here Wo is the peak frequency of the narrow banded spectrum of the wave, and 
D(t) = a[H?(1)] = a'[a*(t) + b*(1)] = D+ k'E(0), (12.64) 
where H(t), hA(t) are the amplitudes of the envelope; a(t) and b(t) are components of 
A(t); a(t) is the phase lag; D is the mean drifting force; and &(r) is the white noise, as will 
be assumed later. 
The equation of swaying motion of the vessel was expressed as 
(M + m)x + F(x,x) = D(t) (12.65) 
or 
X + flx, x) = Bla2(t) + b7(0)] (12.66) 
where (M + m) is the virtual mass of the vessel. Then with a(t), b(t) as the output of white 
noise &(t) through linear filters, the equation of motion was modified to 
z+fiz )= E(t). (12.67) 
The term z is a vector Markov process with four elements [x,x,a,b], and the Fok- 
ker—Planck equation of this four—element variable z(t) was then derived. Then since 
A(t) can be generated as the output of a nonlinear first-order system with a white noise 
input &(t), the process was inverted into a three-element vector Markov pro- 
cess, y(t)[x(t), x(t), A(t)], and the Fokker—Planck equation for this process was also 
derived. 
However, the solution of the multidimensional Fokker—Planck equation has a num- 
ber of difficulties, so Roberts advanced the approximation further, and expressed the 
drifting force as 
DH=D+k'E(s), (12.68) 
the sum of the mean drifting force D, that can be approximated as a’e, plus a white noise. 
Under these approximations, the equation of swaying motion is 
X+ g(x, x) = d+ k&(t), (12.69) 
where d= 2 = Gis = Be, 
(M+m) (M+m) 
k= ia : 
(M +m) 
343 
