connecting the zeros of the process) can be given statistically by a probability density 

 function. If the dominant part of the random process occurs over a narrow interval of 

 frequencies, amplitudes shown in Figure 3 can be given by the Rayleigh probability 

 distribution: 



2 



f(x)dx = -^ e "nr dx (i) 



Thus, the probability that a particular amplitude a. will fall between x and x + dx 

 is given by equation 1 . The first moment about the origin of equation 1 gives the re- 

 lationship between the mean wave amplitude a" and the quantity E: 



2 _xf_ 



T" e " E dx = °' 866 ^" ® 



The quantity E is given by the second moment about the origin of the Rayleigh 

 distribution: 



(3) 



E - a 2 



2 - ° (4) 



2 

 where a is the variance of the process. Now E is related to the energy spectrum of 



the random process by: 



E = 



{Kg dx and J- = /A(f]7 (5) 



2 ° 



where ^A(f)y is called the energy density. From the energy (power) spectra and 



relateastatistical parameters, the process can be completely described for practical 

 purposes. The term "power spectra" is freely used by aero- and hydro-dynamicists 

 among others, although much of the work dealing with the energy spectra of random 

 processes originated in the field of communications engineering (Reference 5). 



10 



