Wave Forces on a Restratned Shtp tn Head-Sea Waves 
APPENDIX A 
Simplification of the far-field expansion in the zero-speed problem 
We will show how (15) can be rewritten into (16) and (17). 
The procedure is based on work by Ogilvie (1969). 
We first introduce 
in (15). If we drop the primes we can write (15) as 
$ p(X ¥izet)| => pbx ynw) a ping (A-1) 
where 
] 
(x, vi z) = 5 dk eik* o* (k) 
4n 
5 rm | a Boney ee 
php O 
Dey, aN a aedia ace at aes (A -2) 
We will let y = 0(1) and we are going to. assume that y > 0. 
We define 
—y obyeen ONE +(v -k 
I(k) = lim ae (A -3) 
! 
