Falttnsen 
problem gives an integral equation for 4, (x) (see (32) ). o, is 
needed in the second order near-field solution. The two-term near- 
field solution is given by (48). 
It should be noted that ''Near-field'' means the region near 
the body, where the distance from the body is 0(€ ). However, we do 
not expect the near-field approximations to be valid near the bow and 
stern. 
We will express the potential of the diffracted wave as fol- 
lows: 
p =F Te y (x, y> z) : (20) 
in the fluid region. The free-surface condition, (11), is: 
Oy ss a 
oe pa =hoeqon ii aaelage: (22) 
The body boundary condition, (2), together with (12), gives 
Oy av. Ov > lr gh vz 
[> Be + n, age ivn, + n, Ser = [ivn, - vn] te 
(23) 
on 2. = hi -6xiy/) 
n,,n,, and n, have been explained before equation (8). A last 
condition on y is that it must match with the far-field solution. 
We will assume that y varies very slowly in the x-direc- 
tion compared with the variation of y in the transverse plane. We 
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