Wave Forces on a Restratned Shtp tn Head-Sea Waves 
Ursell (1968 a)* has given a solution to that problem, but it does not 
appear to match with (31). However, if we say that the one-term near- 
field solution is just the negative of the incident wave (this is a special 
case of Ursell's solution), then (28), (29) and (30) are satisfied, 
and if we require that 
dia {2 ) 
= Ce (32) 
27 se [x ¢| 
k -1/2 kZ -inr/4 
€ e e 
then we see that a one-term outer expansion of the one-term near- 
field solution matches with a one-term inner expansion of a one-term 
far-field solution. So we have the solution 
v (X,Y, Z, € ) 
| 
| 
on 
i] 
— 
nae 
MN 
IN 
7 
@ 
= 
N 
i] 
he 
4 
Se 
AN 
Q 
Str 
Fle 
i] —~ 
AL wt 
(Fy 
eo 
We solve (32) for 0, (x) formally by letting it be an equality for 
all x 2>-L/2. We recognize (32) as Abel's integral equation (see 
Dettman (1965) ), which has the solution 
os 2 in/4 
7 (x)t .2:2ie has Lye) 8° a iG (34) 
This solution is singular at x = -L/2, which is a violation of the 
* Ursell's solution will be needed in the second order term and 
will be discussed then. 
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