Ming-Shun Chang 
long-term variance of response is 
E (X(T), H)/,)] = fof salu an WEL aati eee! oe (1. 3) 
OO 
where p(T, Hib ) is the joint probability density function of T, and 
4/3 
Substituting equations (1.1) and (1.2) into (1.3), one has 
EL(T,.H, 4.1 = Hxp(w) Sf(w) de (1.4) 
0 
where 
ag andi ' fsb cet, pier or 2784, 
% (1./5) 
S5 is the averaged idealized spectrum by definition. 
It is seen from equation (1.4) that the long-term averaged 
variance of the response X(T,, Hy ) is the integral with respect to 
frequency of the product of the frequency response function and the 
averaged idealized spectrum, Sp”. The averaged idealized 
spectrum of the environment determines the long-term averaged 
variance of the response for a given frequency response function ofa 
ship. In addition to the convenience in calculating the averaged long- 
term ship responses, the two-parameter spectrum approach provides 
a rapid method for estimation of the probability of short term average 
ship response and its higher moments. If a probability diagram of 
T, and His is constructed such that 
and 
y = P(A) 73 < b;T,) 
where z and y are the two coordinates of the diagram and the P's 
are the probability functions, then from (1.1) the probability distri- 
bution of short-term ship response, P(X < a), is given by 
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