The Wave Generated by a Fine Ship Bow 
In order to obtain these limits, we manipulate the inverse 
transforms into forms so that the generalized-function procedures 
can again be used. For the vertical component of velocity, for exam- 
ple, we go through the following steps 
> (x, y, 0) bat fete™ a ( - HN cos VK |L|x 
L 
— -H 
ES / al cost y= 5 — 
cosVKE x 
0 
— 2 
|| 4Ue ar ‘ whe Pas /* 
a mae cos Ax cos K aa ce, Pee 
: 2 
co ; Z, -H> /K 
ON i 
oh 2a dite meats = 
Pe: K || 
—%0 
4UaH 
wore ey | as x —» co é CET) 
TwKX 
The interpretation of this result is of some’interest. The 
quantities a and H are each of order ¢. In addition, x =O(e'@ ) 
in the bow near field. Thus, %, =O(e) in the bow near field. Now, 
we have already commented that the solution in the bow near field 
must match the solution given by the usual slender-body theory. In 
fact, the near field of the usual slender-body theory is a far field 
with respect to the bow region ; x =O(1) in the usual theory. From 
this point of view, the expressions obtained above for $, represent 
a one-term inner expansion, and the final formula above is the one- 
term outer expansion of the one-term inner expansion. In matching 
it with the corresponding ''far field'', we must reinterpret the va- 
riables as far-field variables and re-order the expansion. In the pre- 
sent case, this means only that we revise our estimate by consider- 
ing x tobe O(1), in which case we observe that ¢, = O(€2) on the 
plane z=0 as x— ye, This agrees with the well-known result of the 
usual slender-body theory. We shall say more about this presently. 
What is most remarkable about the above result is the man- 
ner in which the flow completely changes its character in the down- 
stream direction. Very close to the bow, the flow appears to have 
been caused by a distribution of vertical dipoles, and so the flow at 
the plane z=0 is almost completely normal to the plane. However, 
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