Vortex Theory for Bodtes Moving tn Water 
D - UNSTEADY MOTIONS WITH SEVERAL VORTEX SHEETS 
As shown in Subsection B, a double model in a uniform 
motion of translation in a direction V, parallel tothe (X, Y) - 
plane of symmetry, sheds three free vortex sheets starting from the 
keel YY, ,» from its image @, , and from the stern-post, respec- 
tively. 
We consider below a body which could not be necessarily a 
double model. Its stern-post is denoted S,S S, ; its stem is the arc 
E,EE,; the segments E,S, and ES, are parts of straight-lines. 
The origin O of the axes OX Y Z moving with the body is in the 
middle section, the X-axis containing the points S and E, and 
the Z-axis being in the longitudinal plane of symmetry of the body 
(See Fig. 5.6). 
At a given instant t', there are between two points B,)Bo 
on the keel ES, (or on the lower stern-post), two bound yortex 
ribbons, one on the port side ),, , with the intensity dX se (B 
and the other one on the starboard side), , with the intensity 
dX 012 (B , , t'), These two vortex ribbons end along the same arc 
B,; ceo , on the upper keel E 4 4 (or on the stern). The BY es 
tion of B, ee depends on B, neces on the abscissa X of B, 
and sSeeibly on, G'. 
ot!) 
In the interval (t', t' + dt'), the increments of the intensities 
of the two Ves vortex ribbons are dt' dX (2F1) 5. 
dt' dX ae Bt’ . If they are not equal to each other a free 
X 
vortex ribbon is shed. It begins on the segment Bje +! Bie, t! and 
ends at By, s Pea the points with subscript e are infinitely close 
to B, ,th BY ths Bist Py, . For the sake of brevity, we con- 
sider this free vortex ribbon ie an arc of free vortex filament 
Bie, t! Boe! with the intensity 
dt' dX woah Deodt, =F} 
at’ idx l 2) BR. tt 
The fluid point P which was at Bie, t! at time -ot' os «sha time t 
ata position Jy ;4:. In the system of axes moving with the body 
3 
— Vy W Ww . 5. 33 
Bie, t' Ix, t! sf YR Pillidett ( ) 
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