Falttnsen 
We will write eer as 
$= @& + ¢ = £2 ca a aoe. (55) 
T I D = toa 
ie) 
where $¢p denotes the diffraction potential. ¢) must satisfy a ra- 
diation condition. As in the zero-speed problem we are going to use 
the method of matched asymptotic expansions to find ¢ D: 
We will assume that the forward speed 
uw = aie Di. een Ie (56) 
In the steady forward-motion problem we know that there is a length 
scale in the x-direction which is connected with the wave length 
zZ rU?/g. So (56) implies that this length scale is large compared 
with the transverse dimensions of the ship, and that it can be of the 
same order of magnitude as the length of the ship. In some way, I 
expect this length scale will enter our diffraction problem and affect 
the rate of change of the variables in the x-direction. But it turns 
out that it will not have any influence on the first two approximations 
of the diffraction potential. The important length scale in the x- 
direction will be connected with the wave length of the incoming wave, 
in the same way as for the zero-speed problem. As we remember 
from equation (10), this wave length is assumed to be of order e. 
If, however, we had assumed that ''a'' were zeroin (56), 
we would have beenin difficulties finding the second approximation to 
the diffraction potential. The reason must be that there then are two 
important length scales of order ¢€ in the x-direction, one connected 
with the wave length of the incoming wave and one connected with the 
forward speed, and it is difficult to separate out the effect of one of 
the length scales from the other. 
Using (7) and (9), we can show that (56) implies that the 
order of magnitude of the frequency of encounter, w will be 
qe ie Ore Sse (57) 
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