That is, it is supposed that the instantaneous apparent vertical accelera- 

 tion a(t) is given by an infinite sum of sinusoids of all frequencies 

 03 combined in random phase e. The amplitude of each sinusoid is assigned 

 by the acceleration spectrum ordinate a(cjD) . The integral in Equation [l] 

 is not an ordinary integral in the Riemann sense; it cannot be formally 

 integrated. It represents a mathematical abstraction which responds to 

 the basic rules of the calculus and that will suffice for this discussion. 



Using the form of Equation [l], a record of vertical displacement 

 [z(t)] may be represented by 



z(t) = \ cos [to t + e (oj) ] V^Co^do) 



[2] 



If Equation [2] is twice differentiated with respect to time, the result 

 is 



I 



TTS =1 — a>^ cos [q t + e{(a)] n/ z (m) do) [3] 

 dt ^o 



Equations [l] and [3] may now be equated to each other, the result being 

 z(ai) = -? aCco) [4] 



Equation [4] states that the energy spectrum of the waves z (oi) , may be 

 derived from the energy spectrum of acceleration by an algebraic operation. 



The errors that exist in a(cu) due to improper measurement of the 

 true vertical acceleration are communicated to z (oj) . In addition, there 

 are errors in aQa) due to the finite length of record and to the analysis 

 technique. Failure to measure true vertical acceleration, plus drift in 

 the electronics, results in an acceleration spectrum a(cjo) which shows 

 finite energy at co = (Figure 8a) which by Equation [4] would propagate 

 to z (cjd) by indicating infinite energy at o) = . This is overcome by 

 arbitrarily cutting off a (en) at a low frequency where the spectral 



10 



