Ogtlvte 
as x-—oco , we find that the normal component of velocity on the plane 
z= 0 vanishes and the flow becomes parallel to the plane. 
We also examine how the wave elevation varies asymptotical- 
ly in the downstream direction. We proceed as with ¢, : We write 
as the inverse transform of the expression in (10) and then mani- 
pulate it so that it appears formally to be a transform with respect 
to x . We obtain in this way 
ras 
ad = 
2a iAx ae l-e ale 
(Gary ie f= = dre sgn cos K ea eee 
co 
2aH a id oie 
a idx 
—_ dd -—— phos ae 
nik e sgnd E aK 
DO 
4aH 5 
+ O(1 = 
pire O(l/x) as x70 : (12) 
It is worth noting that the y dependence enters only in the term which 
drops off inversely with x>. We also observe that [ =O(e 3/2 ) in the 
bow near field, where we assume that x =O(e/2 ) , but when we re- 
interpret x as being O(1) we must conclude that ¢ =O(e?), This is 
in agreement with the well-known results of the usual slender-body 
theory. 
Finally, we obtain an estimate for (x, y,0) as x40 ., The 
transform of this quantity was given in Equation (7). It is clear that 
we cannot follow exactly the same procedure as we did for estimating 
>, of §, since there is a part of the expression in (7) which does 
not even depend on x . However, we can proceed in two steps: 
a) First we consider the part of the transform in (7) that 
does not depend on x, namely, the quantity 
We shall find that this is the transform of 
H 
Rel Ue i df log (y + iz - oH (13) 
-H ze=sQ 
1500 
