The notation here is that used by Pierson (1952, 1955) except 
S(w, 6) is the resolution of the variance of the particle motions into fre - 
quency and direction. For some fixed particle, say, one at the surface 
given by a = 0, 6 =0, 6 =0, the motions x = x(t), y = y(t), and z = z(t) 
form a stationary vector Gaussian process such that the spectrum of 
x(t) 1S 
JS(w, @)(cos6)“do , 
the spectrum of y(t) is 
[S(w, )(sine)“ae , 
the spectrum of z(t) is 
[S(w, 6) de , 
the cospectrum of x(t) y(t) is 
[S(w, 0) cos® sin@ dé , 
the quadrature spectrum of x(t) z(t) is 
[S(w, 8) cos dO, 
the quadrature spectrum of y(t) z(t) is 
[S(w, 6) sin® de , 
and all other cross spectra are zero. 
Other notations are often useful. For example, a notation similar 
to that of Longuet-Higgins [1957] would yield (14) as the complete equi- 
valent of (13). 
xX=a-Za EE IS Seal Charles (omy tear) 
many Kan m n mn mn 
Ss k6 
(14) y=B-2 2s Si sin(k a + kp - Oy ea tf coe) 
ké 
Ze OM eae! Or Comix. Garis wow. ea al 1) 
mn m n mn mn 
In (14), added conditions are that 
2 2 2 
(ne) Lay Sa = Kinn 
and that 
