Ogilvie 



so in the infinite -fluid case was that one could avoid possible ques- 

 tions about the validity of analytically continuing the potential 

 function Inside the body surface. On the other hand, one had then to 

 face all kinds of difficulties in principle in justifying use of matched 

 asymptotic expansions. It was a rather academic exercise. 



The situation may be quite different in the thin-ship problem. 

 The purpose of this chapter is to show one can obtain the first 

 results of thin- ship theory in the same way as for the infinite -fluid 

 problem but that a second-order solution leads to fundamental 

 difficulty. The latter appears to suggest that a combination thin- 

 body /slender-body approach miay be appropriate. A limited amount 

 of other evidence may be cited to support this idea. 



I wish to emphasize that there are no new results in this 

 chapter. It is all a mater of interpretation . Perhaps someone will 

 be able to show that the problem discussed here has a trivial expla- 

 nation. On the other hand, perhaps someone will be stimulated to 

 do further research on the subject. In either case, I shall be happy 

 with the outcome. 



The problem may be partiailly stated just as the infinite- 

 fluid, thin-body problem was stated. Let there be a velocity potential, 

 <|>(x,y,z), which satisfies the Laplace equation, 



[ L] 4>xx "•" ^yy + 4>zz = » 



everywhere in the fluid domain and the body boundary condition, 

 [ H] = ^xW =F <j>y + ^zhz» o"^ y = ± h(x,z) = ± eH(x,z). 



Now we add on the two free- surface conditions: 



[A] ^U^= g; +-^[<l>x + 4>5+ <^S . on z = ;(x.y); 



[B] = ({Jx^x"*" <l>y^y " ^i' °^ ^ = ^(x,y). 



Also, we must specify a radiation condition. 



In the far field, where y = 0(1), we assume the existence 

 of the expansions: 



N 



<(»(x,y,z)~ 2^ (t>„(x,y,z), 

 n=0 



^(x,y)~ ^ ^n(x,y). 



n=0 \ for fixed (x,y,z); 



N 



n;l 



770 



