finite in equation (8, 9). This term, can have negative values which can make 

 the L's and the U's come out negative. The L's in particular can be nega- 

 tive quite frequently because of the operation of the above term on the esti- 

 mated spectrum. The purpose of the smoothing filter is in part to eliminate 

 as much as possible some of the negative values. Usually the negative values 

 are quite small and do not materially affect the analysis. 



In time series theory in general, the spectrum is usually defined so 

 that an integral over a given frequency band represents that contribution to 

 the total variance of the process being studied made by the frequencies in 

 that band. In ocean wave theory, another convenient way to define the spec- 

 trum is so that an integral over a given frequency band represents the sum 

 of the squares of the amplitudes of those simple harmonic progressive waves 

 which lie in that frequency band. This is the definition used by Pierson 

 [1955] and Pierson, Neumann and James [1956]. The E value thus defined 

 is equal to twice the variance of the process under st\idy. 



Equations (8, 6) through (8.14) and equations (8. 18) and (8, 19) are de- 

 rived with ti^e definition of the spectrum used in ocean wave theory. Equations 

 (8.15), (8.16), (8,17), (8.20), (8.21) and (8, 22),have been derived in terms 

 of variance. To place all equations in terms of ocean wave theory equations 

 (8. 15) and (8. 12) should be multiplied by 2 on the right hand side and then 

 all results would be obtained in terms of E values. 



In what follows, all results will be discussed in terms of variances and 



covariances as far as the «4irectional spectra are concerned except that 



71 



