K^ = tan-^ ^ (39) 



These relationships allow definition of an individual wave component in terms 

 of a wave amplitude C^^ and initial phase <j)^^ . These relationships are 

 useful when defining a wave field from a spectrum as is done below. 



31. The spectrum or variance spectral density S^^ of the k'^'^ frequency 

 component of time series x^^ is found by multiplying Equation 32 by its 

 complex conjugate, dividing the result by 2, and dividing again by the 

 frequency bandwidth represented by element k . This frequency bandwidth Af 

 is defined as 



The set of mathematical operations outlined above yields 



1 



2Af 



(41b) 



= ^ (41c) 



= ^ X^X^ (41d) 



where the asterisk (*) denotes complex conjugate. In the above derivation, 

 Equation 38 has been used in the step from Equation 41b to Equation 41c. 

 Equation 41c is essentially the definition of spectral density since the 

 variance of a sinusiodal wave of amplitude C^^ is z'-'xk > ^'^'^ ^^e variance 

 density is the variance divided by the frequency bandwidth represented by the 

 single sinusiodal wave. Equation 41c also shows how the k*^^ spectral element 

 relates to the Fourier series representation of the process. Equation 41d 

 comes about by using the right side instead of the left side of Equation 32 in 

 Equation 41a. Equation 41d shows how the spectral density of element k is 

 related to element k of the DFT. 



32. Clearly, if one knows the spectral density S^^ , the time step 

 At , and the number of time steps N in a time series, the resolution 



17 



