Falttnsen 
assumptions made earlier. However, this is not a serious difficulty 
if we do not try to use our results very near the bow of the ship. The 
near-field expansion of which (33) gives the first term, is not uni- 
formly valid near x = -L/2. In order to examine the solution preci- 
sely in the neighborhood of x = -L/2 we should construct a separate 
expansion for a region in which x + L/2 = 0( a ), for some Y > 0. 
One may expect then that o, (x) is not given in that region by (34) ; 
rather, 9, (x) will decrease continuously to zero at x = -L/2, as 
physical considerations require that it must. Using (34) to express 
0; (x) produces a higher-order, i.e., negligible, error in the velocity 
potential, provided that we restrict our attention to a region in which 
eY = o(x + L/2). 
We wish next to find ¥, , but first we need to say some more about 
the far-field. 
We expect that a two-term far-field expansion is obtained by a line 
distribution of sources of density 
i (wt - vx) 
e 
(0, (x) + 0, (x)) (35) 
spread alongthe line y=z2=0, -L/2 < x < L/2. it is assumed 
that 
) (36) 
A two-term inner expansion of this two-term far-field ex- 
pansion can be obtained from (19), and itis 
x 
i(wt - yx) “1/2 jfk JkZ -in/4 dio (5) 
‘= -€ = s] Vx - 2 
~ia/2 
1780 
