Non-Linear Shtp Wave Theory 
0 0 
ava ERIN Side haw TYE, (55) 
It is easy to show that 
1 1 
(Pio on High r(s) 1 rt (s) 
Sada a Tae caviiviGe Pein eee OSS FEES 
-1 = 
(56) represents the rigid wall solution for the flow past the linearized 
body. It is easy to ascertain that we (f)s= wy’ (f). 
ds (56) 
We have arrived to what we call ''the first small Froude 
number paradox" : the thin body limit (¢-0) of the naive small 
Froude number solution w, (56) is not equal to the small Froude 
number limit of the thin body expansion w; (47, 48). The two solu- 
tions differe in the ''wavy wake" of fig. 6. 
III .5 - Discussion of Results. 
In the preceding sections we have defined the two small 
Froude number paradoxes occurring as F—0O, in the solution of the 
problem which depends on the two small parameters « and F. The 
nonuniform behavior of the solution may be related to the fact that 
in carrying out the naive small Froude number expansion (36) we have 
lowered the order of the boundary condition (32), the derivative dis- 
appearing in the l1.h.s. of (51) and (53), similarly to well known 
boundary layer problems. This observation is strengthened by the 
inspection of the wavy term (49) : the function ett changes its 
order by differentiation and the two terms of (38) become of the same 
order of magnitude no matter how smallis F (this observation has 
underlain Ogilvie's (1968) study). The nonuniformity present in our 
problem is, however, different and more subtle than that of other 
singular expansions (Van Dyke, 1964; Cole, 1968) with a few respects 
(i) we cannot detect the nonuniformity of the solution from the naive 
small Froude number expansion which is well behaved in the entire 
flow domain. We cannot, therefore, rule out this solution at the pre- 
sent stage; (ii) for a submerged body the ''wavy wake" is attached to 
the fictitious image of the body (fig. 6) rather than the body itself. 
It intersects the flow domain <0 only far behind the body and has 
an exponentially small effect upon the body itself; (iii) the wavy term 
eH f/F2 sey e 1¢/Fe has, for F—0, the character of an expo- 
nential boundary layer decay for y <0 anda rapidly oscillating beha- 
vior inthe @¢ direction. It displays, therefore, a complex pathologi- 
cal behavior as F-—0. 
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