Vortex Theory for Bodtes Moving tn Water 
suction side. The transverse cuts of the hullare V or U - shaped, 
the radius of curvature of the U's is small. 
Experiments show that, if a is not too great, then the re- 
lative streamlines on the port side), are less inclined on the 
(X, Y) plane than those located on the starboard side ae - Let E,, 
EGF Ah fs. be a sequence of points on the lower half-stem. On a5 ; 
the relative streamlines coming from the E,'s leave the hull at 
points S;, located either on the lower half-stern-post or on the keel, 
in the (Z, X) - plane. 2p ae , the relative streamlines leave the 
hull at points Sy also located either on the lower half-stern-post or 
on the keel. But the points S} located on the stern-post are below 
the points S, , and the points S, located on the keel are upstream 
of the points S;,. Consequently, the two streamlines arriving ata 
point B on the keel (or on the stern-post) come from two different 
points E. The relative velocities on these two streamlines are equal 
at B, but their directions differ. This entails the shedding of a free 
vortex filament from B. 
Thus there exist three surfaces oF . One, denoted De. Rae: 
generated by the free vortices shed along the keel ; the second one, 
denoted Ee is the mirror image of Ye ; the third surface, denoted 
oa is generated by the free vortices shed along the stern-post et 
is its own image (Figure 5.5). 
Let PD ' denote the complete stern-post S, SS, and B 5 . 
the keel and its image, respectively. Every free vortex filament 
¢ starts either from the upper half of CP or from BR and goes 
at infinity downstream from the body. Because of the steadiness of 
the motion, Y f coincides with a relative streamline. Consequently, 
ae is nearly parallel to the (X, Y) - plane. So does the upper 
edge of aan . The said vortex filament Y ¢ comes back from infinity 
towards the body and reaches it either on the lower half of oF on 
Ro: 
The start point B, of £ 5 and its end B, are mirror images, 
Let S, be the bound vortex BS B, on abet and Ce the bound vortex 
B, By on ‘eo >: As the relative velocity on we is greater than on 
ee the intensity dIy of the free vortex a eee starting from the 
body between two points B, , B} and arriving on B,By is equal to 
the difference dI, -dI, , dI, being the intensity of the bound 
vortex ribbon L, on)), starting from B,By and arriving at 
B, Bi » while ar, is that of the bound vortex ribbon L, on))5 
starting from B,)By and arriving at B, Bi 
at es dt al (5. 25) 
