APPROXIMATION OF SLOWLY-VARY I MG SECOND ORDER FORCE 



(2) 

 If we neglect the first term of F , which represents the high frequency 



components, we are left with the low-frequency second order force 



F, ^^^t) = Re E Z A A " H^^^(a) ,-a) )e'^V^'n^^ (7) 



L m n ' m n ^ ' 



m n 



If the wave frequencies are evenly spaced, with 



0) = mAco (8) 



m 



this can be written the computationally more convenient form 



(9) 



(2) ^-^ 

 F, ^^' = Re Z J 



m=0 



imAut 

 • e 



and H^^^ (w ,u ). 



This ex 



v/here J^ depends on A^^, A^ and H (o) ,cd ). This expression for the slowly-varying 

 force is in the form of a single summation and can be evaluated more rapidly than 

 the previous form. 



Hsu and Blenkarn (1970) proposed an approximate method for calculating the slowly- 

 varying force due to a random seaway. Each successive wave is assumed to apply a 

 force corresponding to a regular wave of the same height and period. They show 

 good comparisons for predicted and measured surge for two cases of moored vessels, 

 which supports this intuitive approach. Pijfers and Brink (1977) have developed a 

 more sophisticated version of this approach, in terms of the square of the wave 

 envelope and the regular wave drift force at a "momentary frequency," which they 

 define. 



Newman (197^*) proposed an approximation based on the assumption 



H^^^(cj ,-0) ) = H^^^(a) ,-w ) + 0(w -w ) (lO) 



m n mm m n 



That is, the contributions to the slowly-varying forces come from wave pairs of 

 nearly the same frequency, and in this case, the second order transfer function 

 can be adequately approximated by the regular wave steady second order force. 

 Newman showed that this was a good approximation to the second order pressure 

 underneath a system of random waves, but that a further approximation to convert 

 the formula into a single summation (but not that given above), did not yield such 

 good results. Loken and Olsen (1979) compared results of this approach with results 

 of the full equation and found generally good results. 



90 



