the function that would have been obtained had the wave pole been stationary. 



Let ^*(t) be the recorded wave height and let /^ (t) be the true v/ave 

 height. Then equation (8. 30) cs,n be obtained. 

 (8.30) ^ ^'^(t) = 91 {^ - z(t) 



It states that if the pole were stationary, (^(t)*. 0) , rf-'^it) v/ould equal 

 9? (■£) S-^id that if the wa.ve pole followed the v/ave profile exa.ctly, 

 (?? (t) = z(t) ), '/[ =:=(t) would be zero. 



From equations (8. 29) and (8. 30) the result is ths.t 



<S-31) ^.(t) = 



'" D(ilj / ^o c°^ ^^t-l 



[K(t.'./^i-o)]0(^^) 



D(|i) 



ao sill [JLt 



where D(]j.) is the denominator of the terms in (8. 23). 



If the wiwz jjole rosponcse is linear, and if the free surface ca.n be re- 

 presented by a itatioaary Gaussian process with the jpsctrum [A(]j,)]'", it 

 then follows tha.t the spectrunn of the recorded function is related to the 

 spectrum of tlic waves by equation (8. 32). The term i:.. '. rackets is just the 



sum of the 3qua.res of €ae two coefficients in (8. 31) 



(8.32) 



^[(iWtx^)^ - 1 + 0(H.)]^" + K^(!..'-/piJ^' 



[((.../M - 1]^ + K (|i./tJ-o) 



[A(ix)]' 



Over the rr.r.ge of frequencies expected in the W3,ve record, the numer- 

 ator of thi,:: expression is a,lways less than tlie denorr!ina.tor . Therefore the 

 vv^aves recorded by the insti'ument will be too low £,nd 'cae spectrum, computed 



from the v/ave record v/ill have to be amplified to get the correct spectrum.. 



"> 2 



The final equation given that [A*(|i)]'' is known permi ts one to find [A([j.)] . 



78 



