Ogilvie 



In the far field, the planing surface appears to vanish in the 

 limit, and so the first term in a far-field expansion must represent 

 just the incident uniform stream. That is, if the outer expansion is 

 represented: 



N N 



F(z;P) ~ 2^ Fn(z;P), w(z;P) ~ )^ Wn(z;P), for fixed Pz 



n=0 n=0 as p -* 0, 



then clearly we have: 



Fo{z;P) = Uz, and Wo(z;p) = U. 



This one-term outer expansion must match the one-term inner ex- 

 pansion, the latter being just Green's solution. This much of the 

 matching procedure is rather obvious, and Green already used this 

 fact to determine the value of c , as given in (5-2). 



The next term in the outer expansion is not quite so obvious. 

 In order to facilitate the matching process, Rispin solved the problem 

 in the t, plane, just as we did above for Green's problem. The 

 free-surface boundary condition on W| is not much different from 

 the familiar linearized condition. One can show fairly simply that: 



Re[4W,,i&|w,]=0 on ,-0. 



where A - a/Tr(b + c) . (The factor A is just the value of dz/d^ 

 far away from the planing surface.) Note that the first term is 

 0(pW|) because of the differentiation, and the second term is the 

 same order because of the g factor. The solution for W| must 

 be analytic in the lower half- space and satisfy this condition on T) = 0, 

 1^1 > 0; note the exclusion of the origin, where singularities may 

 occur. 



As usual, we try to restrict the singularities to the simplest 

 kind possible. In this case, we would find nothing in the near field 

 to naatch with if we allowed all kinds of singularities in W| . A 

 sufficiently general solution is the following: 



W,(C;P) =ie ' '' \ dte ^ 



where C| and C2 are real constants yet to be determiined (in the 

 matching) , 



Rispin discusses more general solutions, which are needed in con- 

 structing higher-order solutions. 



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