f' (z, t)= /^ \ A^W^ (z) cos (o'^ t+ B^WJz) sin w ; H 

 «=1 ' 



N 

 = y [CW^(z) cos(a>'t ~a„(z))] 



with 



n=l 



OJn = . f^ 



277 



n,Q='~T 



n.O 



U 

 1 — — COS 



(44) 



(45) 



The power spectra for infinitely long series of the type given in equations 43 and 

 44 are, respectively 



S(cD— ajn) + S(a) + ojn' 



(46) 



P((y',e)=l/4 y{C^W,^f2))2 [S(aj-aj„j+Sfaj+ a;„^j (47) 



n = 1 



In reality ( see equation 46) , the energy of the internal waves is equal to the 

 square of the sum of the mode amplitudes. The power spectrum of a thermistor 

 chain record leads, however, to the sum of the square of the amplitudes of each 

 mode. The total energy obtained by integration over the whole range is, accord- 

 ing to equation 46, given by 



(z) = 1/2 



Yew 



(z) 



Equation 47 yields, however. 



N 



and therefore 



P' , Jz) = 1/2 y C ^W ^ (z) 



total / ■ n n 



n = 1 



P'tota/^^ ^ ^total^^^ 



(48) 



(49) 



(50) 



That is, the total energy or variance recorded with the thermistor chain is 



different from the actual variance or energy in the ocean. 



Example: 



If 



i (z) = sm ,C = i-ir/n 



20 



