singular Perturbation Problems in Ship Hydrodynamics 



iS rtOO 



N 



+ X. y r t^n(U dC 



N 

 + ^ y r ^il-dC +_, , (2-40) 



The first sum contains terms which are exactly of the form given 

 in (2-34), that is, they represent a lifting line with a strength 

 y Yn(z). The second and third sums represent lines of dipoles 

 oriented vertically and longitudinally, respectively. It is implied 

 above that the sums are asymptotic expansions, in our usual far- 

 field sense. 



We shall presently require the inner expansions of these 

 terms. We obtain the inner expansions by assuming that 

 r = (x^ + y^)''^ = 0(€), which implies that both x and y are small. 



Inner expansion of the lifting-line potential: Each of the 

 double integrals containing a y can be rewritten as a single integral: 



(2-41) 



Now break this into two parts: 



1) The first term in brackets on the right-hand side does not 

 depend on x. As y -♦ (i.e. , for y = 0(e) ), its contribution can 

 be represented: 



where the double sign is chosen according to whether y > or 

 y < 0, respectively, and the special integral sign indicates that the 

 Cauchy principal value is intended. This representation is valid only 

 for |z I < S, but that is no restriction here. It may be noted that 



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