Breslin, Savitsky, and Tsakonas 

 i ' (cli) = - o)^ a ' C oj") + ioj b ' C a;") + c ' 



(18) 



m'(a)) = -co^A'(co) + i(UjB'(co) + C'. 



Equations (15) and (16) now allow one to express the response of the coupled 

 system in terms of wave amplitude by inserting them in (7) and (8) and summing 

 to yield 



— g(^.^. O e . (19) 



T(oj) I 



Thus one may recognize a lumped or effective frequency response function for 

 heave (with freedom in pitch) for the ship- sea system per unit of wave amplitude 

 as 



$,^(^,^,0 = (^ T^^ jg(^.^>0- (20) 



This can be reduced to an amplitude function which depends on w and i and a 

 phase angle which depends upon u> and f (or x); thus 



^,(-,^,0 = A(a.,0 e-^'^('"'^> (21) 



and this is what is determined from either theory or from recorded responses 

 of a model in regular waves. It is important to note that an arbitrariness is in- 

 troduced into the phase by the reference system used or, what is the same thing, 

 by the arbitrary definition of phase. 



Instantaneous Motion in Arbitrary Time- Varying Waves 



The equations for heave and pitch motion are the same as (1) and (2) for 

 regular onset waves, but now the right-hand sides are functions of time explic- 

 itly and are not functions of discrete frequencies. Thus F^e^"* and M^e^'^* are 

 replaced in Eqs. (1) and (2) by F(t) and M(t). The common procedure in solv- 

 ing the equations in this case is to employ Fourier integral transforms which 

 may be defined as follows: 



If z is the Fourier time transform of z(t), then 



CO 



z(co) = f z(t)e"^'"*dt (22) 



- 00 



and the inverse transform is 



466 



