62 



THEORY OF SEAKEEPING 



The coefficients a„ and h„ remain unaft'en'ted by the 

 shift T, pr((\ided that T is sufficiently long. However, a 

 new relationship now appears — that of the dependence 

 of the measured y(t + r) on the record shift r. In order 

 to find the characteristics of this dependence, series 

 (113) and (115) are multiplied and integrated with re- 

 spect to t and taken in the limit as T goes to iiffinity. 

 This defines the autocovariance (autocorrelation) func- 

 tion which b}' the ergodic theorem completely describes 

 the process. 



/?(, 



lim 



1 r^ 



r Jo 



T 



yit) y (1+ t) (It 

 - J2 («"' + ^..') COS ^n r + C (116) 



H = l 



where C stands for correction terms which must he added 

 because series (115) does not represent y(t + r) in the 

 interval (T — r, T) when r > 0, or in the inter\-al (0, 

 — r) if r < 0. The integral expression for R{t) in equa- 

 tion (116) is connnonly known as the "autocovariance 

 function". 



On the extreme right-hand side of eciuution (116), the 

 .sum of the amplitudes «„- + 6„' is the scjuare of the total 

 amplitude of the wa\-e f<n- each harmonic component 

 o),,. The amplitude squared, multiplied by a suitable 

 constant, represents the energy which is contained in the 

 nth wave component. An inter\-al (jf frequency Aco can 

 be visualized as containing a mnnber of harmonics so 

 that the average energy density in the freciuency inter- 

 val Aoj is proportional to 



1 7((i) + Aaj/2 1 



(117) 



The limit of this ratio as Aw -»- was variously designated 

 as w{j) by Rice (K, p. 167), $(aj) by Tukey and Press 

 (1956), n.5 [A(aj)]- by Picrson (1952), and £'(aj) by 

 Longuet-Higgins (1955) anil is seen to be a measure of 

 the total energy in the wa\-e system. In Rice's defini- 

 tion, the frecjuency / = a;,'27r is used instead of the 

 angular freciuency oj. On the basis of the foregoing, 

 equation (IKi) also can bo written'"* 



/?(t) = 



r 



7?(aj") COS uiT (Ico 



(118) 



The cori'cctivc coefficient (' of (M|uation (116) vanishes 

 because of the a.ssumpt ion that T —>■ o= . Tlie coefficient 

 flo is eliminated by measuring the record ordinates, y, 

 from the mean level of the record. 



The value of R(t) can be estimated from wa\'e-record 

 measurements and the integral equation (118) can be 

 solved for E{(^) by using a I-'om-ier transform, yielding 

 the spectral density, 



E{o 



T Jo 



R( 



T) ('<)S (JlT ClT 



di 



(119) 



»« For a more rigorous transition from summation to iiitet!;ration, 

 the reader is referred to Riee (K, p. 167). 



R(0) designates the value of equations (116) and 

 (118) for r = 0. In this ca.se equations (116) and (118) 

 become 



R{Q) = lim ^ r [y[t)]-dt = f E(w)di, (120) 

 r^„ 1 Jo Jo 



This is a manifestation of the important theorem of 

 Parse\'al which states that the average energy in the sig- 

 nal is eciual to the sum of the a^'erage energies in each 

 frequency component . 



Equation (120) is proportional to the mean energy of 

 the wave system, or to the variance of y{t). Because of 

 this relation.ship to the energy, the function E{u) is 

 known as the "power spectrum" or "energy spectrum." 

 The first term is uni\'ersallj' used in electronics and the 

 second has been recommended for use in oceanography 

 and naval architecttu'e. Both of these refer to the dis- 

 triV)Ution fiuiction which is E{(x>) versus frequencies co or 

 /. E{u}) is defined more correctly as the spectral density 

 function. 



In apphcation to the wave spectrum, Pierson, (J, 

 1952) introduced the symbol [,l(a))]- in place of 2E{ui). 

 In this form, the proportionality of the energy to the 

 square of the component wa\-e amplitude is used to em- 

 phasize that the spectrum is e\-erywhere positi\'e. He 

 also introduced the sj'mbol E = '2R{Q) as the measure 

 of the total energy contained in the spectrum. Since 

 Pierson's definiti(jn of E has been widely used in char- 

 acterizing the wave heights, the factor 2 will be hereafter 

 incorporated in definitions of covariance fimctions. The 

 reason will be given in the next .section. 



Lucid and authoritative expositions of the spectral 

 analysis of stationary randon processes are gi\'en by 

 Press and Tukey (1956) and by Blackman and Tukey 

 (1958). 



8.4 Scalar Sec Spectrum — Calculation by Numerical 

 Method: 8.41 Formulation for computing procedure. 

 The continuous time history which is the wave recording 

 may, in theory, be operated upon by equations (116) 

 and (119) in order to obtain the wave spectrum. In 

 practical application, the continuous wave record is re- 

 placed by a sequence of eciually spaced readings of y{t); 

 the approximation can be made as close as desired, by 

 judicious choice of .sampling inter\-al. This sequence of 

 numbers is then converted to a new .sequence composed of 

 numbers denoting de\'iation from the mean of the wave 

 record. This new secjuence of numbers is then operated 

 upon by a discrete approximation to (116) of the form 



R, = T^^— E* y. ^,+,„ (P = 0, 1, ..., »,) (121) 



A - P 5=1 



where 



R,, = autocovariance estimate for lag t = pAt 



At = time inter-\'al between values of y(t) read from 



original record 



//, = value of y(t) at time qAt read from y as origin 



n = number of data points a\'ailable in .sample y{t) 



p = number of intervals defining lag r = pAt 



in = maximum number of lags 



