Let h´ be the depth of the axis of the parabola, m the hydraulic mean depth, f 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 Humphreys and Abbot find their results agree with the expression
Fig. 106 shows the parabolic velocity curves according to the American observers for calm weather, and for an up- or down-stream wind of a force represented by 4.
It is impossible at present to give a theoretical rule for the vertical velocity curve, but in very many gaugings it has been found that a parabola with horizontal axis fits the observed results fairly well. The mean velocity on any vertical in a stream varies from 0.85 to 0.92 of the surface velocity at that vertical, and on the average if v0 is the surface and v_m the mean velocity at a vertical v_m = 6/7 v0, a result useful in float gauging. On any vertical there is a point at which the velocity is equal to the mean velocity, and if this point were known it would be useful in gauging. Humphreys and Abbot in the Mississippi found the mean velocity at 0.66 of the depth; G. H. L. Hagen and H. Heinemann at 0.56 to 0.58 of the depth. The mean of observations by various observers gave the mean velocity at from 0.587 to 0.62 of the depth, the average of all being almost exactly 0.6 of the depth. The mid-depth velocity is therefore nearly equal to, but a little greater than, the mean velocity on a vertical. If v_(md) is the mid-depth velocity, then on the average v_m = 0.98v_(md).
§ 105. _Mean Velocity on a Vertical from Two Velocity Observations._--A. J. C. Cunningham, in gaugings on the Ganges canal, found the following useful results. Let v0 be the surface, v_m the mean, and v_(xd) the velocity at the depth xd; then
v_m = ¼[v0 + 3v_(2/3d)] = ½[v_(.211)^d + v_(.789)^d].
§ 106. _Ratio of Mean to Greatest Surface Velocity, for the whole Cross Section in Trapezoidal Channels._--It is often very important to be able to deduce the mean velocity, and thence the discharge, from observation of the greatest surface velocity. The simplest method of gauging small streams and channels is to observe the greatest surface velocity by floats, and thence to deduce the mean velocity. In general in streams of fairly regular section the mean velocity for the whole section varies from 0.7 to 0.85 of the greatest surface velocity. For channels not widely differing from those experimented on by Bazin, the expression obtained by him for the ratio of surface to mean velocity may be relied on as at least a good approximation to the truth. Let v0 be the greatest surface velocity, v_m the mean velocity of the stream. Then, according to Bazin,