Wave Forees on a Restratned Shtp tn Head-Sea Waves 
The paths in the two integrals pass respectively below and 
above the double pole at w= 0, 
We are going to rewrite (97) so that we can more easily 
evaluate it numerically. We introduce 
me vy sin hu 
as a new integration variable. 
We then introduce closed integration paths in the complex 
a plane properly indented at the branch points of the integrand. 
By using the residue theorem, we can then write (97) as 
0) OO 
: Zz 2 
Ss. ee | i) Oss eee * cos(uy) - » eens [=| : +u 
2 2 
0 a oe +u 
(98) 
The other unexplained terms in (96) are the wave-free potentials 
O,,. They are given by 
O'. (9) = Kom of ( vr) cos(2m - 2) 6 
+ 2K , (vr) cos(2m - 1) +K 
6 
See mi??? cos2m 
2 
eles 
(Ursell denoted these functions by or 
1801 
