singular Perturbation Problems in Ship Hydrodynamios 



**, 



-**/, V (Tn (k,m;€) [4 ,^ 2„2,, 2 , 2. 



2(k +m ) 



. ^ eV^k^ + m'f > . . .] 

 + i <r**(k,m;c) [c |Y| +iy e'|y|'(k^ + m') 



4) Note that: 



<r 



n (k,m;€) _ **,^.^_.^, = „** 



0/1 2 ■ 2 J/2 

 2(k + m ) 



= <j)„ {k;0;m;c) = a„ (k,m;c) . (2-10) 



Also, we observe that, if f (k,m) is the Fourier transform of 

 f(x,z), then (k^ + m^)f**(k,m) is the Fourier transform of 

 - (fj^ + f^z). Defining the inverse transform of a**(k,m;€): 



Q'^(x,z;e) = (J)„(x,0,z;e), (2-11) 



and inverting the above series term-by-term, we obtain: 



1 1 2 2 



<|>^(x,y,z;c) ~ a^(x,z;€) +2 e|Y|a-„(x,z;e) - yje |y| (°'n^^ + ^^ ) 



-TVre^|Y|^(o-n +o-n ) 

 2* 3 1 ' ' "xx "zz 



1 '♦i 



xxzz zzzz 



(2-12) 



^4TnY|[-n_^2-n_^-n,„J ^.'^ • 



This is the inner expansion of a typical term in the outer expansion. 



In order to combine the expansions of the separate terms into 

 a single inner expansion of the outer expansion, let us assume that 

 (r„ and «„ are both 0(e ). (It is not necessary to assume this, it is 

 nnerely convenient.) Then we have for the desired expansion: 



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