586 CARTWRIGHT [CHAP. 15 



values. If, for example, we have ^t) for^ ^ = 0, S, 28, . . . nS, we compute 

 (electronically) for^ = 0, 1, 2, ... w, 



I 9=0 5=0 9=0 J 



which is (25) modified for an artificial non-zero mean of ^(0- The periodogram 

 operation on ipipS) 



yields 



^(S) = ^{'A(0) + (-l)^<A(m8) + 2 2VM) COS (Trrp/m; 



mS sin mcoS tt 



TT mcob mo 



This is not usually satisfactory since /(a>) has large oscillations which tend to 

 zero rather slowly as |a»| increases. A much better filter is obtained by taking 



^''lil = J^ 



(^-1)77 



wS 



mS 



P^\^i 



+ ie 



ip + ^Y' 



mS 



which gives 



„ w8 sin mojS 



lOe — 



277 m(oS[l — (ma)8/77)2] 3m8 



This is a somewhat broader filter whose oscillations are, however, negligible 

 outside a>= ± 27T/m8. (For other filters, and in fact for a full discussion of the 

 problem of spectral measurement, see Blackman and Tukey (1958).) 



The same shape of filter applies to cross-spectral estimates between two 

 signals ^i{t) and ^2{t), though the variability is more complicated (Goodman, 

 1958). In analogue form one process is to measure the power of the sum ^i(i) -f- 

 ^2(0 after passing through various filters. This is an estimate of {Ei{or) + E^i^) + 

 2[E i{a)E 2{or)]y2 cos 6{(r)] where d{u) is the phase angle between the signals. To 

 obtain the sign of 6, a phase increment can be introduced to one of the signals. 

 In digital analysis the filtering procedure is similar to that outlined above ; the 

 co-spectrum is obtained from the cosine periodogram and the quadrature 

 spectrum from the sine periodogram of the co variance function of l,i{t) and 

 ^2(0> with 2) ranging from —m to +m. 



8. Second- Order Approximations to Energy Spectra^ 



Finally we consider some approaches that have been made to account for 

 the neglected second-order terms in the basic equations of wave motion (section 



1 It is necessary that S be less than half the smallest time of oscillation of l,{t) to avoid 

 the "aliassing" effect (Blackman and Tukey, 1958). 



2 Since this section was written, the analysis of non-linear interactions in random wave 

 systems has been considerably extended by Phillips (1960, 1961). Interested readers 

 should also study the papers by Hasselman, Phillips, and Tick, and relevant discussions, 

 presented at the National Academy of Sciences Conference on "Ocean Wave Spectra", 

 Easton, Maryland, U.S.A., May 1961. 



