166 



INTENSITY FLUCTUATIONS 



When the sound intensity is measured well inside 

 a so-called shadow zone, there is usually no self- 

 correlation for step intervals even as short as one 

 second. As illustrated in Chapter 4, Figure 2B, the 

 amplitude, or intensity, varies so rapidly that no 

 correlation can be expected between consecutive sig- 

 nals with the usual keying intervals. However, the 

 same figure illustrates another possibility for treating 

 coherence in a quantitative manner. If we consider, 

 instead of a sequence of signal pulses, the amplitude 

 fluctuation in a continuous signal, we may define the 

 self -correlation coefficient of the amplitude as a func- 

 tion of the continuously variable interval t as follows : 



pW = 



TJo 



A{t)A{t + T)dt -A 



(31) 



r ~ A' 



10 



-2 



-8 



-10 



0.1 



0.2 0.3 



T IN SECONDS 



0.4 



0.5 



Figure 6. Self-correlation coefficient of a 10-sec sig- 

 nal received in the shadow zone. 



one run for sound transmitted into the shadow zone. 

 The appearance of this function is similar to those in 

 Figure 5, except for the enormous change in the time 

 scale. Figure 7 shows the self-correlation coefficient 

 which has been predicted theoretically for the in- 

 tensity of reverberation produced by a square-topped 

 single-frequency signal of length to- The expression 

 obtained by Eckart' for this coefficient is as follows : 



P(r) 



i'-^y 







for |r| ^ io 



for T ^ U. 



(32) 



Hidden Periodicities 



In the preceding section, the coefficient of self- 

 correlation was introduced primarily as a mathe- 

 matical measure of the coherence of the transmitted 

 signal or, in other words, as a measure of the rapidity 

 of fluctuation. In addition, the self-correlation coef- 



1.0 



where the interval of integration T must be large tog 

 compared with the step interval t. Figure 6 shows the 

 self-correlation coefficient which was found during 



Figure 7. Theoretical self-correlation coefficient of 

 the intensity of reverberation from a square-topped 

 signal; 



ficient provides a powerful tool for discovering "hid- 

 den periodicities." A hidden periodicity is essentially 

 nothing but a tendency of the fluctuation pattern to 

 repeat itself with a fixed period, a tendency which is 

 modified by nonperiodic disturbances. Consider, for 

 instance, an ordinary pendulum which is subject to 

 random forces. This pendulum will be moved to carry 

 out periodic swings, but the periodicity will not be 

 strict since both amplitude and phase of its vibration 

 are subject to random changes. But if we were to plot 

 the motion of the pendulum for a long time (large 

 compared with its period), we should find that the 

 self -correlation coefficient will have a maximum (al- 

 though not quite 4-1) for an interval equal to the 

 period of the pendulum and a minimum (although 

 not quite — 1) for an interval equal to one-half the 

 period of the pendulum. Extending the self-correla- 

 tion analysis to longer intervals, we should find 

 another minimum at 3/2 the period, again a max- 



