Singular Perturbation Problems in Ship Hydrodynamics 



T53 = - p(iw)^ \ dS n <j> - p(ia)U) \ dS (n3<j>3 - n,<j),); 



Tgg = - pdo))^ y dS n5it»5+ U^y dS (n3<^3 - n,<J),). 



These results have been obtained with no assumptions made 

 about the shape of the body. The only assumption was that the sinu- 

 soidal oscillations had very small amplitude. 



Now, finally, let us assume that the body is slender. The 

 only effect is that we lose the terms containing n|4>,. For a slender 

 body, n^ and n^ are 0(1) as the slenderness parameter, c, 

 approaches zero, whereas n| is 0(e). From (2-73), we see that 

 <^. is therefore higher order than ^3 and 1^^ by a factor of e. 

 Thus: 



n dS n,<|>,/j dS n^<^A = 0(€^). 



Seldom in practical problems do we ever retain terms with such a 

 great difference in orders of magnitude, and so we neglect the terms 

 containing n^^^ if the body is slender. 



T33 will be 



In the ship-motion problem, the quantity corresponding to 



where ajj and bj, are the heave added-mass and damping coeffi- 

 cients , respectively. The other Tjj 's have a similar interpreta- 

 tion in terms of pitch added-moment-of-inertia and damping coef- 

 ficients , cross -coupling coefficients, etc. We note that there are 

 three kinds of terms here: 



a) Terms Independent of U. These are all of the same form: 



(0) 2 r 



Tji = - (icu)^ \ dSnj^j. (2-78) 



b) Terms proportional to U. These occur only in the cross 

 terms, Tji , with i^ j. For a slender body, we have: 



In the ship-motion problem, (j)j is complex. Here, of course, 

 ^1 is purely resil, and so there is no analog to bj». 



729 



