HYDRAULICS.] HYDROMECHANICS 497 the maximum velocity should be at the surface, and then c=0, and the equation becomes Assuming this for the present, the mean velocity is _/ (3). The bottom velocity is, putting the depth h for y in (2) ; 1 Gi z ~ r ~ and therefore It is now understood that the motion in a stream is much more complete than the viscous theory just stated assumes, The retarda tion of the stream is much greater than it would be in simple motion of that kind. This has already been partly explained in the introduction to the present article. Nevertheless the viscous theory may probably be so modified as to furnish ultimately a true theory of streams. 99. Experimental Observations on the Vertical Velocity Curve. In obtaining the vertical velocity curve from direct observations in streams, a preliminary difficulty arises from the fact that the velocity at any given depth is not constant, and hence the motion in the strict sense is not steady. The velocities taken on a given vertical section at any given moment do not form when plotted any regular curve. But if a series of observations are taken at each depth and the results averaged, the mean velocities at each depth when plotted give a regular curve, agreeing very fairly with the parabola, which might be expected from the theory above. Hence it may be in ferred that the velocity at any given point fluctuates about a mean value, the fluctuations being due to irregular eddying motions superposed on the general steady motion of the stream, and having an effect which disappears in the mean of a series of observations. For certain purposes these irregular motions may be ignored, and the constant mean velocity substituted for the actual varying ve locities at each point. In the next place, all the best observations show that the maximum velocity is to be found, not at the free surface of the stream, but at some distance below it. 1 Influence of the Wind. In the experiments on the Mississippi the vertical velocity curve in calm weather was found to agree fairly with a parabola, the greatest velocity being at 1 yhs of the depth of the stream from the surface. With a wind blowing down stream the surface velocity is increased, and the axis of the parabola approaches the surface. On the contrary, with a wind blowing up stream the surface velocity is diminished, and the axis of the para bola is lowered, sometimes to half the depth of the stream. The American observers drew from their observations the conclusion that there was an energetic retarding action at the surface of a stream like that due to the bottom and sides. If there were such a retarding action the position of the filament of maximum velocity below the surface would be explained. _ It is not difficult to understand that a wind acting on surface ripples should accelerate or retard the surface motion of the stream, and the Mississippi results may be accepted so far as showing that the surface velocity of a stream is variable when the mean velocity of the stream is constant. Hence observations of surface velocity by floats or otherwise should only be made in very calm weather. But it is very difficult to suppose that, in still air, there is a resist ance at the free surface of the stream at all analogous to that at the sides and bottom. Further, in very careful experiments, Boileau found the maximum velocity, though raised a little above its posi tion for calm weather, still at a considerable distance below the surface, even when the wind was blowing down stream with a ve locity greater than that of the stream, and when the action of the air must have been an accelerating and not a retarding action. Professor James Thomson has given a much more probable explan ation of the diminution of the velocity at and near the free sur- ! face. He points out that portions of water, with a diminished velocity from retardation by the sides or bottom, are thrown off in eddying masses and mingle with the rest of the stream. These iddymg masses mo.lify the velocity in all parts of the stream, but hare their greatest influence at the free surface. Reaching the free surface they spread out and remain there, mingling with the water at that level and diminishing the velocity which would otherwise be found there. 100. Influence of tlie Wind on the Depth at which the Maximum Velocity is found. In the gaugings of the Mississippi the vertical velocity curve was found to agree well with a parabola having a Pilot first showed, in 1732, that the velo ity in a stream diminishes from the imace downwards; about sixty years after, Woltmann concluded that the Vertical velocity curve was a parabola. horizontal axis at some distance below the water surface, the ordi- nate of the parabola at the axis being the maximum velocity of the section During the gaugings the force of the wind was registered on a scale ranging from for a calm to 10 for a hurricane. Arrang ing the velocity curves in three sets (1) with the wind blowing up stream, (2) with the wind blowing down stream, (3) calm or wind blowing across stream it was found that an up stream wind lowered, and a down stream wind raised, the axis of the parabolic velocity curve In calm weather the axis was at ^ths of the total depth from the surface for all conditions of the stream. Let K be the depth of the axis of the parabola, m the hydraulic- mean depth, /the number expressing the force of the wind which may range from +10 to- 10, positive if the wind is up stream negative if it is down stream. Then Messrs Humphreys and Abbot find their results agree with the expression -0-3170-06/. m Fig. 115 shows the parabolic velocity curves accordin" to the American observers for calm weather, and for an up or down stream wind of a force represented by 4. BM 101. Bazin s Formulae, for the Variation of Velocity in a Vertical Longitudinal Section of a Stream. M. Bazin assumes that the ver tical velocity curve is a parabola, and has investigated numerical values for the constants from his own and other experiments. Assuming the general equation already found, 98, 1 Gy (} -^^r J UA v will haVe the maximum value V, for a value of h of y which makes m, , . zero. That is, dy ck Gz , and the maximum velocity is V = i H ^h Inserting these values of t and c in (1), or putting Gili? , where h is the whole depth of the section, (2), h h where M is constant for any given stream. Let ^ = x and 7- = o h h t- = V-M(z-a) 2 (2a). Then the mean velocity on the vertical is =/" {V-M(a?-a) s }die-V-M(J- a + a 1 ) . ft Let v n be the velocity at nh feet from the surface, ri_,, the velocity at an equal depth from the bottom, Let ft = , and put t>j for the velocity at mid dopth, then so that the mid depth velocity differs from the mean velocity by the small quantity -^ M only, whatever be the position of the axis of the parabola. Messrs Humphreys and Abbot have based on this property a method of rapidly gauging rivers which will be described hereafter.
xrr. 6*
