singular Perturbation Problems in Ship Hydrodynamics 

 the body boundary condition is now: 



[H] $1^ +^2^ +CE>3^ +^4y + ... 



~e'[UG^ +(^i,G, +$,^G,) + ($2^G, ^ %G^) +...] 



on Y = G(x,z). (2-24) 



The solution for <b\ is generally an expression linear in Y, but, 

 for the same reasons as in the symmetrical-body problem, only the 

 "constant" term can ultimately be matched to the far-field solution, 

 and so we take for ^| : 



$,(x,Y,z;e) = A|(x,z;€) f 0(e). 



The superscript ± has been attached to the solution to indicate that 

 this quantity may be different on the two sides of the body. This was 

 not necessary in the previous problem, because of the symmetry, but 

 in the present near-field problem the body completely isolates the 

 fluid on its two sides and there is no reason to assume that A| is 

 the same on both sides of the body. (It turns out, in fact, that 

 Ai" = - A| .) 



One next obtains: 



^2(x,Y,z;€) = A2(x,z;e) + B^{yi,z',^) Y . 

 From the body boundary condition, the following is true: 



<|)2y(x,G,z;€) = B2(x,z;e) = €^UG^(x,z) . (2-25) 



Thus , we find that 



B2(x,z;e) = B2(x,z;e) = B2(x,z;e). 

 Similarly, one can proceed: 



^3(x,Y,z;e) = A3(x,z;6) + B3(x,z;c)Y - -| e V(Af + Af ), 



where 



± 2 



691 



Bt(x,z;e) = e^ (GA^ ), + (GAf, ) J . 



