Breslin, Savitsky, and Tsakonas 



which simply states that the total heave response is the algebraic sum of two 

 convolutions of the force and moment time histories with the appropriate impul- 

 sive response functions. 



However, one does not have at his disposal the force and moment time his- 

 tories, but rather only the wave input time history. It is, therefore, necessary 

 to eliminate the explicit dependence of the result on f and M and to determine 

 how one operates on the known (or given) wave record to determine the motions. 

 For simplicity, the following development is applied only to part of the response 

 Zf to illustrate the procedure. 



It is noted from (16) that the normalized force at any discrete frequency is 

 known and hence one can express the force as a function of time and the instan- 

 taneous surface wave t?o(t) by convoluting the wave with the force transfer 

 function, or 



F( 



t) = ^ J Vo(r';^) \ f'(a;') g(a)',^, C) e^" ^*'" ) do^'dr' . (32) 



Upon insertion of this into (30) and through the use of (27), one finds that this 

 part of the response takes the form: 



'^ (27T)- 



00 f CO 00 1 /-" ^ 



T i 7) (t ^) f g e ^ ^ dr ^ -— - e ^ ^ dwdr. 



!^) -'.= -00 Kco -"-00 J -co T(^) 



Interchanging the order of integration and noting that the t- integral is simply 



giC'^ "^)'^dr = 27rS(aj'-aj) 



- 00 



where S is the Dirac delta function, yields 



CD 00 CO 



T =-00 co=-o^ CO = - CO 



and, because of the delta function property the ^'-integral produces f (w) g(a;) e' 

 to give 



If'" . ' ^x r" f (^) g(^, -f. O S(co) i^(t-T'), , , /oo^ 



a completely similar result would be secured for the motion constituent z^ so 

 the total motion can be written 



468 



