apart. For example, if waves with frequencies greater than 0.5 cycles per second are 

 absent, the wave record can be sampled every second. However, if appreciable wave 

 energy is present at frequencies above f cycles per second, this energy will contribute 

 (by aliasing) to the spectrum of the lower frequency waves. Thus, the sampling inter- 

 val must be determined by the high frequency cut-off. 



b. The range of wave frequencies shown on the abscissas of Figures 4, 5, 10, 

 and 1 1 represent the most important part of the spectral energy density associated with 

 surface wave heights. Similarly, the range of frequencies in Figure 9 are associated 

 with the dominant rolling motion of the submarine. Now, If a random process is char- 

 acterized by oscillations about a slowly varying centerline, the spectral estimates at 

 frequencies near zero may have appreciable values. However, since this spectral 

 energy is not associated with frequencies in the gravity wave spectrum, which are the 

 only frequencies of interest in this report, the graph Is arbitrarily cut off at the lowest 

 important frequency. Similarly, spectral densities at the high frequencies were not 

 plotted since spectral densities decreased rapidly as the Nyquist frequency was 

 approached. 



3. Comparison of Spectral Estimates 



When the estimated analog spectrum of a process is to be compared to a 

 digital estimate of the same spectrum, the equivalent spectral windows must be the 

 same. Figure 9 illustrates 4 digital estimates of the analog spectrum shown In Figure 

 8. The analog spectrum in Figure 8 had 58 degrees of freedom, and the four digital 

 spectra had 31, 42, 63, and 125 degrees of freedom. Therefore, the digital spectra 

 with 63 degrees of freedom were superimposed on Figure 8 for comparison. 



IV. POWER SPECTRA OF SURFACE WAVE, SUBMARINE, 

 AND FLUID MOTIONS 



A. General 



Wave, ship, and fluid motion data analyzed in this report were taken while the 

 REDFIN was executing a series of maneuvers which included hovering at different 

 relative headings. During each hovering run, the REDFIN attempted to hold a keel 

 depth of 100 feet at a fixed relative heading while making recordings of surface wave 

 height (Sonic Scanner), roll angle, transverse relative flow velocity, longitudinal 

 relative flow velocity, and in situ pressure fluctuations. Each recording was at least 

 one-half hour in duration. 



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