SEAWAY 



67 



Computed spec+run 

 with aliasing 



True shape of high 

 frequency end of 

 spectrum 



Spectrum of energy 

 aliased about nyquist 

 frequency 



Nyquist frequency (t^") 



0.12 0.16 QZO 



f=l/T [sec''] 



Fig. 74 Example of aliasing a wave spectrum. Hatched area 

 shows erroneous energy contribution 



frequencies. Informatimi at ./', + ,/' caiiiiot be rescjh'ed 

 because of the discrete sanipliiifi ami the energy sjiectrum 

 is "folded" about /,. The Nyfiuist fre([U(>iicy is often 

 called the folding frequency or cut-off frc(|uciicy. l"ig. 

 74 illustrates the erroneous distoi'tioii in the wave 

 spectrum due to a large At. 



"Selection of an optimiun At tlepends on the highest 

 frequency in the wave spectrum above which no ap- 

 preciable wave energy can be aliased back into the spec- 

 trum." This glib statement, which appears often in 

 the literature, is pi-edicated on the assumption that there 

 is a priirri knowledge of the shape of th(> spectrum. Most 

 often this is not the case and a simple rule of thumb, 

 commonly u.sed, suggests the selection of that At which is 

 one half the smallest "apparent" period in the wave 

 record. This is .still somewhat subjecti\'e in that it in- 

 volves a definition of what is a cycle of wa\-e motion. In 

 the illustration of the wave-record analysis of I'ierson 

 and Marks, the pressure transducer had a natural cut- 

 off at 0.25 cps becau.se of the attenuation of wave energy 

 with depth. This only obtains with pressure trans- 

 ducers. It is how'ever possible to ijrefilter either at the 

 transducer or recorder and this is often worth while in 

 eliminating .spurious noi.^e. It also affords a natural 

 choice for the Nyc[uist frecjuency. An example of elec- 

 tronic prefiltering will be given in Section 8.43. 



The number of lags to be computed, equation (1215), 

 for the autocovariance function determines the number of 

 estimates of spectral density (actually m + 1) that will 

 be made in accordance with equation (122). These 

 spectral densities will he spaced at distances of/ = /,/m, 

 [see equation (123), for the eciuivalent expres.sion]. Ac- 

 cording to Fleck (1957): ". . .Since the effective width 

 of the filter corresponding to the inunerical procedure for 

 computing the spectrum is 2Af, the \-alue of m must l)e 



1.2 

 i.c 



0.8 



. o.e 



0.4 

 0.2 







0.10 0.12 0,14 O.lfi 



f = l/T [sec-'J 



0.18 



o.eo 



Fig. 7 5 Spectra of a seakeeping event computed from the same^ 

 record for 30 and 60 lags (courtesy David Taylor Model Basin) 



chosen so that 2A/ is the minimum resolution interval 

 desired or some smaller interval. Then m = 1 2A/A/. 

 It is recommended that m be at least twice this minimum 

 value to insure adeciuate resolution." 



From equation (12G) it is seen that the greater the 

 resolution (greater m) the smaller the number of de- 

 grees of freedom (/) and consequently the less confidence. 

 That is, the more nearly we represent the sample record 

 through high resolution of its spectrum the gi-eater is the 

 probaliility that we tleviate from tlie true spectrum tjf the 

 parent population. Comprf)mise must be made between 

 resolution and confidence. In most wide-band spectra it 

 has been found that (iO lags is more than adetiuate for a 

 good description of the record and 30 lags is u.sually 

 ample. In the case of a swell ,sp(>cti'um, adeijuate resolu- 

 tion of the .steep sides will i'e(|uire ,s(.imewhat more lags; 

 twice the number needed for a wide band spectrum may 

 be a good rule <jf tluunb. I'ig. 75 shows a wide-band 

 spectrum (not waves) computed for 30 and (30 lags. 

 The 90 per cent confidence limits, Table 12, for each of 

 these spectra will contain the other spectrum completely. 

 No significant peaks have been masked in going from 60 

 to 30 lags and it is seen that it would be worth while to 

 use 30 lags and impi-o\'ed confidence, for this specti'um 

 anyway. 



Thepurposeof the foregoing was to ac(|uaint the reader 

 with some of the .subtle aspects of spectium analysis. 

 One cannot ru.sh headlong into the computations with- 

 out consideration of what is desired and of the penalties 

 that must be paid to achieve the desired spectrum. 



8.43 Additional sources of error. There are some 

 additional sources of difficulty connected with arri\ing 

 at the correct wave spectrum of the sample. The.se are 

 involved with the collection and preparation of data dis- 

 cussed in Section 8.2 rather than with the computational 

 procedure discussed in Section 8.42. 



A'ery often, spurious information appears in the 

 .spectrum due to d-c ilrift in the recording electronics. In 

 this case the wave record drifts away from the preset zero 

 of the recorder and the spectrum manifestation of this 

 event is a spike, at or near zero frequency, corresponding 



