TM Wo. 377 



For the sake of argument, assume an ocean with a large system of ideal, 

 small amplitude, sinusoidal waves propagating in the same direction. The waves 

 are of identical wave length, wave height, and frequency, but all are of random 

 phase. What then is the probability distribution of the height of the free surface 

 level 71 (t), or of the magnitude of a velocity component v(t), when measured at a 

 fixed location? If the density of the phase angle given by R^^rC*) is uniform 

 (i.e., has equal probability of occurring anywhere within the range 0, 27T , as 

 shown by the upper plot of figure V-13); then, with classical sinusoidal waves, the 

 probability density of the amplitude A or the velocity function u (given by Py^O*)) 

 will be as depicted in the lower plot of figure V-13. (See appendix A.) This 

 distribution is far different from the Gaussian in that the maximum values occur at 

 the peaks of the waves. 



Applying this argument to the case of real waves: if there is a dominant class 

 amongst the myriad of waves present (represented by a peak in the spectra), a class 

 with similar heights, frequencies, and wave lengths, but of random phase; then there 

 will be a tendency to produce a maximum in the distribution function associated with 

 either the velocity or the free surface functions of the waves. This maximum would 

 occur on the CJ^vr scale around the mean amplitude value of the particular function. 



In short, if there are in the ocean a quasi-infinite number of waves of all 

 frequencies, amplitudes, and wave lengths, such an assortment of waves has an 

 approximate Gaussian distribution function. If, on the other hand, there is a 

 dominating sub-group of waves having similar parameters, their distribution 

 function tends to appear like Py,r(x) in figure V-13- This latter effect would 

 produce a net distribution that is symmetric about the mean, but with a bi-modal 

 tendency as shown in figures V-9 5 V-ll, and V-12 (upper histograms). 



Velocity Auto-Covariance Spectra 



The spectrum analysis of the wave particle velocity records can, with proper 

 interpretation, present the most meaningful part of the data analysis. Net only 

 can the auto- spectra of the time series records help in the interpretation of the 

 physical processes being studied, but they can often reveal something about the 

 reliability of the data and of the instrument used to obtain it. 



Auto-spectra plots of the wave observations and tabulations of the statistical 

 data are presented in appendix B. Along with table IV-3, which summarizes both the 

 statistical and environmental data for each observation, these should provide 

 ample information for reference. 



For purposes of orientation, examine first a typical auto-spectrum - serial 

 069 (BBELS-11, 3m III) made 30 March 1965. This auto-spectrum ^ w (see P lot p "°69 

 in appendix B) is of the vertical velocity component w measured at about 3 meters 

 below the trough level. It was computed using 1059 data points with M = 50 lags. 

 A logarithmic ordinate is used in the appendix B plots to show the frequency 

 contribution over the whole range of measured spectral density. The ordinate $ w 

 is in units of cm^ sec" per unit frequency bandwidtn. In this case, if M = 50 



105 



