Wave Forces on a Restratned Ship tn Head-Sea Waves 
Ww 
Cle) ee (G=i2) 
l 
Oo 
Z 
Uk Z il y Vz f a 
Gm Ly f fe e . e wo 
+ exponentially small terms 
Using (C-2), (C-3), the different approximations we have ob- 
tained for I(k) and the fact that exponentially small terms in I(k) 
will give exponentially small terms in the expression for © , we can 
easily obtain (61). 
APPENDIX D 
Inner expansion of far-field source solution for the forward-speed 
problem, 
We will show how a two-term inner expansion of (61) can be 
written as (64). 
We let y be of ordere and we reorder the terms in (61). 
We will assume as for the zero-speed problem (see Appendix 
B) that o (x) and o'(x) are continuous in the interval -L/2 < x 
=1/2. Outside -L/2< x. < L/2, ¢() = 0. Mt can then be 
shown (see Lighthill (1958) ) that k? o*(k) remains bounded as 
k ++... Using this it can be shown that the contribution from the 
inner expansion of Ta . ly and the fifth integralin (61) will be 
of higher order of magnitude than the terms we will retain in the 
inner expansion of (61). 
In the third and fourth integral in (61) we make an expansion 
of the integrand. The integration limits + e774 -B)y in) then play an 
important role. In the first two integrals in (61) we first introduce 
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