SEAWAY 



61 



either a graphical presentation of relative spectral density 

 versus frequency (usual output of analog computers) or a 

 set of numbers representing relati\'e spectral density 

 (usual output of digital computers). Sucii a spectrum 

 determination pro\'idcs information on the general dis- 

 tribution of energy. Further statistics are derived from 

 the total energy in the spectrum as well as from the var- 

 ious even moments of the spectral-density function. 



This review of data collection and data reduction il- 

 lustrates that a variety of errors may be introduced into 

 the signal long before it is analyzed. It is generally 

 agreed however, that if the precautions discussed are 

 taken, the net erroi- in the signal, at the time of analysis, 

 may be neglected when compared to the computational 

 errors and the m(5re serious error due to samj^ling vari- 

 ability. These last two errors will be discussed in detail 

 in subseciuent sections. 



8.3 Scalar Sea Spectrum — Theoretical Determination. 

 Wave records, of finite length, can be analyzed by famil- 

 iar Fourier-series methods. This however is found to be 

 impractical, at least by numerical methods. Complete 

 analysis of a record maj'' rec(uire a coni])utation of as 

 many as 300 harmonics with little contribution to the 

 total energy in the spectrum in the first hundred har- 

 monics. Usmg analog-filter techniques, such an analy- 

 sis is not nearly so cumbersome. Blackman antl Tukey 

 (1958) point out that such elementary (Fourier series) 

 methods frequently fail because indefinitely long rec- 

 ords may not be available. The high-precision, high- 

 resolution analysis of such short records results in 

 spectra which may be quite misleading because they 

 represent very well the fluctuations of the particular rec- 

 ord being analyzed rather than the ensemble of functions 

 it is desired to represent. A quote from Lee and Wiesner 

 (1950) will illustrate further the shortcomings of Fourier 

 techniques : 



". . . we wish to know the spectrum of the fluctuating 

 voltage. If only Fourier series and Fourier integral 

 theories for periodic functions and transients are at our 

 disposal we are not sufficiently equipped to solve a prob- 

 lem of this sort. The reason is simply that these theo- 

 ries, as they stand, are not applicable to functions which 

 are specified in terms of statistics and probability and 

 which are not representable by specific analytic expres- 

 sions giving their precise values for all values of the in- 

 dependent variable. Howe\'er, the extension of the 

 Fourier theorems to the harmonic analysis of random 

 processes through the medium of correlation functions 

 has enabled us to obtain a solution to our problem with 

 surprising ease." 



Since we are dealing with a single function y(t), we 

 shall speak of the autocovariance (autocorrelation) func- 

 tion as the descriptive property of such a record. How- 

 ever, Wiener (1930) showed that the autocovariance 

 function and the energy spectrum are Fourier trans- 

 forms of each other. This being the case, both must 

 yield the same information, in principle. In practice, 

 confidence criteria derived for the autocovariance func- 

 tion are very complicated and difficult to apply, while 



the work of Tukey (1949), on sampling effects, finite 

 length of recortl and computational procedtire, is easily 

 applied to the energy spectrum. In addition, the energy 

 spectrum, being in the frequency domain (rather than 

 the time domain), appeals to the naval architect who 

 deals with transfer functions and ship responses. The 

 energy spectrum has gained wide use ;uid acceptance in 

 such fields as meteor(jlogy, oceanography, seismology, 

 and aerodynamics. 



The application of harmonic analysis to random 

 stationarj' processes appears in many places in the litera- 

 ture. A particularly clear development is given by 

 Rice (K), who extended the ideas of Wiener (1930) 

 to obtain derivations of certam important statistical 

 properties of Gaussian random processes. The pro- 

 cedure of Rice in defining the energy spectrum, via the 

 covariance function, will be paraphrased here to apply to 

 ocean-wa\"e records. 



Let a sample wave record, T seconds kmg, such as is 

 shown in Fig. 67, be considered periodic," with period T, 



Fig. 67 



and absolutely integrable m that period, i.e., 



• r 



I 



Jo 



uiiyii < 



then ij{t) can be represented by a Fourier series 



yit) =^+ i (a,, cos a>J + K sin o>J) (113) 



where a)„ = '2-nn/T and n is an integer. The coefficients 

 a„ and h„ are the amplitudes of the nih cosine and sine 

 waves. These amplitudes are evaluated in the theory of 

 Fourier series by the integrals 



".. = Tp I 2/(') CO* '^"' f" 

 J Jo 



(114) 



2 r^ 



''.. = 7r y(') sni uij (It 



J Jo 



Let the record now be shifted by an amount of time r. 

 Then for the interval —T<t< T — t 



y(t + r) 



flo 



^ H = l 



\a„ cos aj„ {t + t) + b„ sin o}„ (I + t)\ (115) 



" The condition of periodicity will not disturb us, because we 

 are interested only in the properties of y(t) in any interval exactly 

 T seconds long. 



