60 



THEORY OF SEAKEEPING 



spectrum formulations of Section 6. In fact, it is very 

 likely that a great many wave spectra of different sea 

 conditions will need to be measured and interpreted be- 

 fore a particular equation will be settled upon as ade- 

 quately descriptive of all possible wave conditions. It 

 should be mentioned that some oceanographers are at 

 present engaged in collection of wave data on a large 

 scale for the primary purpose of settling the wa-\-e- 

 spectrum formulation question. A symposium dedicated 

 to the wave spectrum will be held in 1961, and perhaps 

 some unification of wave theories will result. 



The intent of this section is to quit the realm of ques- 

 tionable sea spectrum prediction and to accjuaint the 

 reader with this alternative formulation of the spectrum 

 of the seaway: namely, that derivable from direct ob- 

 servation of waves and analysis ^-ia the concepts of ran- 

 dom process theory. Naval architecture is indebted to 

 Prof. W. J. Pierson, Jr. for recognition of the apphcabil- 

 ity of random-process theory (well known in the field of 

 communications), to ocean waves and ship motions, and 

 for making these methods available to na-\-al architects 

 for practical use (Pier.son, 1952a; St. Denis and Pierson, 

 1953). 



Several ciuotations from Press and Tukey (1956) will 

 partially summarize what has been discussed m these ui- 

 troductory remarks and will suggest what is to follow: 



"Fourier methods in particular have found wide re- 

 gions of application in aeronautics and have provided 

 many useful results .... The mtroduction of a chance 

 or random element leads naturally to the application of 

 probability and statistical notions which in combmation 

 with classical Fourier techniques have in recent j^ears 

 evolved into a general technique for the analysis of prob- 

 lems of this type. These techniques stem from the field 

 of random-process theory and from generalized harmonic 

 analysis. Of particular appropriateness for present 

 purpo.ses are the methods centered around the use of 

 power spectra .... Fourier techniques are described, as 

 those techniques, which when fused with statistical 

 theory, form the basic mathematical structure for spec- 

 tral methods." 



8.2 Evolution of the Description of the Seaway from 

 Wove Measurements. Most of the discussion in Section 8 

 will be de\-oted to the formal reduction of wave data to 

 energy spectra and the problems which arise from the 

 nature of the data and from the various means of data 

 handlmg. In order to pro\'ide the reader with a suitable 

 perspective that will prepare him for exposure to some of 

 the subtleties of random-process theorj', it will be worth 

 while to discuss the various paths which may be taken, 

 in proceedmg from the observation stage to the final form 

 of sea-state representation. 



There are five basic steps involved in the reahzation of 

 statistical information from the energy spectrum of 

 waves: 1) Observation, 2) recordmg, .3) preparation, 

 4) reduction, 5) presentation. It is not necessarily the 

 case that all five steps must apply (for example, one may 

 go directly from the transducer to the analyzer), but 

 these exceptions will be rare occurrences pertaining to 



operational use of the energy .spectrum rather than to 

 the energy spectrum as a descriptive tool in a particular 

 mvestigation. 



8.21 Observation. Transducers used m wave ob- 

 ser-\-ation are more \aried than those for any other event 

 related to seakeeping. The reason is that waves are 

 harder to measure than ship motions. One can find in 

 the literature such waves sensors as: graduated staffs 

 fixed to piers, pressure transducers on the bottom, float- 

 ing (capacitance, resistance, inductance) poles, floating 

 accelerometers, .stereo-photography, combmations of ac- 

 celerometer and pressure transducer (shipboard), etc. 

 None of these is without error due to distortion and 

 noise, of some magnitude, but most are fairly reliable 

 and find application m different investigations. It is m- 

 cumbent upon the user to ascertain that the error in sens- 

 ing, of his "wave observer," is either sufficiently small to 

 neglect or to qualify his results in terms of that error. 



8.22 Recording. Almo.st always, the raw wave data 

 will be analyzed in the laboratory, rather than at the 

 transducer, hence the necessity for a memory or storage 

 system. The information observed and interpreted by 

 the transducer is delivered to the memory, the recorder; 

 that is, the .sensor observes the event and may reproduce 

 the ob.servation as a continuou.sly fluctuating voltage. 

 This voltage is tran.smitted to a recorder which may re- 

 convert the data to a trace on paper, modulated electro- 

 magnetic signal (tape recorder), trace on film (camera 

 recording output of oscilloscope), secjuence of numbers, 

 or any one of a number of other memory media. Chart 

 paper and magnetic tape are the two most popular wave- 

 recordmg methods in use. 



Like the wave-sensing equipment, recorders are not 

 without some error owing to distortion and noise; for 

 after all, a recorder is first a transducer and second a 

 memory. The qualification of re.sults due to recording 

 error apply in the same way as to the wave observing 

 transducer. 



8.23 Preparation. The data, at this point, are still 

 in their origmal form and must be prepared for input to 

 an analyzer, either digital or analog. The digital input 

 will u.sually be in the form of equally .spaced measure- 

 ments of deviations from the mean of the wave record, 

 recorded on punch cards or tape. The analog mput is 

 usually the fluetuatmg ^•oltage presentation on magnetic 

 tape, sped up many times, so that the frequency range 

 of the input signal is amenable to survey by filters of 

 practical bandwidth. 



8.24 Reduction. The data are now ready for spectrum 

 analysis. Analog computers are usually required to 

 Ijerform the operations of filtermg, squaring and smooth- 

 ing, while digital computers perform calculations for the 

 autocoA^ariance function, Fourier cosine transform and 

 smoothing. Both techniques are compatible and will 

 produce the same re.sults, if properly treated. In fact, 

 the two techniques may successfully be switched, (that is, 

 digital filters versus electronic autocorrelators), though 

 this is not particularly convenient. 



8.25 Presentation. The output of the analyzer is 



