C,9 • SPECTRAL ANALYSIS IN ONE DIMENSION 

 first consider 



1 r^ 



a(K, Z) = Pi- / u{x)e'^dx (9-1) 



^TT J -X 



and then try to adopt a suitable limiting process as X -^ oo . In fact, we 

 want to consider first the amplitude not at k but associated with a finite 

 range of values of k. We integrate Eq. 9-1 between k and k -f A/c, obtaining 



LA{k,X) = j^^^^ a{K,X)dK 

 1 P w(a;) 



^ , e'^^c^xFe^^^*)^ — 1] 



2t J -X X 



Here, we may take the limit as X -^ oo , and obtain 



AA(/c) = ^ / ^ e"^dx[e'^^''^- - 1] (9-2) 



Zir J -co X 



since the integral is now convergent. 



We now form the expression for the measure of energy AA{k) • AA*(k) 

 and calculate its statistical average. Then 



hr^j- 



AA(/c) • AA*{k) =±-i — [6^^^"^== - l]dx 



u^R(x' - a;)[e-^(^''^^' - l]dx' 



where R is the statistical correlation between u{x) and u(x'). The inner 

 integral can be transformed by replacing a;' — a; by ^. Then it becomes 



and we obtain 



AA(k)AA*{k) 



n''' C"^ dr /"" p-«f 



= ^ ^ [ei(A.)x _ 1] / _1^ i2(^)d^[e-«^'')--«^'')? - 1] 



We shall now divide both sides by Ak and replace {Ak)x by a new varia- 

 ble f. Then we obtain a measure of the "density of energy": 



AA(k)AA*(k) _u^ r dt ^^,, _ ,j ^-^ i^(^)d^[e-r-..(.. - 1] 



A/c 4x2 7_=, f ^ V-= f + KA/c) 



It is easy to see that the right-hand side has a limit as A/c -^ 0. We there- 

 fore have the interesting situation that AA(k) • AA*{k) is of the order of 

 A/c and not of the order of (A/c)^. Let the Hmit be denoted by Fi{k)/2. Then 



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