The Wave Generated by a Fine Shtp Bow 
We assume everything that is necessary for the existence of 
a velocity potential, which we write in the following form 
Ux + (x, y, z) 
As usual, the potential satisfies the Laplace equation in the fluid do- 
main ; 
[L] Oat pts teenct oe adage 
[4/] [o/e*J[o/e 7] 
The expressions in square brackets give the orders of magnitude in 
the bow near field of the terms immediately above. Although we do 
not yet know the order of magnitude of ¢ , it is already clear that we 
can ignore the term ?¢,. in finding the first approximation to the so- 
lution in the bow near field. 
The boundary condition on the hull can be written 
D's ea < y ZZ 
[e ] [¢e Ve] [?/e] [¢/¢] 
Dropping the one term which is clearly of negligible order of magni- 
tude, we can rewrite this condition 
+ 
Sn = oer Ub 
Pete te Ong ae esein Ed oe = 2 “Y= )Gole) 
Vl + be Ls: be 
Since the operator 0/dn is similar to, say, 3/dr with respect to 
its effect on orders of magnitudes, we can now conclude that either 
¢=O(e*) or the first approximation to ¢ satisfies a homogeneous 
boundary condition on the hull. Let us suppose that the former is true. 
If this is wrong, we shall discover that fact when we consider the other 
conditions on ¢ 
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