Wave Forees on a Restratned Shtp tn Head-Sea Waves 
Using (16) and the symmetry properties associated with 
sources, we can now write down a two-term inner expansion of the 
far-field source solution as 
+ oy ly| o (x) - etl aoe 
We note that this expression represents waves propagating 
along the ship in the same direction as the incoming wave. So to the 
lowest order it seems as if the disturbance created by the ship is 
carried along the ship in the direction of positive x-axis (see Fig. 1). 
Looking on what is happening at some cross-section x, we should 
therefore expect an integrated effect of what is happening at sections 
from the forward perpendicular (x = -L/2) up to the cross-section 
we consider. And this is what the lowest order term in (19) repre- 
sents. The two last terms in (19) are the highest order terms. The 
last term represents an integrated effect from a given cross-section 
to the after end of the ship. This can be better understood from the 
mathematical expressions in Appendix B. 
III.2, The near-field problem and the matching. 
We are now going to formulate the near-field problem and 
perform the matching between the near-field and the far-field solu- 
tions, A one-term far-field solution is found to be due to a line of 
sources with source density 
i (wt - vx) 
a, (x) € 
spread along the line y=z=0, -L/2 < x <L/2 (see Figure 1), 
and a one-term near-field solution will be found to be the negative of 
the incident wave. The matching between the lowest-order term in 
the near-field problem and the lowest-order term in the far-field 
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