Ming-Shun Chang 
cross spectra of the surface elevation and its space derivatives. The 
floating buoy built at the oaeoniie To: mere of Oceanography was de- 
signed for this approach. Barber 8] related the Fourier coefficients 
to the cross spectra of the surface elevations measured at several 
points in space. However, in the latter case the relations become 
more complicated and the lower order harmonic coefficients can not 
be determined without assumptions regarding the higher order har- 
monic coefficients. 
From the Fourier representation of S( k;,) with respect 
to the wave-number component k, , Barber and Pierson|? have 
shown that S( R; Wo) can be approximated directly from an array of 
probes which lies in a direction parallel to the x, coordinate. The 
Fourier coefficients obtained from this approach have a one-to-one 
correspondence with the cross spectra of measurements having space 
separations of ¥ = (nD,0), n=1,2, ...N; where Y is the separa- 
tion vector and D is the fundamental separation of the probes. Thus, 
from an array of probes which has separations D, 2D, ... ND, one 
is able to approximate the directional spectrum up to the Nth har- 
monic. The derivation of this is given below. 
By multiplying equation (2.1) by the corresponding equation 
for u(t+7, % + Y) and taking the expected value, one has 
Ett) Mt 1 tas +f), = Sf ailk ; x-wt)-i[K'. (24+F)- (tt )] 
(2.3) 
where 7 is the time lag. Since the wave field is assumed stationary 
as well as homogeneous, the correlation function E [n (t, %) n (t+ 1, xt) 
in the above equation is a function of t and 7 only; it is independent 
of t and x. If the correlation function is denoted by R(t, ¥f), equa- 
tion (2.3) can be written as 
(2. 4) 
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