The Wave Generated by a Fine Shtp Bow 
may well be the cause of the increasing error at high Froude number. 
If Pay is below some moderate value, it can be seen from 
Figure 7 that our method of nondimensionalizing the data seems to be 
still valid even when the Froude number drops below the level at 
which the analysis is valid. The reason for this is not clear, but the 
fact may be useful in reducing experimental data, even in cases in 
which the present analysis is obviously invalid, 
The inaccuracy of the wave height predictions at high Froude 
numbers can probably be ameliorated if not completely removed by 
the introduction of a more precise method of solution of the problem. 
In principle, it appears to be possible to solve the bow-flow problem 
without introducing the thinness assumption, and some efforts have 
already been made to do just this. At the moment, however, we have 
no results to show for this effort. 
VI. CRITIQUE OF THE ANALYSIS 
Intuitively, we visualize a ''slender body" as a body of which 
the length is much greater that the transverse dimensions. In addi- 
tion, if we want to be a bit more precise, we require that there be no 
sudden changes in cross-section size or shape. 
For such bodies, slender-body theory is likely to lead to 
reasonable predictions concerning a fluid flow around the body - 
provided we do not examine too closely what is happening near the ends 
of the body. The last qualification is necessary because slender-body 
theory is based on one major assumption which is usually violated 
near the body ends : It is assumed that the rates of change of all flow 
variables are much greater in the transverse directions than in the 
longitudinal direction. For a body with cusped ends, this assumption 
is valid even in the region near the ends, but the assumption is not 
valid near the body ends for most bodies of practical interest. The 
result is that slender-body theory typically predicts some kind of sin- 
gular flow near the body ends. 
Such a result is not.necessarily unacceptable. If the singu- 
larities are integrable in some appropriate fashion and if the solution 
is approximately correct in most of the flow region, the presence of 
singularities in the mathematical solution may not even be serious. 
If one is very careful in obtaining the singularity strengths, one can 
even make some reasonable calculations concerning the flow around a 
blunt body in aninfinite fluid. At cross-sections not too near the ends, 
the presence of singularities in the solution for the body end regions 
manifests itself as a perturbation of the longitudinal velocity compo- 
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