= Iisys 
Construction of z = z(x) for the linear model 
With 6 and t equal to zero in (21), the result is 
5 Sh ap oe SS Cho pyc Epbal(e Ghar G)) 
1 n n n 
(32) 
N 
TT 
= ar 
Z1 Zar cos(k_a ce) 
The quantities xy and z, forma vector Gaussian process.* The spectrum 
1 
of x] is S(k), the spectrum of z, is S(k), the co-spectrum is zero, and 
the quadrature spectrum is “aes The coherency is one. 
Tick [1961] has developed procedures for the construction of vector 
Gaussian processes with required spectra, cross spectra, and coherencies. 
A vector process with a reasonable spectrum and with the proper cross 
spectra was available, and it was a simple matter to prepare graphs of 
x(a) and z(a) as a function of a. 
The results are shown in figure 1, where x, (a), z(a), anda + x, (a) 
are all graphed as a function of a. For any value of a inthis figure, a 
pair of values for x and z result that can be plotted as a point in the x, z 
plane as shown immediately below. The points in the x, z plane then trace 
out the indicated curve that represents z = z(x). Only two crests are shown 
intrgure: ll: 
Figure 2 shows the result of preparing much longer graphs of 
x(a) and z(a). Eight crests are shown in the figure. In many ways, this 
artificial record appears to be much more realistic than one that might be 
obtained with a linear Gaussian model. 
These results show that, although the spectrum of the orbital motion 
could be band limited (i.e. , identically zero outside of a certain wave num- 
ber range), the spectrum of z = z(x) could be very rich in higher harmonics 
of the orbital motion spectrum. With one or more simple discontinuities in 
slope, as would be achieved at the sharp crests, the asymptotic form pre- 
dicted by Phillips [1958] would be found as applied to the long crested case. 
However, it would occur at wave numbers much higher than are presently 
capable of being resolved in spectral analyses. 
One of the difficulties with the Eulerian models is that they do not 
indicate within themselves unrealistic wave forms. A spectrum that would 
yield an impossible sea condition does not appear to be different from one 
that would yield a realistic sea condition. The linear Lagrangian model is 
*In fact, Zz is the Hilbert transform of x): 
