singular Perturbation Problems in Ship Hydrodynamics 



. £ = U- v2 + [ (iu;)^^3 + (lc.U)(i|;3 + V . V <h)] ^3(t) 



+ [ (ia))^<^5 + (icoU)(iij5 + V . V ^5) - U^(qj3 + V . V<!»3)] ^5(t). 



In the terms containing |j , one can use primed and unprimed co- 

 ordinates interchangeably, since the difference leads to terms of 

 higher order. 



The force (moment) corresponding to the j-th mode of 

 oscillation is given by: 



Fj(t) = j dS njp(x,y,z,t) = 2, ^jj 4) (t) + Fjq , 

 ^ i 



where S is the surface of the body at any instant and Fjq Is the 

 steady force component. (For j = 1,2,3, the latter is zero.) The 

 "transfer functions" Tji are: 



T33 = - p y dS nj (ia))%3 + (icoU)(qj3 + ^-V ^^]) 



T = - p \ dS n [ (iw)^4>5 "^ <i'*»U)(i|j5 + V . V (^g) - 1/(^3 + V . V 



T53 = - p y dS n^ (iu))%3 + (icoU)(^3 + V . V <j»3)] ; 



T55 = - p r dS ng[ (ioo)%5 + (icoU)(4^5 + V . V <t>5) - U2(4i3 + V . V 4.3)] . 



s 



These formulas can be simplified considerably, even before 

 we introduce the slenderness approximation. We use two theorems: 

 One is an extension of Stokes* theorem, proven by Tuck (see 

 Ogilvie and Tuck [ 1969] ): 



\ dS nj( V V4>i) = - \ dS mj<^i . 



The other theorem is Green's theorem; in applying it, we note that 

 all of the functions decrease sufficiently rapidly far away that there 

 is no need to account for effects at infinity. Thus, in T33 and 

 Tjg we have: 



(^1 and ^3 appear to represent dipoles at infinity; thus, both are 

 proportional to l/r^ as r -* co. <^5 appears to represent a quad- 

 ripole, thus is proportional to l/r' at infinity. 



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