A = 

 2 



~Pien and Lee 



_co 2 (M+A )_jojB 22 _oj 2 (a Mz )_j 



coB, 



_w A 26 .j W B 26 



_a; 2 (A 42 -M Zg )_j W B 42 ^u 2 & 4 + \^^-J^% A -^ A 46 "J w ^6 



~ W A 6 2 -J<° B 62 



-oj A & ^ _jwB 



64 





J 66 



After an inversion of the matrix in Equation (8), we can 

 obtain the absolute motion amplitudes by 



C" 



ko 

 k = 2, 3, .... 6 



■ {< « k / + < «*/ } 



(9) 



and the phase angles with respect to the wave motion at the origin 



by 



a = tan 

 k 



-1 



{ " *ks / £kcl 



(10) 



k = 2, 3, .... 6 



Once the motion amplitudes £■ are obtained as previously 

 described as a function of wave frequency, we can obtain various 

 statistical averages of the displacement and velocity of the ship, usinj 

 the method introduced by St. Denis and Pierson (1953). We can show 

 that the statistical averages of the short-term response of ships to 

 sea waves can be given by 



( f° ) = C \ 



k stat ave 



( k° ) = c \£ 



v ^k stat ave y 



k = 2, 3, 



, 6 



where 



474 



