Ogitvie 



'^xx '^z g X 'xz'x 2U *■ 'z * 



. U/,2. 1 ,1 g 1. ^ 



Z U 'x 'zz U 'xx 'x ^ U ^ ^y 



(4-7) 



2 



In condition [ A] , we note that differentiation of the e terms with 



respect to y yields: 



= gZ2y ± u\,. 



Therefore, in the complicated free-surface condition above, (4-7), 

 only the first two terms involve y; all of the other terms are func- 

 tions of just X. From (4-7) and (4-6), we can thus write the follow- 

 ing: 



= [h^^^(x,0) + K\^{x,0)]\j\y\ + (a function of x) . 



This must be true for any y, and so we obtain the condition: 



= hxxx + Kh.^2> on z = 0, 



If the ship is wall-sided at z = 0, the second term is separately 

 zero, and so we would have to require that h.^^^ =0 at z = 0. 



Now this is clearly unacceptable. Why should our theory 

 work only for such a special case? (The waterline is made of circu- 

 lar arcs in this case,) 



As a result of our having stretched the coordinates , we came 

 to the prediction that the fluid velocity near the thin body consists 

 of a tangential component which is essentially independent of the 

 local conditions plus a normal component which depends only on 

 local conditions. Near the free surface, such results are simply 

 untenable. 



I present here a formalism which apparently avoids this 

 difficulty. Again, I point out that no new results are obtained. 

 However, it does seem possible that the procedure might be fruitful 

 if studied further. 



The idea is to define a third region, complete with its own 

 asymptotic expansions of ^ and ^, This region will be essentially 

 the same as the near field in a slender-body analysis, that is, it is 

 a region in which y = 0(e) and z = 0(e) as € — ^ 0. It follows from 

 this assumption that 3/9y and d/dz both have the effect of changing 

 orders of magnitude by a factor 1 /e , What is most important is 

 that this region is interposed between the thin-body near field and 



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