Non-Linear Ship Wave Theory 
0 an 
Wie Wat Bow, ts: (45) 
for F* = o(1). To carry out the expansion of (44) we have to find the 
asymptotic expansion of w for large arguments. 
It can be shown that the function w(f ) = e's E; (if) has 
the following asymptotic expansion for large¢é 
— k! 
SL ED) Sh Ree eeae rag (-7 <arg ¢< —5) 
peo At Sy) 
foe) a4 
aise, c= Tana 4 Dacian | (-6<arg (<7) (46) 
k-0 (i ¢) 
where 6 is anarbitrarily small angle (see Erdely, 1956). 
Hence, by using (46) we find that w, has the following 
expressions 
1 1 
il pa Pee eee de 
(-7 <arg(f - ih+ 1) < 6) (47) 
and 
One 1 1 i 
wee f Ga Tease ea 0 
=1 
1 
4 Niuean Bye yin 
=| 
(- 6<arg(f - ih+ 1) <7) (48) 
Hence (47) is valid in the f lower half plane, excepting a 
"wake'' attached to the image of the body across the free-surface 
(fig. 6). In this wake we have to add the last "wavy" term of (48) to 
(47). In particular (47) is not uniform along any line parallel to y= 0 
and below w=h. In other words, no matter how smallis F, itis 
always possible to find, for W<h, a sufficiently large $ such that 
the wavy term of (48) becomes arbitrarily larger in comparison with 
the first term. In fact (48) is a uniform asymptotic expansion in the 
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