A THEORY OF FLUID FRICTION. 79 
Even if the elevation of the surface is small, for a given reduction in velocity, 
the total loss of head will be less than if there were no elevation of the surface, 
and the reduction in velocity will occur at a less distance abaft the cutwater. At 
low speeds, it is probable that the influence of the rise of the surface will be appre- 
ciable, and this fact may account for the observed irregularity in the resistance 
of models when towed at the lower range of speeds. 
The increase in “position” head of a lamina of water for each foot of depth 
for each per cent loss of velocity is shown in Fig. 7, Plate 60. The loss of velocity 
head for each per cent loss of velocity for an initial velocity of 10 foot-seconds is 
shown on the same figure. Corresponding curves for the whole width of the wake 
are shown in Fig. 8, Plate 60, based on the assumption that the velocity curve across 
the wake is a cubic parabola. As the elevation of the surface cannot be main- 
tained in the open, the curves giving the increase in position head are of interest 
only as showing the initial relation between the two effects. 
If we neglect the effect of eddies and of elevation of surface, and if F, the co- 
efficient of liquid friction of water, is considered known, as well as f, the coefficient 
of friction of the surface of the plane in water, the resistance may be calculated 
by determining the length of the surface corresponding to a definite change in ve- 
locity due to the friction. For instance, from equation (2) we determine the values 
of tan a corresponding to velocity Vo and to velocity 90 per cent of Vo, the latter 
value being 81 per cent of the former. Considering the velocities between the sur- 
face of the plane and the unaffected water to be represented by the ordinates of a 
cubic parabola, the breadth b of the water affected by friction at the distance / from 
the cutwater, at which the velocity of rubbing is .goV», is found by construction as— 
3x(%o—-90%) _ 3% 
tan O& sistant aes 
ws (3) 
The corresponding value of H —h, the velocity head remaining in the cross- 
2 
section of the wake at this point, can be calculated, and is found to be .95 a The 
2 
loss of head due to friction, therefore, in this length /, is 04952 This loss of 
head is also— 
= JS v2. d= Stan a. al (4) 
If the intervals in the velocity are taken sufficiently close, this value of h may 
be taken as that due to the value of tan a corresponding to a velocity slightly less 
than the mean velocity. For the particular instance, we have— 
Ae ee (s) 
JD 6 AEST Chey 
