176 



THEORY OF SEAKEEPING 



spaced points provide the data for analysis, the ap- 

 proximate covariance functions associated with the co- 

 and quadrature spectra are given by 



F„ 



G„ = 



(40) 



-s, + s. 



Fp and G^ are the digital forms for the expressions inside 

 the parentheses of ecjuation (39). 



where 



N-p j=i 



S-. = vf- e' >M^.) zit,) 



iV P g=l 



(41) 



The raw estimates of the cross spectrum are then given 



by 



L(c,.) = - ( F„ 

 m 



2 V F, cos "^ + F„ cos irh 



(42) 



L{qj = - (g, + 2 "e G, sin ^' + (?„. sin irh 

 m \ p=i m 



The rough estimates are smoothed according to ecjua- 

 tion (1-125). 



f/ ((•„.-. qj« = 0.5 Lo + 0.5 U 



U{c,,. qjh = 0.25 L,_: + 0.5 L, + 0.25 L,+i (43) 



U{Cy,. qj„ = 0.5 L„_i + 0.5 L„ 



3.62 Analog methods. The analog-filter method of 

 analysis in which a seakeeping parameter is recorded in 

 the form of a fluctuating voltage signal on magnetic tape 

 has been discussed in Section 1-8.5. The extension to 

 cross-spectrum analysis of two signals is quite similar. 

 Each signal is scanned simultaneously and separately 

 bj' identical filters. The outputs of these filters are in 

 one instance multiplied to yield the co-spectrum and at 

 the same time the output of one filter is phase shifted 90 

 deg and then multiplied by the output of the other filter 

 to yield the quadrature spectrum. This is analogous to 

 the cosine and sine transformations in (39). The clear- 

 est description of this procedure was given by Smith 

 (1955) and his paper is included as Appendix D. The 

 difficulties associated with this method concern the 

 matching of filters and the electronics associated with 

 phase shifting. 



Another (rather unicjue) method involves the use of one 

 filter and the sum of the input signals as recorded as well 

 as the sum of one signal and the derivative (or integral) of 

 the other. Consider the simultaneous sum of the two 

 signals y(t) and z{t). The covariance function of such a 

 record is 



Rir) 



hm 



T Jo 



[!j{t) + m] [yit + t) +Z{t+ T)]dT 



= lim ^, I ' [!i(t) y(t + t) + z{t)z(t + r) 



T^a 1 Jo 



+ z{n y(t + t) + zU + t) ,j(t)]dt (44) 

 The Fourier transform of i?„+..(T) results in 



^,+Ao^J = 'i'M) + '^.-M + '^ejo,,) (45) 



In a similar fashion, the Fourier transform of the covari- 

 ance of the sum of one signal and the derivati\'e (or in- 

 tegral) of the other is: 

 Differentiation 



*.+.("«) = ^„M + oj/*,(aJ,) + (2/cOe)g„+,(a),) 



Integration 



<i-„+fM) = *„(^.) + (l,V.-)*.(^o) + (2/a,,)g„+,(a).,) (46) 



If then, the spectra of the two signals are determined 

 separately and sul)tracted from (45) and (46) the result is 

 the CO- and quadrat lu'e spectra, respectively (after their 

 coefhcients are cancelled). The price of eliminating 

 matched filters and phase shifting is the additional time 

 required to make the foregoing four analyses. Such a 

 method is the basis of a cross-spectrum analyzer reported 

 by C'hadwick and Chang (1957). Tucker' (1950, 1952) 

 reported on a photoelectric correlation meter based on 

 somewhat the same principle. 



The literature abounds with papers on spectra and 

 cross-spectra. To start the reader off, some current 

 works will be cited that quite well co\'er the field of spec- 

 trum analysis. The reader can follow his own inclina- 

 tion from the many references given in each of these 

 papers: Liepmann (1952); Press and Houbolt (1955); 

 Press and Tukey (1956); Rosenblatt (1955); Goodman 

 (1-1957); Pierson (1957). 



Generalized harmonic analysis is treated well in the 

 following books : 



H. M. James, W. B. Nichols, and R. S. Phillips, 

 "Theory of Servo-Mechanisms," McGraw-Hill Book 

 Co., 1947. 



H. S. Tsien, "Engineering Cybernatics," McGraw- 

 Hill Book Co., 1954. 



J. Halcombe Laning, Jr. and Richard H. Battin, 

 "Random Processes in Automatic Control," McGraw- 

 Hill Book Co., 1956. 



3.7 Rolling of Ships in Natural Irregular Waves. 

 The only material on spectral analysis of ship rolling in 

 irregular waves can be found in papers by Cartwright and 

 Rydill (1957); Kato, Motora and Ishikawa (1957); 

 and Voznessensk}' and Firsoff (1957). 



Kato, Motora, and Ishikawa experimented with the 

 rolling of a ship model in natural beam waves and wind. 

 The model 2m (6.55 ft) long was placed about 100 ft off- 

 shore in water about 10 ft deep. The model was re- 

 strained bj' strings and springs to remain in a fixed posi- 

 tion. Wa\-e height, model roll angles, and wind-speed 

 fluctuation were recorded. The model's natural rolling 

 period was varied by changing ballast disposition. The 

 response-amplitude operators (the real parts of the fre- 



