TM No. 377 



phenomena are amenable to time studies (whether the sampling interval be 

 one millisecond or a million years), and workers in these fields now fre- 

 quently utilize such methods in many applications. (See, for instance, 

 Platzman and Rao, 196^, and Hamon and Hannan, 1963.) 



For examination of the time series data, the methods of power density 

 spectral analysis were used. There are many reasons for applying these 

 methods in lieu of the classical Fourier techniques for evaluating sinusoidal 

 amplitude coefficients. Barber (1961) presents a lucid discussion of these 

 advantages when dealing with random functions. His discussion, as it might 

 apply to ocean waves, is briefly summarized below. 



If a sample of the wave velocity component described as the random 

 oscillation u(t) is measured over the period T, it can be represented 

 (see James and James, 19^9) by a Fourier series of even-cosine and odd- 

 sine functions as : 



u(-t)= 2- A ^ c • (in-i) 



where ^ is the appropriate weighting constant for each term. It can be 

 shown that no correlation must be expected between successive amplitudes 

 Ajj and A^-i • The amplitudes are complex numbers whose phases and arguments 

 occur in a random fashion. 



If different samples ( 77j ~?Lj 73, - ^ *• 7^, »« - ) are taken off the infinite 

 spanning function u(t), no correlation is necessarily expected between the 

 different values of any one harmonic Hence, the lvalues of A„ can be con- 

 sidered as random choices of a family of complex Fourier coefficients. The 

 variance of a sequence of values Gil *■ may be large, but any individaul iL 

 calculated for a particular Fourier component of u(t) may or may not be" 

 indicative of the total energy associated with a finite band of energy 



centered around A . 

 n 



In the analysis of a quasi-random function such as u(t), the significant 

 quantity is a statistical parameter. This parameter should be associated 

 with the variance of the family from which the amplitudes of A n are derived. 

 If the sample is of sufficient time length and has a large number of values 

 (i.e., a large number of n's in equation (lll-l)}, then a large number of 

 different Fourier components are produced. It therefore follows that a large 

 number of adjacent harmonic amplitudes (A-n,Ar>+i t " -) are associated with a 

 spectral band of similar variance or energy content. One may, in effect, 

 estimate the variance or energy content as a function of frequency by cal- 

 culating the mean squares of a number of neighboring harmonic amplitudes. 

 The concept of the auto covariance function can therefore be used, and the 

 function can then be formulated into a power density spectrum. In other 

 words, a time series can have associated with it a spectral function, or 

 simply a spectrum that displays in a histogram-like form the variability 

 of the function (i«,e«, the contribution to the variance associated with a 



k& 



