Wave Forces on a Restratned Shtp tn Head-Sea Waves 
i( wt - vx) Vz abe tik 
$ y(*, y, 2; t) ~e e = we 
Nr *-L/2 ed ow, 
Vw 
(65) 
- vly| ¢ (x) - (x) 
ave 
As for the zero-speed problem we see that the two-term inner 
expansion represents waves propagating along the ship in the same 
direction as the incoming wave. Arguing as for the zero-speed pro- 
b lem we should therefore expect an integrated effect along the ship as 
the lowest order term in (65) represents. 
We note that (65) does not reduce to (19) when U = 0. It 
should not be expected that (65) reduce to (19) when U =0 since we 
have assumed T = = > 1/4 and since this assumption has been an 
important part in our analysis. We should note that the last term in 
(65) represents a distrubance arising from upstream while the last 
term in (19) represents a disturbance arising from downstream. 
IV.2. The near-field problem and the matching 
We now formulate the near-field problem and perform the 
matching between the near-field and the far-field solutions. A one- 
term far-field solution is found to be due toa line of sources with 
source density. 
i(wt - vx) 
7) (x) e 
spread along the line y=z=0, -L/2< x < L/2. (See Figure 1). 
As in the zero-speed problem, a one-term near-field solution is found 
to be the negative of the incident wave. The matching of the far-field 
solution and the near-field solution determines o, (x) in a similar 
r791 
