The Wave Generated by a Ftne Shtp Bow 
analysis, while we also investigate possible modifications and check 
them against experiments, 
In the analysis as presented here, one may observe that the 
solution in the ''bow near field'' was never matched directly with the 
usual far-field solution of slender-ship theory. As Hirata [4] dis- 
covered, this is no small task. I have not yet carried out this match- 
ing, but Iassume that it would lead only to a modification of the far- 
field source density in the neighborhood of the ship bow. Presumably, 
the source density curve would be rounded over ina region of length 
O(€ V2 ) near the bow. Thus it would be possible to compute the wave 
resistance of this shape of ship. (We have no assurance that the value 
computed would be accurate, but it would be a big improvement over 
ordinary slender-ship theory, which would give an infinite value of 
wave resistance for the ship with wedgelike bow ! ) 
We have made only a fewcrude attempts to predict what 
happens just ahead or the edge of the wedge, and these attempts have 
not been described. Using a very heuristic mathematical mode, I 
concluded at one time that the rise in water level ahead of the bow 
should be independent of forward speed (for a given wedge angle), and 
it was this tentative conclusion that led us to examine our photographs 
carefully, after which we came to a conclusion that there must be 
some truth in the crude analysis, since the water rise is in fact quite 
insensitive to forward speed. 
The fact that the analysis is linearis, of course, a great 
help in obtaining a solution, but the most casual observation of the 
physical situation (as in Figures 3 and 4) suggests that linearisation 
may be a great over-simplification. In defense of the linearization in 
this analysis, I offer just two comments : 
(i) It always seems reasonable to try a linear analysis of any 
problem. One must in any case trust experimental evidence for the 
justification of an analysis. In the present problem, it is evident that 
the linear analysis is not grossly wrong. 
(ii) In many mathematical analyses of fluid mechanics problems, 
apparently unacceptable singular solutions often become very useful 
when they are properly interpreted. I have already mentioned the ap- 
pearance in slender-body theory of flow singularities which result 
from the invalidity of the assumptions in the regions near the body 
ends. Perhaps an even more interesting situation arises in some pro- 
blems in which we find that the solution to a linearized problem re- 
presents approximately the correct flow patterns - but in slightly 
wrong places. Our bow-flow solution, for example, is not so singular 
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