Understanding and Prediction of Ship Motions 

 For the other convolution, we must perform a little manipulation: 



Sjj a^(T)Mj^(t-T)dTU SJJ K(r)-a^(oo)H(T)] M.^( t - T)dTl + a^(oo)c3J| Mjt,(r)dr 



M.,(w) 

 = yij^(oj) X 3{a^(t) - ai^(co) H(t)} + ai^(co) x ^.^ 



. M3k(a;) x ^- + a^(a,) x ^-— 



The two kinds of convolution integrals can now be combined, as in Eq. (AlO), 

 and their sum will have as its Fourier transform 



a^(a;) 



Mj^(w) + ioj L.^(co) 



The above derivation amounts to a demonstration that the integrals in (13), 



t 



J ^k('^> K.^(t-T)dr, 



do indeed exist and have conventional transforms. 



The transform of (Xj - X^^) can now be written out explicitly: 



6 

 k=l 



- L {-^Vjk + i^-b.^ + c.^ + icL.^(c) + M.^Co;)}^;^ . (B2) 

 k=i 



This can be substituted into the transform of (13), and we recover (15) — with 

 three changes: 



1. a.^(a})/'i.co replaces ol^(co). 



2. L.j^(aj) + Mj^(aj)/iw replaces Kji^(aj). 



3. There is an extra sum on the left-hand side: 





77 



