F(2)(t) = Re E E A A H^2)(^ )e'K-'^)^ 

 ^ ' m n m n 



m n 



+ Re E E A A " H^^^cu ,-(. )e'^V%^'^ (4) 



m n m n 

 m n 



In these H and H are the first and second order transfer functions respectively 

 and are complex. 



For the usual representation of the seaway as a continuous spectrum, there are 

 analogous representations of the forces as single and double integrals over frequency 

 (Neal, 197^). In a digital time domain simulation, the seaway will be represented as 

 a discrete spectrum, and the form used here is appropriate. The continuous form 

 can be derived from a general representation of a nonlinear functional as a Volterra 

 functional series. Neal (197^) gave a discussion of this and further references. 

 An important requirement for the validity of this form is that of the continuity of 

 the functional relationship between the input and the output. This means that such 

 phenomena as hysterisis loops may not be modelled by this form of representation. 



Two special cases are of interest. The first is the second order response to 

 two waves of complex amplitude A. and A„ and frequencies w. and oj^ 



■-(2) / N r> TA n .,(2) / N 2iaj,t 

 F^ Mt) = Re{A.A. H' '(aj.,w.)e 1 



» A .1 (2) / \ 2ia)„t 

 + A2A2 H' (0)2,0)2)6 2 



+ 2 A.A„ H (oj, ,a)„;e 1 2 



+ A^A, H^ Mwp-o)^) + A2A2 H^^' (0)2,-0)2) 



+ 2 A,A2" H^2^o),,-a)2)e'^'^r'^2^h (5) 



It is clear that if we can calculate the response to two waves for all combinations of 

 frequencies, we have all the information required to predict the second order force 

 in a random sea. 



The other special case of interest is the second order response to a 

 regular wave of amplitude A and frequency o) 



F^^^t) = Re{A^ H^^^o),o))e^'"^ + A A" H(a),-o))} (6) 



The second term gives the well-known steady drift forces in regular waves. 



89 



