singular Perturbation Problems in Ship Hydrodynamiae 



frequently drop the explicit mention of the c dependence. 



As € ~^ , the body shrinks down to a sheet of zero thickness 

 aligned with the incident flow. Thus, the first term in an asymptotic 

 expansion of the velocity potential in the far field is just the incident- 

 stream potential. In general, let the far-field expansion be expressed 

 as follows: 



N 

 <|»{x,y,z;c) ~ ) <J>j,(x,y,z;c), where <j>^^ = o(4> ) as 



n=0 



€ — for fixed (x,y,z). (2-3) 



Then we have: 



4> (x,y,z;€) = Ux, (2-4) 



The far field is the entire space except the y = plane. 

 Since the potential <^(x,y,z;€) satisfies the Laplace equation through- 

 out the fluid domain, the individual terms in the above expansion 

 satisfy the Laplace equation in the far field: 



Kx^Sy'^*"zz=^ for |y|>0, (2-5) 



At infinity, we expect (on physical grounds) that: 



V(<^ - Ux) ^ 0. (2-6) 



Therefore, for n > 0, every <^n must be singular on the y = 

 plane or be a constant throughout space. The latter would be too 

 trivial a result to consider, and so we assume that <^n ^s indeed 

 singular on the y = plane. 



But what kind of singularities will be needed? Because of 

 the symmetry of the problem, it is not difficult to show that a sheet 

 of sources will suffice. One can use Green's theorem to show this. 

 Alternatively, one can use transform methods for solving the Laplace 

 equation, which is practically equivalent to solving by separation of 

 variables. Whatever method is used, the result is the same; 

 6 (x,y,z;€) has a representation: 



>-»oo .-»oo 



4>„(x,y,z;€) = - 4^J 3 " . 2 ^,/^ > (2-7) 



V.00 ^'-OD [ (x-l) + y + (z-C) J 



where (r„(x,zj€) is an unknown source-density function. The outer 



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