singular Perturbation Problems in Ship Hydrodynamics 



N 



|y(t;e) - 2, Yn(T;e) | = o(Y,^) as e — for fixed t. 

 n=l 



In the latter, we evaluate the difference on the left-hand side for 

 smaller and smaller values of t (= er) as € ""*" 0; in other words 

 we restrict the range of t more and more as € —*' . This is in 

 contrast to the interpretation of the outer expansion, in which we 

 simply fix t at any value while we let e — ^ . In even more physi- 

 cal terms, we may say that the inner expansion describes the solu- 

 tion during the time when the e^' term is varying rapidly, and the 

 outer expansion describes the solution when the e^' term has 

 effectively reached zero and the e'^^ term is varying significantly. 

 This separation into two distinct regimes is characteristic of prob- 

 lems in which we apply the method of matched asymptotic expan- 

 sions. Of course, the real key to the success of the method is in 

 the procedure by which the two aspects of the solution are matched 

 to each other. After all, they do represent just two aspects of the 

 same solution. 



Usually, we insist that our asymptotic expansions be 

 consistent. A precise definition of this term is awkward, but per- 

 haps it is clear if we state that each term in such a series depends 

 on e in a simple way that cannot be broken down into simpler terms 

 of different orders of magnitude. For example, the following two 

 series are equal: 



,rl .12,13, -I 



, r 1 2 , 1 3 , T 



On the right-hand side, let: 



1 12 12 



fo(€) = 1 + I c + i 6 + ^ e' + . . . ; 



Then we can write: 



673 



