71 



THEORY OF SEAKEEPING 



We =2 Tl/Tj, [sec 



Fig. 83 Spectra of same seakeeping event resulting from 

 analysis by different filters (from Marks and Strausser, 1959) 



posed on the smooth and rough digital spectra. The fit is 

 excellent, as the graph shows. The SEADAC analysis 

 took 5 niin. The record was 20 min long and the spec- 

 trum was first computed for 60 lags numerically and then 

 analyzed by a 5-cps filter in the SEADAC. The areas, 

 as tabulated, are in good agreement. 



Subsequent tests, for the purpose of studying resolu- 

 tion, resulted in Fig. 83 where the same record was sub- 

 jected to analysis by filters of different width (corre- 

 sponding to different lags numerically). The 5-cps 

 analysis corresponds, of course, to the 60-lag analysis 

 shown in Fig. 82. The 2-cps analysis shows more resolu- 

 tion than is probably required to define the ensemble 

 properties, since the 60-lag analysis has about 50 degrees 

 of freedom. The 10-cps analysis has lost most of the de- 

 tail in the spectrum and the use of still broader filters 

 will result in complete loss of detail as the filter's charac- 

 teristics become apparent in the spectrum. 



Choice of filter bandwidth is subjective in the same 

 way as is choice of lags in the numerical process. By 

 and large, the problems of experimental design are the 

 same as for numerical computation. Pierson (1954a) 

 discusses at length the mathematical theory of analog- 

 spectrum computers and Spetner (1954) treats the errors 

 resulting from analysis of a finite length of data spliced in 

 a loop. 



8.6 Information Derivable From the Wave Spectrum. 

 The wave spectrum, once it is computed, may be an end 

 in itself or a means to an end, depending on the problem 

 associated with it. Visual examination reveals certain 

 useful information, .such as: 



1 Significant range of frequencies outside of which 

 there is no appreciable energy contribution to the seaway. 



2 Frequency of maximum energy. 



3 Existence of low-frequency swell. 



4 Energy content in difTerent frequency bands. 



Beyond these directly observable "statistics," in- 

 formation obtained from the wave spectrum requires ad- 

 ditional calculations, some simple, .some lengthy. Chief 

 among the easily obtainable and most desired informa- 

 tion is the wave-height distribution. Since energy and 

 wave height (squared) are related, it is natural that aver- 

 ages of wa\'e height should relate to the total energy in 

 the spectrimi. The first computational step then is an 

 area measurement to determine the total energy; that is, 



iJ = m„ = 1 ^ y„2 = r E{u>)d^ (127) 

 n Jo 



Jasper (1956) found that wave and seakeeping data 

 (in the form of peak-to-peak measurements) behaved ac- 

 cording to the Rayleigh distribution, equation (112). 

 Longuet-Higgins (1952) assumed a Rayleigh distribution 

 and a narrow frequency spectrum to derive the following 

 statistical relationships: 



H = 1.772V-B 

 ^(1/3) = 2.832VE 



H{i/io) = smovE 



(128) 



already given in Table 7. The work of Longuet-Higgins 

 lends itself to extraction of a number of pre.sentations of 

 wave height such as the percentage of waves between 

 height strata and the average height of the highest wave 

 out of a total of iV-waves. This information is conven- 

 iently tabulated in Pierson, Neumann and James (H). 



The symbol E was introduced by Pierson (1952) to 

 designate the double mean square of the water-surface 

 elevations in waves. The integral in equation (127) 

 represents the total area imder the curve of spectral 

 densities. The use of the integral expression is permis- 

 sible because the smoothed spectrum is assimied to rep- 

 resent the true continuous spectrum of waves (or of the 

 ship's motion) under consideration. A summation is 

 used when the true spectrum is approximated by the 

 average energies in a series of frequency bands. The 

 "root-mean-square" (abbreviated mis) wave amplitude 

 is now defined simply as y/E. It is the amplitude of an 

 imaginary simple harmonic wave which has the same 

 mean energy per unit of sea surface as the complex wave 

 system represented by the spectrum. 



Cartwright and Longuet-Higgins investigated a cor- 

 rection factor to take into account the broadness of the 

 spectrum. This factor 



(1 



2U/2 



multiplies the quantities in (128). The broadness pa- 

 rameter, e, depends on the even moments of the spectrum 

 and is gi\-en by 



mom4 



where 



(128a) 



m„ 



/; 



''E{oj)di^ 



