singular Perturbation Problems in Ship Hydrodynamics 



solution of the $| problem discussed above. In fact, one finds 

 explicitly that: 



Ai W = J -rn- ® ^2, (k) ; B,, (x) = J -j^ e bg, (k) , 



-00 



Thus, the two-term inner expansion contains enough information to 

 determine the strength of the dipoles which appear in the third term 

 of the far-field expansion. The same inner expansion would deter- 

 mine the strengths of quadripoles in the fourth term of the far-field 

 expansion, etc. , etc. 



On the other hand, the far -field expansion (even at the second 

 term) contains much information about three-dimensional effects, 

 information which is largely lacking in the near-field expansion. I 

 have already pointed out that only the "constant" term contains im- 

 portant information about 3-D effects in the two-term near-field 

 expansion. The rest of the <J>| solution depends on just the shape 

 of the local section and the local rate of change of section shape and 

 size. If higher-order near-field terms are found, it will be seen 

 that they are influenced even by the two-term outer expansion. In 

 fact, the "constant" term in $| can be interpreted as a modification 

 to the incident stream, caused by the presence of all the other cross 

 sections of the body. The effects of this extra incident flow on the 

 transverse velocity field are not perceived until one finds a higher 

 order expansion of the solution in the near field. 



The briefest account of slender-body theory would be seriously 

 lacking without mention of the possibly catastrophic effects of body 

 ends. If a body has a blunt end, then s(x) increases linearly in 

 some neighborhood of the end. Accordingly, s'(x) is discontinuous, 

 jumping from a value of zero just beyond the end to a finite value at 

 the end. This is an obvious violation of our assumptions about 

 "slenderness. " But trouble develops even without a blunt-ended 

 body. For example, if the tip is pointed (but not cusped) , there will 

 still be a stagnation point right at the point. Thus this case violates 

 the assumption that longitudinal perturbation of the incident flow 

 velocity is a second- order quantity. 



Sometimes these end effects can be overlooked with impunity. 

 There are major examples later in this paper. However, even when 

 we have such luck, we must be prepared to have higher-order expan- 

 sions go awry, 



2.32. Small- Amplitude Oscillations at Forward Speed . In 

 this section, we consider the same kind of body as in Section 2.31, 

 namely, a slender body which is aligned appr ox innately with an 

 incident stream. However, now we formulate a time- dependent 

 problem in which the body performs small- amplitude oscillations 

 while it moves through the fluid. 



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