SEAWAY 



69 



order of importance but, because of the particular se- 

 quence of discussion in Section 8. The importance of 

 sampHng variabiUty was hinted at earlier by the oft-re- 

 peated qualifying statement, ". . ., if the record is long 

 enough." Simply, this means that no matter how care- 

 fully the spectrum analysis is carried out and how sin- 

 cerely it is qualified, the results will only define the state 

 of the sea within the limitations of the length of the 

 sample. The length of sample has been discussed in 

 Section 8.43 but it will be worth while to illustrate the 

 effects of variable sampling length and at the same time 

 to review the consequences of choice of computational 

 parameters. 



Consider the record of a particular seakeeping event 

 (not waves) which was originally 30 min. long. Ten dif- 

 ferent spectrum analyses of this basic record were made 

 (Marks, 1959). The results appear in Fig. 79. Five 

 analyses were made with m = 30 lags and five with m = 

 60 lags. The five analyses in each set were made for dif- 

 ferent lengths of the record as specified in the figure. 

 Since the problem of aliasing is straightforward, a good 

 choice of At eliminates consideration of this a.spect here. 

 It is obvious that the more detailed spectra, computed for 

 60 lags, Fig. 79(A), have much more variability than the 

 spectra computed for 30 lags, Fig. 79(B). > This is to be 

 expected, in view of eciuation (126) and Table 12 which 

 predict wider confidence bands (less degrees of freedom) 

 for the spectra associated with 60 lags. 



It should be pointed out here that the entire discussion 

 relating to degrees of freedom and confidence bands 

 always refers to sampling from a population in which the 

 sample is one member of an ensemble that might ha\'e 

 been taken from the population. Description of the 

 population is derived through estimation, from the 

 sample, of the average statistical properties of the en- 

 semble. Care must be exercised in analyzing such a 

 sample to be certam that the results reflect the char- 

 acteristics of the ensemble rather than those of the par- 

 ticular sample. Longer records and less resolution tend 

 to give results in this direction. However, if the wave 

 record is the population, such as is often the case when 

 generating the same irregular waves in the towing tank 

 for successive tests, then the foregoing discussion does not 

 apply. The "sample" record completely represents the 

 waves and its energy spectrum will describe the process; 

 the greater the resolution (more lags) , the better the de- 

 scription. 



Examination of the indi^'idual sets of spectra reveals 

 that the shortest records result in highly variable esti- 

 mates of spectral density while the longer ones tend to- 

 ward stability, with the spectrum of the 30-min record 

 appearing to give a good approximation of the averages 

 of the other records, and which may now be considered to 

 be a good estimate of the population from which the 

 sample record was taken. Comparison of the spectra of 

 the longest records for 30 and 60 lags, Fig. 75, shows little 

 variability as a fmiction of the number of lags, a further 

 indication of .stability, in this case. 



Blackman and Tukey point up the difficulties of using 



f-l/T sec- 



B-m 30 Lags 



012 0.14 



f--l/T [se 



OZO 



Fig. 79 Spectra computed from various lengths of same sea- 

 keeping record (courtesy David Taylor Model Basin) 



theoretic'al methods to estimate variability and end in 

 saying: ". . . The precision of final over-all values is 

 ordinarily far more wisely judged from the observed con- 

 sistency of repeated measurements than from any theo- 

 i-etical variability based on a Gaussian assumption." 



This is sound advice, for towing tank work, where 

 short samples demonstrate high variability and where re- 

 peated tests in different waves reveal a safe number of 

 such tests to provide a stable ensemble average. For 

 full-.scale trials (wave mea.surements at sea), it is hardly 

 possible to repeat tests and so experience gained by the 

 type of experiment discussed in this section is used in the 

 design of criteria for sampling at sea. 



The work of Tucker (1957) lends itself well as an 

 example in the study of sampling variability. Two con- 

 tmuous records, each of about 10 hr duration were made 

 and the mean-squared wave height for consecutive 10- 

 min intervals was computed and the results plotted in 

 Fig. 80. The means for groups of 20 of these estimates 

 are drawn in as reference levels. The upper figure dem- 

 onstrates nearly stationary conditions over the 10-hr 

 recording period yet the mean-squared values for 10-min 

 samples, which may be taken as a measure of the total 

 energy in the related spectra, demonstrate large \'aria- 

 bility. Averages of pairs of 10-min squared means of 

 the wave height are seen to reduce the variability 

 bounds. Averages of trios decrease the variability even 

 more. It is clear then that in conditions which are 

 known to be stationary for 10 hr, a 10-min sample will 



