Dagan 
f, he whe (x —co) (13) 
¥, =tu, t (Ixl<1, y= -bh +0) (aa) 
Equations (7) - (14) have been written, after a simple integration, in 
terms of f, , f, rather than w, , w (the derivation may be 
found, for instance, in Wehausen & Laitone, 1965). 
The main purpose of this section is to derive the second 
order solution f, , and the associated wave resistance, with appli- 
cation to a few particular shapes. 
Such computations have been carried out previously by Tuck 
(1964) for a submerged cylinder and Salvesen (1969) for a submerged 
hydrofoil. The subsequent developments in this section are a continu- 
ation of their work. The cylinder is an extremely blunt shape whose 
three dimensional counterpart is a sphere. In the case of the hydrofoil 
it was found that a major nonlinear contribution comes from the 
vorticity associated with the Kutta-J oukovsky condition. We have been 
interested to extend the previous computations to the case of elonga- 
ted two-dimensional bodies ressembling ships and we have also con- 
sidered only the contribution of thickness, since the trailing edge 
condition has no direct counterpart in three dimensions. In contrast 
with the previous works, we have been able toderive f, ina closed 
analytical form and we are inclined to believe that the method em- 
ployed here may be efficiently used in three dimensional cases. 
To obtain simple results we replace the body by a distribu- 
tion of an arbitrary number n of discrete sources of strength q', 
located at z = x') - ih' (j=1, ...., n) (fig. 1). Under this scheme 
the two boundary conditions (10) and (14) become 
3 
€. 
EN aac In(z - z,)] + O{ ew) [zeal eee (15) 
where €)= q' /U'T'=2At./T'=2At, is the change of the rela- 
tive thickness at z) . Obviously, by (15) the actual body is replaced 
by one undergoing abrupt thickness changes at each source, but by 
taking the spacing Kia 7 Xj sufficiently small in the portions of 
steep variation of t , we can achieve any desired accuracy. Again, 
in view of our interest in three dimensional applications, we do not 
take into account the trailing edge Kutta-Joukovsky condition. Our 
problem reduces, therefore, in determining f,; and fz , subject to 
(7), (9), (11), (13) and (15). The profile of the far free wavesis 
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