Vortex Theory for Bodtes Moving tn Water 
To calculate Dr » one can observe that a vortex filament 
Keer) is equivalent to a uniform dipole distribution on any open sur- 
face the edge of which coincides with fYy(t'). For that surface we may 
take the surface Sy. t! consisting of the five following parts : 
i 
(i) The part dy (X) of a behind the arc BOB : 
sie 
(ii) The part of ys interior to the contour S I I Br See: 
i O 2 4h se Ga Oe 
2 ? ? 
(iii) The part of Pe interior to the contour s\J L J B, Ss, ; 
E Set Meet ae (5.4: 
(iv) The part of pas interior to the contour I s Sd I : 
ot! o 1 -4, t! = t! 
(v) The part of the surface S! generated = the arc Jy tly ¢ 
when X varies from - 5 to the value X defining the vortex 
Z 
filament Fy (t")s 
= this surface let us select a unit vector ? normal to it ; for instance, 
Vison 2 1, 2m the outward direction with respect to the hull. On the 
fifth part of Sx t vy is thus directed toward Be ete!’ Det). M*, ‘M7 
be two points cnheitely close to each other, M* being on one side of 
the surface and M™ on the other one ’so that M~M’ is in the direction 
of Y. The circulation of the velocity induced b x(t') ina circuit 
starting from Mt, turning around x(t') and ending at M is equal 
to the intensity of X, t! and equal to the density of the normal dipole 
distribution on the surface. Hence the velocity potential due to the 
dipole distribution is 
did ?y 4 (M) = = 4,4 d x [A r (Xt!) -2,(x, >) Aff ees, ~ ana &5(M"). (5.42 
X, x 
The total potential outside W, and its boundary is 
max - 
smo =f 1 fe x, tM) : (5. 43) 
! 
According to (5.41) the contribution from Bis 
1 - 1 
Feuate (X, t) = eee t) xp aia (M') ; 5. 44 
4 In 2 lhe Onan M'M 1 ( ) 
