Og-ilvie 



the limit case. Thus, we shall be able to apply the results of 

 slender-body theory to bodies which are not especially slender. In 

 such cases, we may expect that the predictions will be less accurate 

 than the predictions that we would make for a much more slender 

 body. But we never know a priori how slender the body must be for 

 a certain accuracy to be realized, and it would be wrong to assert 

 that the theory applies only to needle-like bodies. All that we can 

 say is that it would be more accurate for such bodies than for not- 

 so-slender bodies. 



1,3. Multiple- Scale Expansions 



In the problems of the previous section, we had two greatly 

 contrasting scales for the independent variable. The fact that enabled 

 us to obtain two separate expansions was that each of the scales 

 dominated the behavior of the solution in a particular region of space 

 or a particular period of time. The major practical concern was to 

 ensure that the separate expansions matched, because they really 

 represented just different aspects of the same solution. 



The present section is devoted to problems in which there 

 are again two greatly contrasting scales. However, in these prob- 

 lems, it will not be possible to isolate the effects of each scale 

 into a more or less distinct region of space or time. The effects 

 of the two scales mingle together comipletely. However, we may 

 still expect to be able to identify these effects somehow, just because 

 the two scales are so different. 



There are classical problems of this kind, the most famous 

 being related to nonlinear effects on certain periodic phenomena. 

 Cole [ 1968] discusses a number of these problems. Perhaps the 

 simplest example of all is a linear one: Find approximate solutions 

 for small 6 in the problem of a linear oscillator with very small 

 damping, where the differential equation might be written: 



y + 2ey + y = 0, 



To be specific, let the solution satisfy the initial conditions: y(0) = 1 

 and y(0) = 0. Physically, we expect that the system will oscillate 

 with gradually decreasing amplitude. It would be desirable if the 

 approximate solution at least did not contradict this expectation. 



We might try representing y(t;c) by an asymptotic expansion 

 with respect to c: y(t;€) ~ 2j yn(t;e'). We would find immediately 

 that the first term in this expansion is just: yo(t;€) = cos t. This 

 seems quite reasonable, since it represents a steady oscillation at 

 the frequency approximate to the undamped oscillator. The second 

 term in the expansions would be obtained from: 



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