578 CART\^TIIGHT [CHAP. 15 



The above is somewhat an ideahzation of the practical problem of estimating 

 E"{u, v), since very specialized equipment borne by aircraft is needed to make 

 sufficiently extensive surveys of l,{x, y, t), and all proposed methods (see dis- 

 cussions by Longuet-Higgins and Pierson follo\Wng Munk et ah, 1957) have 

 their limitations. More conventionally, we have to deal with a small number of 

 fixed detectors recording continuously in time. Suitable anah^ical methods for 

 such records are mostly due to Barber (1954, 1957, 1958, 1959). The first stage 

 is to perform a cross-spectral analysis with respect to frequency between the 

 various records ; this, in effect, filters the complex covariances between pairs of 

 records into conveniently narrow bands of frequency. We may briefly explain 

 the cross-spectral process \>y denoting a pair of simultaneous related records 

 derived from our wave system by 



Lit) = y Zra cos (cT„f-l-A„-he„) 



n 



M{t) = 2 m„ cos {(Jnt + [Xn + en), (20) 



n 



where A„ and [Xn are phase angles depending on the type of detectors and their 

 position relative to the wave system, and e« are as usual random. Then the 

 cross-spectral components of these two records are : 



Cii{a)Sa = 2 ¥n^> Cmm{cr) Sa = 2 |w„2, 



da da 



Cim{cr) So- = 2 pK^ra COS (A„ — /x„), Qim{cr) Sff = 2 l^w^w sin (A^ — /Xn). 



6ct da 



Cii and Cmm are the energy spectra mth respect to frequency alone of the 

 individual functions, while Cim, Qim are their "co-spectrum" and "quadrature 

 spectrum", sometimes called the real and imaginary parts of the cross-spectrum, 

 and respectively equivalent to the covariance of L and M with and without a 

 phase shift of 7r/2. In practice, as sho\^Ti in section 7 of this Chapter (page 584), 

 we obtain weighted sums centred on the interval ha according to the type of 

 filter used, and not a simple sum as indicated in (20), but the principle is the 

 same. 



The cross-spectral estimates divide the directional spectrum into narrow 

 rings of nearly constant k = k{a), and for any one of these rings we are con- 

 cerned therefore only with a function of 6, E{k, d) = Ek{d) say. It is convenient 

 to expand Ek{B) in a Fourier Series : 



Ek{e) = lAo{k)+ 2 {Ar(k) cos rd + Br{k) sin rd), 



r=\ 



where 



Ar{k) -f iBr{k) = i I '" Ek{d) €*'•« dd. (21) 



