SEAWAY 



63 



70 80 90 

 Lag in Seconds 



Fig. 68 Non-normalized autocorrelation function reduced to unit amplitude by division by E,, ,„,^ for pressure record 

 of 10-18-51 from 2258 to 2323 EST at a depth of 32.5 ft MS off Long Branch, N. J. (from Pierson and Marks, 1952) 



The number 2 is inserted in (121) to account for the 

 symmetry of the aut(jco\'ari;uu'e function. It accounts 

 for the products y,ij^-p in computing the total \-ariance 

 in the record. The aut()co\'ariance function of a wave 

 record appears in I'ig. 68. The energy spectrum of the 

 finite sample, ^/(O, may he estimated from the discrete 

 values of the autocovariani'e function through a discrete 

 Fourier cosine transform which is the same as u.sing the 

 trapezoidal rule for integration to ai5]ir(iximate equation 

 (119). The result is 



ra - 1 



/?o + 2 J] /?,, COS TT /;////» + A\„ COS irli 



(/) = 0, 1, . . . , m) (122) 



where the L;, are estimates of the spectral densities at 

 frequencies 



o) = 



1 



m 



= hXw 



(123) 



The iiuaiitity Tr/niSt in (12;!) is the intcr\-al between 

 spectral estimates computed in (122) and is bound up in 

 that equation as the frequency bandwidth. Therefore, 

 if the Li are summed, the result is an estimate of the 

 total energy of the spectrum computed from the sample. 

 This important property will lie used in Section 8.6 when 

 statistics are obtained from the computed spectrum. 



The process of discrete cosine transformation can Ije 

 thought of as a weighting or filtering process of the form 



/: 



J'J(o)) G (co— a),,)(/i 



The filter operator, G'(co - 

 the sample, E(<x>), has the 



o),,), on the 

 form 



(124) 

 true spectrum of 



(l(u} — CO,,) 



M 



sin [2TViAt{io — coj] 



tan[7rA/(u) — (XI,) ] 



In the ideal case, it is desired that G(w — co,,) be constant 

 in the interval wj — Aaj/2 < cu^ + Aa)/2 and zero else- 

 where. Then, Lj would be an accurate estimate of the 

 energy density in each frequency band. Actually, the 

 function G(a) — coj) has the form shown in Fig. 69 and is 

 a poor ajDproximation to the ideal filter. Because of the 

 high-amplitude side lobes, it is expected that spectral- 



2 4 G 8 10 12 



Deviation from Center Frequency 

 (In Units of Aoo) 



Fig. 69 Filter response (/\1 = 20) (from Fleck, 1957) 



energy estimates at any freciuency will l)c affected by 

 energy in neighlxiring fre(iuencies. Also, the negative 

 lobes yield "negati\-e" quantities of energj'. The work 

 of Pierson and Marks (1952) and Fig. 70 show the com- 

 putation of "negati\-e" energy at certain freciuencies. 



A .simple smoothing o]3eration on ec[uation (122) will 

 improve the shape of the filter, Fig. 69, and results in bet- 

 ter estimates of the specti'al density. The smoothed 

 spectral estimates take the form 



E... 



0.5L„ -I- 0..3/.1 

 0.2o L,,_i -I- 0..5L,, 

 0..5L,„„i -I- 0.5L,„ 



h 



1, 2, 



0.2.-,L,,+ , 



- 1) 



(12.5) 



This form of smoothing (0.25, 0.5, 0.25) is called Han- 

 inng (after van Ilann). Another form, used extensively 



