The Wave Generated by a Fine Shtp Bow 
where s(x) is the area of the immersed part of the cross-section at 
: a Yo is the Bessel function of the second kind, and Ho isa 
Struve function, (Notation is the same as in [9] .) Near the bow, we 
note that 
Srey i 
ZiOl SS tae eae OT ee en 
where the '"'...'' denotes some smooth function of x. The first and 
second derivatives of s(x) can then be expressed as follows 
s' (x) 
BRON Jong) Oe eee AS 
SU = 3 Za@kpese)) 4) cee |3 
where 6(x) is the Dirac delta function. 
The function F(x) represents the effects of interactions bet- 
ween the various cross-sections. For a body in an infinite fluid, we 
would have just the first term in the integrand, and one can show ea- 
sily that it represents the flow on the x axis caused by a distribution 
of sources both upstream and downstream of the point under conside- 
ration. The other two terms represent the effects of the free-surface, 
and they combine with the logarithm term in such a way as to cancel 
any flow upstream of a source. Tuck [8] has shown this explicitly. 
The integrand of the F(x) expression has a wavelike nature for < x 
but not for §> x. 
We are interested in how the above solution behaves as x -_,0. 
In fact, the easiest procedure for determining this behavior is totreat 
x as being O(e 2 ) and re-order all quantities accordingly. When we 
do this (after much algebra, expanding of the Bessel functions, etc. ), 
we find that 
3 
4UH K 3 
Pc) — [toe x +f log +1, for x = O(ef2) , 
where Y is Euler's constant. The problem for g@ becomes, for 
x =O(e1/2), a wedge-flow problem, with a rigid wall in place of the 
free surface ; its solution is 
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