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FREQUENCY IN CPS 



Figure 8 — Spectral Analysis of Waves Produced with Actuator Installation 

 to Simulate Neumann Spectrum with Peak Frequency of 0.65 CPS 



SYNTHESIS OF A NEUMANN SPECTRAL SHAPE 



The basic building block of an analog computer is the integrator, with transfer function 

 in the Laplace transform notation of 1/s. Through proper interconnection of these elements, 

 an overall transfer function can be synthesized which is the ratio of arbitrary polynomials in 

 s, assuming that the degree of the denominator polynomial is not less than that of the numer- 

 ator. The Neumann spectrum has the form A'(<u) =(Cj/c<j^ )exp (-Cj/w^)? which is certainly 

 not suitable for direct simulation using the analog computer. 



The Neumann spectrum has several interesting properties that are not readily apparent. 

 Equating the derivative of the spectrum to zero, we find that the peak power density occurs 



&t (o = VCjTs with a value of Cj exp (~^/a;°). Normalizing with respect to the peak value, 

 we obtain N{a)/N{(o^) = i Wp/w!''' ! exp [ 3] 1 i exp [-3/(w/&jp)^]i , which is only a function 

 of the ratio of frequency to peak frequency. This indicates that if plotted on log-log graph 

 paper, all Neumann spectra would have the same shape; and if normalized to the peak fre- 

 quency and peak amplitude, they would plot on the same curve; see Figure 7. This suggests 

 that a proper presentation of experimental sea spectra should be in logarithmic form for 

 comparison with the Neumann hypothesis. 



This normalization property of the Neumann spectrum also has significance for analog 

 computer simulation. Because spectra should be provided over a range of peak frequencies 

 in the basin to simulate various sea states, it is of considerable convenience that the shape 



