66 



THEORY OF SEAKEEPING 



Fig. 73 Aliasing a short wave record (from Press and Tukey, 1956) 



/ = 



?tt/4 



m/2 



•2n 

 m 



(12(j) 



Given /, then the use of Tal)le 12 yields information on 

 the reliability of the estimate for each frequency l)and ob- 

 tained by the analysis of one record. 



"If a given analysis has, say, ten degrees of freedom, 

 then the number obtained from (125) can be multiplied 

 by 2.6 and by 0.55. The true value of the power in the 

 band under analj'sis will lie between the.se extremes 90 

 pet. of the time. Stated another way, given a great 

 many analyses of different short samples from a long run 

 of time series, then the set of bands given above will en- 

 close the true power in the liand 90 times out of 100 if the 

 sample length is such that ten degrees of freedom are ob- 

 tamed and if a very large number of samples are evalu- 

 ated. 



"In this analysis, n = 750 and m = 30. The result 

 from (126) is approximately 50 degrees of freedom. 

 Therefore, if each ordinate of the smoothed spectrum is 

 multiplied by 1.45 and by 0.74, it can be said that the 

 true pressure power spectrum lies between the.se bounds 

 (dotted lines hi Tig. 71) 90 pet. of the time." 



Eciuation (126) was first introduced by Pierson 

 (1952a) as a deduction from the work of Tukey (1949), 

 and Tukey and Hamming (1949). A paper by Black- 

 man and Tukey (1958, Section B.9) leads to the expres- 

 sion 



/ 



n — m/6 

 ?n/2 



The difference is insignificant. Both ecjuations tend to 

 2n/m as n becomes large and any small differences in / 

 would hardly be noticed in Table 12 where the confidence 

 bands vary slowly above / = 10. 



It is clear that the confidence limits obtained through 

 eciuation (126) and Table 12 define the probable ensemble 

 spread and consequently ciualify the usefulness of the 

 spectrum as a descriptive tool. The narrower the con- 

 fidence limits, the more reliable the spectrum. This im- 

 plies the desirability of a large number of degrees of free- 

 dom (/) in (126). We are now in a position to evaluate 

 the three basic questions in terms of the desired proper- 

 ties of the spectrum. 



On the (luestion of record length, one extreme rules out 

 \'ery short records. It is cleai- that one or two wa\-e os- 

 cillations can hardly serve our needs for description of 

 the seaway. The other extreme, that of an infinitely 

 long record, is ideal, in principle, since it can completely 



and accurately describe the sea surface, yet is equally 

 ridiculous in that such a record cannot lie recorded and if 

 it coukl it cannot be analvzed. Compromise retjuires 

 the convergence of these two sampling concepts to an 

 nptimum record length that embodies sufficient length 

 lor an adefjuate description of the seaway yet not so long 

 that the additional length, beyond a time T, returns lit- 

 tle, in minimizing sampling variability, for a great deal of 

 computational effort. I'inally, the record, however 

 long, must be stationaiy (or nearly so) throughout. 

 The .seaway is u.sually stationary for hours at a time and 

 in this respect wave analysis is more fortunate than ship 

 motions analysis where many different samples must be 

 taken in the same interval of time. Nevertheless, it has 

 been .shown in the exam])le of Pierson and Marks that a 

 25-niin record yields 50 degrees of freedom, a worth- 

 while mark at which to aim. Since there is some free- 

 dom in the recording time of wa\'es (unless the sea state 

 changes rapidly), one .should aim for a minimum of 20 

 mill and maximum of 1 hr of wave-recording time. 



Once the record is obtained, its length is fixed, but the 

 number of discrete points in the sample is not determined 

 until the sampling iiiter\-al A/ is specified, because there 

 are 



n = {T/M) + 1 



sample p(jints, where T is the record length. It is ob- 

 \-ious that a At which is .small, compared with the short- 

 est period that contributes energy to the spectrum, will 

 have misleading effects on equation (126). In fact, / 

 can be made as large as desired, by letting A^ become suf- 

 ficiently small. This is contrary to the intent of (126) 

 which implies some measure of independence between 

 the ?i-discrete values taken from the original record. 

 The "small" At does, in fact, provide a better approxi- 

 mation to the record compared to an o])timuni At (which 

 will be defined) but adds little to the energy spectrum of 

 that record, for large additional computational effort. 

 The net effect is the computation of more .spectral es- 

 timates of zero value for the higher freciuencies and no 

 improvement in resolution. 



On the other hand, if At is large, compared with the 

 shortest period that contributes energy to the spectrum, 

 the //-discrete values do not ade(iuately represent the 

 higher freciuencies in the record (.see Fig. 73) and the 

 energy in harmonic components above /„ where /, = 

 l/2At is the Nyquist frequency (Shannon, 1949), is 

 aliased (transposed) into the energy content of the lower 



