singular Perturbation Problems in Ship Hydro dynamics 



<^Jx.y.z) = -^ y J dk e"'''Kn{|k|r)[al(k) cos n0 + b*„(k) sin n0] . 



n=0 -^ 



(2-59) 



where amn{k) and bmn(k) are unknown functions. The most general 

 far-field expansion comprises the incident-flow potential, Ux, and 

 a sum of terms like the above, that is, 



M 



<Kx,y,z) ~ Ux + y <^n,(x,y,z) for fixed (x,y,z) as e -^ 0. 



'"=' (2-60) 



It will be necessary to have the inner expansion of the outer 

 expansion. This means that we must interpret r to be 0(e) in 

 the above expressions, instead of 0(1) as heretofore, and re- 

 arrange terms according to their dependence on e. The easiest 

 procedure is to replace any of the Kp functions in (2-59) by its 

 series expansion for small argument. We obtain formulas such as 

 the following: 



n = 0: 

 ^y"dke'^^K^(|k|r)a*,(k) 



log r r ji "« * /I X 1 r*" ji '•« * /u\ 1 „ c|ki . 



(2-6la) 



n > 0: 



^r dke*'*K(|kir)a*(k) (T)-^ 

 Sir J n ' ' ' mn' ' Vsm/ 



n-l 



00 



Physically, the n = integral represents the potential for a 

 line of sources. This can be seen directly from (2-6la): As r -^ , 

 the function is proportional to log r, which is the potential function 

 for a source in two dimensions. However, the strength of the 

 apparent 2-D source is a function of x. In fact, the integral defin- 

 ing that strength is identical to the integral which gives the inverse 



L (x) be the function having a^^^Jt 



of a Fourier transform. Let a he) be the function having a Jk) 



715 



