singular* Perturbation Problems in Ship Hydrodynamics 



Y,(z) = 0(e); ^,(z), \,{z), V2(z) = O(e^); 



also, all other terms in (2-40) are o(c ). These statements can 

 all be proven. The description of the problem is greatly simplified, 

 however, by their being assumed now. 



We can write the three -term outer expansion now: 



<^x,y,z) ~ Ux + 4),(x,y,z) + <j>2(x,y,z), (2-44) 



where 



''-s *^o [(x-l) +y + (z-;) J 



The inner expansion of the one-term outer expansion Is, of course: 



4Kx,y,z)~Ux, [0(€)] (2-45a) 



to any number of terms. (Recall that x = 0(€) In the near field.) 

 The inner expansion of the two -term outer expansion is: 



4>(x,y,z) ~ Ux +ivi(z) [i-^ta^''j] tO(€)] 



(2-45b) 



Finally, the inner expansion of the three-term outer expansion Is: 



(Kx,y,z) ~ Ux +1 v,(z) [l - i tan-' X] + Y\^^-) ^ ^^i-) [ o(c)] 

 ^ ^ "" ""-^ 27r(x2+y2) 



(2-45c) 



703 



