the circuit is also closed and the bell rings. It is easy to distinguish the continuous See also:ring when the ground is reached from the short ring when the counter signals. A convenient winch for the wire is so graduated that if set when the axis of the meter is at the water surface it indicates at any moment the depth of the meter below the surface. Fig. 144 shows the meter as used on boat. It is a very convenient instrument for obtaining the velocity at different depths and can also he used as a sounding instrument. § 143. Determination of the Coefficients of the Current Meter.—Suppose a series of observations has been made by towing the meter in still water at different speeds, and that it is required to ascertain from these the constants of the meter. If v is the velocity of the water_and n the observed number of rotations per second, let v=a+,t3n (I) where a and $ are constants. Now let the meter be towed over a measured distance L, and let N be the revolutions of the meter and t the time of transit. Then the speed of the meter relatively to the water is L/t=v feet per second, and the number of revolutions per second is Nit =n. Suppose m observations have been made in this way, furnishing corresponding values of v and n, the speed in each trial being as uniform as possible, En=ni+n2+ . . Ev=vi+v2+ Env =nevi +n2v2 + fn2=nl+ni+ [En]2=[ni+n2+ ]2 Then for the determination of the constants a and $ in (I), by the method of least squares En2~v — ` nEnv a= mEn2—[2n]2 mInv — = ntl n2 — [En]2 ' In a few cases the constants for screw current meters have been determined by towing them in R. E. See also:Froude's experimental tank in which the resistance of See also:ship See also:models is ascertained. In that case the data are found with exceptional accuracy. § 144. Darcy See also:Gauge or modified Pitot See also:Tube.—A very old instrument for measuring velocities, invented by See also:Henri Pitot in 1730 (Histoire de l'Academie See also:des Sciences, 1732, p. 376), consisted simply of a vertical glass tube with a right-angled bend, placed so that its mouth was normal to the direction of flow (fig. 145). The impact of the stream on the mouth of the tube balances a See also:column in the tube, the height of which is approximately h=v2/2g, where v is the velocity at the depth x. Placed with its mouth parallel to the stream the water inside the tube is nearly at the same level as the I l ~ surface of the stream, and turned with the -— > I, mouth down stream, the ii fluid sinks a depth Ijl~ y i ! h' =v2/2g nearly, though the tube in that case interferes with the free A B C flow of the liquid and result. Pitot See also:expanded the mouth of the tube so as to form a See also:funnel or bell mouth. In that case he found by experiment h= 1.5v2/2g. But there is more disturbance of the stream. Darcy preferred to make the mouth of the tube very small to avoid interference with the stream and to check oscillations of the water column. Let the difference of level of a pair of tubes A and B (fig. 145) be taken to be h=kv2/2g, then k may be taken to be a corrective coefficient whose value in well-shaped instruments is very nearly unity. By placing his instrument in front of a boat towed through water Darcy found k= 1•o34 ; by placing the instrument in a stream the velocity of which had been ascertained by floats, he found k= I •oo6 ; by readings taken in different parts of the section of a canal in which a known volume of water was flowing, he found k=0.993. He believed the first value to be too high in See also:con-sequence of the disturbance caused by the boat. The mean of the other two values is almost exactly unity (Recherches hydrauliques, Darcy and Bazin, 1865, p. 63). W. B. See also:Gregory used somewhat differently formed Pitot tubes for which the k = I (Am. See also:Soc. Mech. Eng., 1903, 25). T. E. See also:Stanton used a Pitot tube in deter-See also:mining the velocity of an air current, and for his instrument he found k=1•o3o to k=1.x32 (" On the Resistance of Plane Surfaces in a Current of Air," Proc. Inst. Civ. Eng., 1904, 156). One objection to the Pitot tube in its See also:original form was the great difficulty and inconvenience of reading the height It in the immediate neighbourhood of the stream surface. This is obviated in the Darcy gauge, which can be removed from the stream to be read. Fig. 146 shows a Darcy gauge. It consists of two Pitot tubes having their mouths at right angles. In the instrument shown, the two tubes, formed of See also:copper in the lower part, are See also:united into one for strength, and the mouths of the tubes open vertically and See also:horizon-See also:tally. The upper part of the tubes is of glass, and they are provided with a brass scale and two verniers b, b. The whole instrument is sup-ported on a vertical rod or small See also:pile AA, the fixing at B permitting the instrument to be adjusted to any height on the rod, and at the same time allowing free rotation, so that it can be held parallel to the current. At c is a two-way See also:cock, which can be opened or closed by cords. If this is shut, the instrument can be lifted out of the stream for reading. The glass tubes are connected at top by a brass fixing, with a stop cock a, and a flexible tube and See also:mouthpiece m. The use of this is as follows. If the velocity is re- quired at a point near the surface of the stream, one at least of the water columns would be below the level at which it could be read. It would be in the copper part of the instrument. Suppose then a little air is sucked out by the tube m, and the cock a closed, the two columns will be forced up an amount corresponding to the difference between atmospheric pressure and that in the tubes. But the difference of level will remain unaltered. When the velocities to be measured are not very small, this instrument is an admirable one. It requires observation only of a single linear quantity, and does not require any time observation. The See also:law connecting the velocity and the observed height is a rational one, and it is not absolutely necessary to make any experiments on the coefficient of the instrument. If we take v=kJ (2gh), then it appears from Darcy's experiments that for a well-formed instrument k does not sensibly differ from unity. It gives the velocity at a definite point in the stream. The See also:chief difficulty arises from the fact that at any given point in a stream the velocity is not absolutely constant, but varies a little from moment to moment. Darcy in some of his experiments took several readings, and deduced the velocity from the mean of the highest and lowest. § 145. Perrodil Hydrodynamometer.—This consists of a frame abed (fig. 147) placed vertically in the stream, and of a height not less than the stream's depth. The two vertical members of this frame are connected by cross bars, and united above water by a circular bar, situated in the vertical plane and carrying a horizontal, graduated circle ef. This whole See also:system is movable round its axis: being suspended on a See also:pivot at g connected with the fixed support mn. Other horizontal arms serve as guides. The central vertical rod gr forms a torsion rod, being fixed at r to the frame abed, and, passing freely upwards through the guides, it carries a horizontal C D See also:needle moving over the graduated circle ef. The support g, which carries the apparatus, also receives in a tubular See also:guide the end of the torsion rod gr and a set screw for fixing the upper end of the torsion rod when necessary. The impulse of the stream of water is received on a circular disk x, in the plane of the torsion rod and the frame See also:abcd. To raise and lower the apparatus easily, it is not fixed directly to the rod mn, but to a tube kl sliding on mn. Suppose the apparatus arranged so that the disk x is at that level the stream where the velocity is to be determined. The plane abcd is placed parallel to the direction of motion of the water. Then the disk x (acting as a rudder) will place itself parallel to the stream on the down stream side of the frame. The torsion rod will be unstrained, and the needle will be at zero on the graduated circle. If, then, the instrument is turned by pressing the needle, till the plane abcd of the disk and the zero of the graduated circle is at right angles to the stream, the torsion rod will be See also:twisted through an angle which See also:measures the normal impulse of the stream on the disk x. That angle will be given by the distance of the needle from zero. Observation shows that the velocity of the water at a given point is not constant. It varies between limits more or less wide. When the apparatus is nearly in its right position, the set screw at g is made to clamp the torsion spring. Then the needle is fixed, and the apparatus carrying the graduated circle oscillates. It is not, then, difficult to note the mean angle marked by the needle. Let r be the See also:radius of the torsion rod, 1 its length from the needle over ef to r, and a the observed torsion angle. Then the moment of the couple due to the molecular forces in the torsion rod is M =Etta/l; where Et is the modulus of See also:elasticity for torsion, and I the polar moment of inertia of the section of the rod. If the rod is of circular section, I =11-r4. Let R be the radius of the disk, and b its leverage, or the distance of its centre from the axis of the torsion rod. The moment of the pressure of the water on the disk is Fb = kb(G/2g)irR2v2, where G is the heaviness of water and k an experimental coefficient. Then Et I a/l = kb (G/2g)1R2v2. For any given instrument, v=c/a, where c is a constant coefficient for the instrument. The instrument as constructed had three disks which could be used at will. Their radii and leverages were in feet R= b= 1st disk . 0.052 0.16 2nd „ . 0.105 0.32 3rd „ . . . 0.210 o•66 For a thin circular See also:plate, the coefficient k =1.12. In the actual instrument the torsion rod was a brass wire 0.06 in. diameter and 61 ft. See also:long. Supposing a measured in degrees, we get by calculation V=0.33531 a; o.115-V a; 0.042s/ a. Very careful experiments were made with the instrument. It was fixed to a wooden turning bridge. revolving over a circular channel of 2 ft. width, and about z6 ft. 'circumferential length. An allowance was made for the slight current produced in the channel. These experiments gave for the coefficient c, in the formula v =c' a, 1st disk, c=0.3126 for velocities of 3 to 16 ft. 2nd „ 0.1177 „ „ I; to 31 3rd 0.0349 , less than 11 „ The instrument is preferable to the current meter in giving the velocity in terms of a single observed quantity, the angle of torsion, while the current meter involves the observation of two quantities, the number of rotations and the time. The current meter, except in some improved forms, must be withdrawn from the water to read the result of each experiment, and the law connecting the velocity and number of rotations of a current meter is less well-determined than that connecting the pressure on a disk and the torsion of the wire of a hydrodynamometer. The Pitot tube, like the hydrodynamometer, does not require a time observation. But, where the velocity is a varying one, and consequently the columns of water in the Pitot tube are oscillating, there is See also:room for doubt as to whether, at any given moment of closing the cock, the difference of level exactly measures the impulse of the stream at the moment. The Pitot tube also fails to give measurable indications of very low velocities. Additional information and CommentsThere are no comments yet for this article.
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