Ochi 



The variance of wave motion, a^ , is simply determined from the energy 

 spectrum for a given sea state. Variance of vertical motion at an arbitrary 

 point, X , can be evaluated by the following formulae if the variances of motion 

 at two different points along the ship length, a-^ and a^ are known. 



''~^^' a' + 2p ^""-^^ ^^"'^ 



where 



(A.3) 



x,a,b = distances between points X, A, and b from the aft perpendicular 

 (see Fig. 21), 



o-^ = variance of vertical motion at point A, 



cr^ = variance of vertical motion at point B, 



Pgb = correlation coefficient of vertical motion at two different points, A 

 and B. 



Thus, the relative motion at arbitrary point along ship length can be ob- 

 tained from Eqs. (A.2) and (A.3). However, two correlation coefficients, p^^ 

 and p^^, involved in these equations must be determined experimentally. 



The correlation coefficient, p^^, can be obtained by the following formula 

 with the aid of auto and cross- spectral analysis of the vertical motions at points 

 A and B. 



- ^"''ab _ /(JC,b(^e)dc^e)' + (/Qab(^e)d^e)' (A.4) 



where 



C^y^(co^) = energy density of cospectrum, i.e., energy density of the real 



part of the cross- spectrum of vertical motions at points A and B, 



Qg(^(ajg) = energy density of quadrature spectrum, i.e., energy density of 

 the imaginary part of the cross- spectrum of vertical motions at 

 points A and B, 



$gg(ajg) = energy density of the auto-spectrum of vertical motion at point A, 



*bb(^e) ~ energy density of the auto-spectrum of vertical motion at point B. 



In the above formula, the definition of the variance and covariance given by 

 St. Denis and Pier son was used. If the acceleration is measured instead of the 

 vertical motion at one point (say, point A), Eq. (A.3) is still valid, since the ac- 

 celeration spectrum can easily be converted to the motion spectrum. The fol- 

 lowing relations are used in Eq. (A.4) in this case. 



582 



