Wave Forces on a Restrained Shtp in Head-Sea Waves 
The integration limits Sy “(Ul “6,) 
role in obtaining (B-1). 
have played an important 
Note that the first brackets in (B-1) contain the lowest-order 
terms, the second brackets the next-lowest-order terms. 
Because we,want to apply Fourier-transform techniques, we 
want to set ¢ “(1-9 equal tooo. For the three higher-order terms 
in (B-1) we could do that ; the effect would be only to introduce 
higher-order, negligible effects. But we must be careful with the 
lower order terms. But assuming that o(x) and o'(x) are continuous 
in the interval -L/2 < x <L/2, it can be shown that, k2o° (k) remains 
bounded as k— too. This enables us to replace e -(1-5)) by in the 
first two integrals too. 
Let us now define 44 hot 
V k 
E(k) IKI (B-2) 
= k >0 
V | 
F *(k) denote the Fourier transform ofa function F(x). So 
et n/4 
F(x) =—————-_ H (x) (B -3) 
rie 
where H(x) is Heaviside step function. 
We also define F 
= u< 0 
e*q) =] 01 (B -4) 
eee wd >,0 
Vu 
So 
re 1/4 
G(x) =—————— H(-x) (B-5) 
We can now write (B-1) as 
1817 
