Vortex Theory for Bodtes Moving in Water 
rection and, on >» fo in the negative z-direction, it is easily found 
that 
for B on B 
PE = Bia) (5.28) 
: e ; fox) on 
Equation (5.23) holds in all the cases. 
Let us consider two values a , a. Of a , with ada 
The dissymmetry of the flow is more strongly marked forasa ’ 
The part of free vortices coming from BR and arriving on? & is 
greater in the second case than in the first one, while the part of 
those coming from the upper part of cP and arriving on the lower 
partis smaller. This entails a rapid variation with a of the posi- 
tion of the lift, that is of the y-component of the hydrodynamic force 
exerted on the body. 
It has been assumed that the line “Bh is in the (Z, X) plane, 
In fact, if the bottom is flat, the line WR becomes a curve with posi- 
tive values of Y. However this phenomenon cannot alter seriously 
the velocity induced on the hull by the vortex distribution. 
One among the advantages due to the substitution of a normal 
dipole distribution for the vortex distribution is that one needs not 
know exactly the direction of the free vortex filaments. 
When a is too large, the relative streamlines on Poe tend to 
pass from ae to oF and separation occurs on the suction side. In 
such case, the above considerations do not hold. 
C - UNSTEADY MOTIONS IN THE CASE OF A UNIQUE FREE VORTEX 
SHEET 
Let us consider a point Mg on me ¢ attime t, and let a t? 
denote the position of Py at t'< t. Let P be the fluid point located 
at Mr at t. The condition 
B. (t') M -| ¥. (pyr) a (5.29) 
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