The Wave Generated by a Fine Shtp Bow 
we accept slender-ship theory, we have already recognized that the 
overall Froude number does not characterize the ratio of inertial and 
gravitational forces uniquely throughout the fluid region, 
This idea was also discussed in the earlier work [1] already 
mentioned. There I pointed out that special order-of-magnitude con- 
sideration should be given to conditions near the ship bow. Because 
of the presence of the free surface, the fluid particles just a very 
short distance ahead of the bow are quite unaffected by the oncoming 
ship, until -suddenly ! - those particles are in the region of highly 
accelerated flow around the ship bow. The effects of water displace- 
ment by the moving ship are much greater than the effects of gravity, 
which normally hold the water surface horizontal, and so the presence 
of the free surface is momentarily simply equivalent to a pressure- 
relief surface. All of this can be implied by saying that the flow near 
the ship bow is a high-Froude-number flow. 
Thus we come to the concept that the bow flow is a high- 
Froude-number problem, even if the ship speed is moderate. The 
previous argument then suggests that we try to relate the Froude- 
number aspect of the bow flow to the slenderness parameter. In this 
paper, I have done this in a very pragmatic way : 
In the usual slender-body theory, we assume, in a symbolic 
notation, that d/dx = Q{(1) but that d/dy and 0d/dz = O1/e), where 
x is the longitudinal coordinate, This means that rates of change in 
the longitudinal direction are smaller than rates of change in the trans- 
verse direction by an order of magnitude e . (It is this very gradual 
variation in the longitudinal direction that leads to the typical feature 
of the slender-ship near field, namely, that the free surface acts as 
a rigid wall. Rates of change are so gradual that gravity dominates 
and holds the free surface horizontal.) This intuitive picture is for- 
malized in the mathematics by stretching coordinates in the transverse 
directions by a factor 1/e 
Now we suppose that, near the bow, rates of change of the 
flow variables should be greater than those usually assumed in slender- 
body theory. We may expect to introduce such a notion formally by 
stretching the x coordinate from the bow sternward, But what should 
be the degree of stretching ? Let us define a new longitudinal coor- 
dinate, X=x/e™, with x and X both measured from the bow in 
the downstream direction. If n= 0, we have the usual slender-body 
theory, and if n= 1 we have the original problem in three dimensions, 
(In the latter case the stretching is isotropic.) Therefore we seek a 
value of n such that 0<n<l. It turns out that a nontrivial problem 
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