The Wave Generated by a Fine Shtp Bow 
Zz 
log Cx + O(1/x ) as SS Me 
~ 
4UaH 
oa 
where C isa constant which cannot be determined from this analysis. 
From the two-part analysis above, we obtain our desired re- 
sult, the estimate of the potential on z=0 as x +o: 
H 
$(x, y, 0) ~Rel Sf df log yan - 
-H 
as xX — © , (14) 
4U aH 
2 
log Cx +O(1/x ) 
Thus, we see that the potential represents the source distribution al- 
ready discussed, in addition to which there is a term which becomes 
infinite logarithmically when x goes to infinity. These results will 
both appear in a proper perspective when we consider what the usual 
slender-body theory predicts near the bow. Both of the explicit terms 
above are O(e€‘loge ) in the bow near field. 
The appearance of the constant, C, in the above result is an 
unfortunate consequence of our use of generalized-function theory. In 
general, the value of the constant may even have to change as the for- 
mulas are manipulated. Froma strict mathematical point of view, it 
is quite improper to leave a final formula in sucha shape that it can 
be interpreted only in terms of generalized functions, especially when 
it is supposed to have direct physical significance. Fortunately, this 
is not so much of a problem for us here as might be supposed. The 
quantities with real physical significance are $, and § , and their 
estimates are not at all murky. 
V. THE USUAL SLENDER-BODY SOLUTION 
If one stretches coordinates near the body in such a way that 
and then treats derivatives with respect to the new variables as if they 
had no effect on orders of magnitude, one obtains the usual problem 
and solution of slender-body theory. Without going through the forma- 
lism of such changes of variables, we write down directly the boundary- 
value problem that results for the wedge-like body that we are consi- 
dering in this paper. The first approximation to the near-field pertur- 
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