singular Perturbation Problems in Ship Hydrodynamics 



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 X02 



FsO.71 FO.40 



£--0 30 £=0 30 rF=U/VloWl 



N I [£=t/b 



x-r" 



gt/U^ = 



N,/(H,iH2^9) 



Hj/fHi+Hz+Ws) 

 Hj/tHi+H^+Hj) 



From Solv*>«i (1969) 



Fig. (5-8). First-, Second-, and Third- 

 Order Wave Heights at Low 

 Speeds . 



(t is the thickness of the foil, as in the last section.) Then the 

 figure shows the three ratios , Hn/(H| + H2 + H3) , for n = 1,2,3; 

 that is, each curve shows the relative contribution to the wave 

 height of one of the first three terms in the wave-height expansion. 

 As speed decreases (toward the right-hand side of the figure), the 

 second-order part comes to dominate the linear-theory part, and 

 then the third-order part dominates the first two. It seems quite 

 likely that the fourth-order term would take over if the graph were 

 extended, then the fifth-, sixth-, .. . order terms. 



Salvesen's analysis is based on the condition that t (or, 

 more properly, t/b, where b is the body depth) is very small; the 

 Froude number is simply a parameter unrelat ed t o t, which is 

 equivalent to saying that Froude number = U/V(gb) is 0(1) as 

 t/b -♦ 0. Perhaps it is not surprising if Salvesen's expansion is not 

 uniformly valid with respect to Froude number. That is all that 

 Fig. (5-8) really says. 



The reason for its nonuniformity has already been mentioned: 

 In the expansion of the solution near the free surface, it has been 

 assumed that the lowest-order approximation is just the uniform- 

 stream term, Ux; all other terms in the expansion of the potential 

 must be very small compared to this term. And this is nonsense if 

 we consider the limit process U -* 0, Of course, we might have 

 been luck/: It could have turned out that the velocity perturbation 

 approached zero more rapidly than U. But it does not. And so we 

 have here a genuine singular perturbation problem. 



Let us consider a sequence of steady-motion experiments, 

 each lasting for an infinite length of time. We arrange the sequence 

 of experiments according to decreasing values of body speed, U, 



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