singular Perturbation Problems in Ship Hydrodynamics 



Yl + y, = - 2cyQ = 2e sin t, with Yi (0) = Y| (0) =0. 



It is impossible to obtain a steady- state particular solution of this 

 problem. In fact, the solution is: 



Yl (t;€) = €[ sin t - t cos t] . 



Thus, we obtain an expansion in which the second term grows linearly 

 with time. One might expect that succeeding terms will grow even 

 faster. This expansion is correct, and, for small values t, it could 

 be used for numerical predictions. But we would certainly prefer 

 to obtain an expansion which is uniformly valid, even for very large 

 t. 



The exact solution is easily found, of course. It is: 



Y(t;c) = e*** [ cos V(l-€^)t + ^ sin V(l-C^)t] . 



The approximate solution becomes worse and worse with increasing 

 t because the frequency is wrong and because the exponential factor 

 is expanded in a power series in t. If we watch the oscillating mass 

 on a time scale appropriate to the period of the oscillation, we do 

 not see the exponential decay and the slight shift of frequency caused 

 by the damping. On the other hand, if we watch for a very long time, 

 the effects of damping accumulate gradually. Thus , the effects of the 

 "slow-time" scale, 1/c , persist throughout the history of the motion 

 as observed on a real-time scale, but these effects never occur 

 suddenly. It is this fact which enables us to separate them out of 

 the real-time problem. 



There seems to be less reliable formalism available for 

 handling such problems than in the case of the method of matched 

 asymptotic expansions. More is left to the insight and ingenuity of 

 the individual problem solver. In the example discussed above, the 

 procedure is fairly clear: Expand y(t;c) in a series such as this: 



y(t;c) ~ yo(t,T;e) +yi(t,T;€) + ... , 



where we define: 



t = €t ; t = T + f,(T;e) + i^{r\€) + , . . , 



* 

 Strictly speaking, the series really is uniformly valid except at 



t = oo. 



677 



