The energy or mean square in this element is a^. Assume the ele- 

 ment is a part of a continuum of elements for -co < f < +00 and <^ 

 < 2tt. In this case a^ must be an infinitesimal energy associated with 

 the frequency differential, df, and space frequency differentials, d£, 

 and dm, which are related to the direction 6 of the wave element as 

 before. Let the power spectrum, S(Jl,m,f), be defined with units of 

 amplitude squared and divided by unit spacial frequency, i, unit 

 spacial frequency, m, and unit time frequency, f. The power spectrum 

 is then a spectral density value at (Jl,m,f). In this case we must have 

 the infinitesimal energy, a", defined by 



«J = S(i V.UU>n ^4 (2.15) 



The real valued, nonnegative function S(Jl,m,f) is a power (energy 

 density) spectrum of the standard type in three-dimensional frequency 

 space (Jl,m,f). Intuitively, we can write an infinitesimal wave ele- 

 ment as 



[Exb (i 2lf ( ^^x + ^^H ^ ^t t «»<^)p5(i^,>l^.0^i'^>^^^' » (2 . 16) 



where the positive square root is assumed. To arrive at a model of 

 ri(x,y,t) for the continuous case, we need only form a triple "sum"_of 

 the infinitesimals or, to be precise, the triple integral iKx^y^t), = 



For different sets of values of the phase relation a(£,m,f), the 

 wave system, n(x,y,t), has a different shape, even when S(Jl,m,f) is 

 fixed. In fact there is a wide range of possible shapes of n(x,y,t) 

 for a given S(Jl,m,f). The above development is more intuitive than 

 mathematically rigorous, it has been shown by Pierson (1955 - pp. 126- 

 129) that if 2TTa(£,m,f) is a random function such that for fixed (£,m,f) 

 phase values of the form 27ra MOD 2Tr between and 2tt, are equally 

 probable and all phase values are independent, then Equation (2.17) 

 represents an ensemble (collection of all probable) n(x,y,t) for a 

 given S(£,m,f). The random process represented by the ensemble is then 

 a stationary Gaussian process indexed by the three dimensions (x,y,t). 

 Detail discussions of the above can be found in St. Dennis and Pierson 

 (1953 - pp. 289-386) and Kinsman (1965 - pp. 368-386). The fact that 

 a particular sea-way can be considered as a realization of a stationary 

 three-dimensional Gaussian process has been verified. Refer to Pierson 

 and Marks (1952). 



The model in Equation (2.17) as a stationary random process will be 

 assumed in following discussion. 



