Eggers 



The difficulties with the term 1 2 made me resign and decide on approach 

 (c), for which I was certain that the sum of ij and the integral of <p^^^b{X,Y) must 

 be independent of x^ by Gauss' theorem. 



Following now Prof, Wehausen's suggestion, using conditions A and B, I can 

 finally prove that 



( 1 ) d 



,(!)' ,(1)/ fl) 



^x ^xz/^0 - ^'x 



_d_ 

 dY 



,(1) ,(1) 



The second term vanishes under Y integration due to condition C, if we may dis- 

 regard a contribution from the ship's load waterline. Thus definitely approach 

 (b) gives one and the same result for any value of x^ behind the ship, as it 

 should, and our result, Eq. (49), can be obtained by approach (b) as well. 



A critical review of our analysis showed that this equivalence even holds if 

 we do not disregard the influence of the waterline. We had, however, to acknowl- 

 edge that extending the integration over 5^!,^^ in Eq. (19) to the waterplane area 

 meant a change of order e^, which, after a correction to be made in Eq. (19), 

 again made all line integrals cancel in the final expression for R^^\ Eq. (27). 



For the case of infinite tank width, it may be argued that 1 2 tends to zero 

 as Xg-»-oo, though for the contribution of the cusp region this is not necessarily 

 evident. But even in this case it is only the addition of the expression for 1 2 to 

 1 1 which enables us to introduce the reversed potential into the integrand and 

 thus obtain the convenient expression (Eq. (49)) amenable to numerical evalua- 

 tion. 



REPLY TO THE DISCUSSION BY YIM 



There is at least an equal amount of respect on my side for Dr. Yim's 

 courage to attack the problem for the domain bounded by the first-order free 

 surface, where the analysis is definitely less elegant to perform. I sympathize 

 with his idea of using coordinates adjusted to the wave profile, and I wonder if 

 to first order this is just equivalent to distorting the ship's geometry as was 

 done by Hanaoka. 



We can hardly expect one-to-one correspondence between our components 

 of potential and resistance; however, if Dr. Yim finds one of his line integral 

 terms missing in my analysis, perhaps a hint may help: A recent study of 

 Bessho reveals that if only the boundary condition on the body surface below 

 Z = is prescribed, different second-order potentials may exist for a floating 

 body of finite beam. Within his scheme, my potential appears as the limiting 

 case of that due to a flat-topped body, and so does Sisov's. 



For a wall-sided ship, the second summand of my double integral term in 

 Eq. (13) could be transformed to line integrals representing contributions from 

 the bottom and from the load waterline. But I am not convinced that the differ- 

 ence of these line integrals should dominate against the remaining double inte- 

 gral over the vertical surface, even if the ship has shallow draft. 



678 



