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9. Failure of the Absolute Channel 



If it is decided that the absolute transducer has failed, the 

 water surface displacement and power spectrum can theoretically be 

 estimated from the x- and y-axis differential transducers that are 

 deemed as operational. This procedure necessarily assumes that there 

 is only one wave direction per frequency. The record of the absolute 

 transducer has been developed in the time domain to that of Equation 

 (A. 40) : 



n(x,y,t) = — cosCk X + k y - at) 

 2 X y 



so that the power spectrum of the absolute transducer is as: 



2 2 



^nn^"^ " V ^°s^k^x + k^y - at) = |- (4,46) 



and the power spectrum of the x and y differential gauges are 

 correspondingly: 



2 

 S (a) =^k cos^O (4.A7) 



X X 



2 



S (a) = ^ k sin^G 

 W 8 (4.48) 



The power spectrum of the absolute record, then, might be developed 

 from the sum of the x and y differential power spectra divided by the 



