510 HYDROMECHANICS [HYDRAULICS. the data are determined for plotting the path CP of the float and determining its velocity. By dropping in a series ot floats, a mim- ber of surface velocities can be determined. When all these hav.. been plotted, the river can be divided into convenient compartments. The observations belonging to each compartment are then averaged, and the mean velocity and discharge calculated. It is obvious that, as the surface velocity is greatly altered by wind, experiments of this kind should be made in very calm weather. The ratio of the surface velocity to the mean velocity in the same vertical can be ascertained from the formulae for the vertical velocity curve already given ( 101). Exner, in Erbkam s Zeitsclirift for 1875, has given the following convenient formula. Let v be the mean and V the surface velocity in any given vertical longitudinal section, the depth of which is h If vertical velocity rods are used instead of common floats, the mean velocity is directly determined for the vertical section in which the rod floats. No formula of reduction is then necessary. The observed velocity has simply to be multiplied by the area of the compartment to which it belongs. 135. Mean Velocity of the Stream from a Series of Mid Depth Velocities. In the gaugings of the Mississippi it was found that the mid depth velocity differed by only a very small quantity from the mean velocity in the vertical section, and it was uninfluenced by wind. If therefore a series of mid depth velocities are determined by double floats or by a current meter, they maybe taken to be the mean velocities of the compartments in which they occur, and no formula of reduction is necessary. If floats are used, the method is precisely the same as that described in the last paragraph for sur face floats. The paths of the double floats are observed and plotted, and the mean taken of those corresponding to each of the compart ments into which the river is divided. The discharge is the sum of the products of the observed mean mid depth velocities and the areas of the compartments. 136. Boileau s Process for Gauging Streams. Let U be the mean velocity at a given section of a stream, V the maximum velocity, or that of the principal filament, which is generally a little below the surface, W and w the greatest and least velocities at the surface. The distance of the principal filament from the surface is generally less than one-fourth of the depth of the stream ; W is a little less than V ; and U lies between W and w. As the surface velocities change continuously from the centre towards the sides, there are at the surface two filaments having a velocity equal to U. The deter mination of the position of these filaments, which Boileau terms the ganging filaments, cannot be effected entirely by theory. But, for sections of a stream in which there are no abrupt changes of depth, their position can be very approximately assigned. Let A and I be the horizontal distances of the surface filament, having the velocity W, from the gauging filament, which has the velocity U, and from the bank on one side. Then c being a numerical constant. From gaugings by Humphreys and Abbot, Bazin, and Baumgarten, the values c = 919, 922, and 925 are obtained. Boileau adopts as a mean value 922. Hence, if W and w are determined by float gauging or otherwise, A can be found, and then a single velocity observation at A feet from the filament of maximum velocity gives, without need of any reduction, the mean velocity of the stream. More conveniently W, w, and U can be measured from a horizontal surface velocity curve, obtained from a series of float observations. 137. Direct Determination of the Mean Velocity by a Current Meter or Darcy Gauge. The only method of determining the mean velocity at a cross section of a stream which involves no assumption of the ratio of the mean velocity to other quantities is this a plank bridge isfixed across the stream near its surface. From this, velocities are observed at a sufficient number of points in the cross section of the stream, evenly distributed over its area. The mean of these is the true mean velocity of the stream. In Darcy and Bazin s experi ments on small streams, the velocity was thus observed at 36 points in the cross section. When the stream is too large to fix a bridge across it, the observa tions may be taken from a boat, or from a couple of boats with a gangway between them, anchored successively at a series of points across the width of the stream. The position of the boat for each series of observations is fixed by angular observations to a base line on shore. 138. Harlacher s Graphic Method of determining the Discharge from a Series of Current Meter Observations. Let ABO (fig. 150) be the cross section of a river at which a complete series of current meter observations have been taken. Let I., II., III. . . . be the verticals at different points of which the velocities were measured. Suppose the depths at I., II., III., . . . (fig. 150), set off as vertical orclinates in fig. 151, and on these vertical ordinates suppose the velocities set off horizontally at their proper depths. Thus, if v is the measured velocity at the depth h from the surface in fig. 150, on vertical marked III., then at III. in fig. 151 take cd = h and ac v. Q ffl Fig. 150. Then rf is a point in the vertical velocity curve for the vertical III., and, all the velocities for that ordinate being similarly set off, the curve can be drawn. Suppose all the vertical velocity curves I. . . . V. (fig. 151), thus drawn. On each of these figures draw verticals corresponding to velocities of x. 2x, 3x . . . feet per second. Then for instance cd at III. (fig. 151) is the depth at which a velo city of 2x feet per r rr -nr second existed on * -^ a, IF v p. the vertical III. in fig. 150, and if cd is set off at III. in fig. 150 it gives a point in a curve passing through points of the section where the velocity was 2# feet per second. Set off on each of the verticals in fig. 150 all the depths thus found in the corresponding diagram in fig. 151. Curves drawn through the corresponding points on the verticals are curves of equal velocity. The discharge of the stream per second may be regarded as a solid having the cross section of the river (fig. 1 50), as a base, and cross sec tions normal to the plane of fig. 150 given by the diagrams in fig. 151. The curves of equal velocity may therefore be considered aa contour lines of the solid whose volume is the discharge of the stream per second. Let fi be the area of the cross section of the river, O lt O 2 . . . . the areas contained by the successive curves of equal velocity, or, if these cut the surface of the stream, by the curves and that sur face. Let x be the difference of velocity for which the successive curves are drawn, assumed above for simplicity at 1 foot per second. Then the volume of the successive layers of the solid body whose volume represents the discharge, limited by successive planes passing through the contour curves, will be a:(0 + Qj) , ^(Oj + 2 ) , and so on. Consequently the discharge is The areas fi^f^ . . . . are easily ascertained by means of the polar planimeter. A slight difficulty arises in the part of the solid lying above the last contour curve. This will have generally a height which is not exactly x, and a form more rounded than the other layers and less like a conical frustum. The volume of this may be estimated separately, and taken to be the area of its base (the area Q n ] multiplied by 3 t 2 its height. Fig. 152 shows the results of one of Professor Harlacher s gaugings worked out in this way. The upper figure shows the section of the river and the positions of the verticals at which the soundings and gaugings were taken. The lower gives the curves of equal velocity, worked out from the current meter observations, by the aid of vertical velocity curves. The vertical scale in this figure is ten times as great as in the other. The discharge calculated from the contour curves is 14 1087 cubic metres per second. In the lower figure some other interesting curves are drawn. Thus, the uppermost dotted curve is the curve through points at which the maximum velocity was found ; it shows that the maximum velocity was always a little below the surface, and at a greater depth at the centre than at the sides. The next curve shows the depth at which the mean velocity for each vertical was found. The next is the curve of equal velocity corresponding to the mean velocity of the stream ; that is, it passes through points in the cross section where the velocity was identical with the mean velocity of the stream. XII. IMPACT AND REACTION OF WATER. When a stream of fluid impinges on a solid surface, it presses on the surface with a force equal and opposite to that by which the velocity and direction of motion of the fluid are changed. Generally, in problems on the impact of fluids, it is necessary to neglect the effect of friction between the fluid and the surface on which it moves. 139. During Impact the Velocity of the Fluid relatively to the Sur
face on ichich it impinges remains unchanged in Magnitude. Con-
