HARMONIC ANALYSIS. 



91 



nates in such a " harmonic plot" should not be connected unless by straight 

 lines to guide the eye. Smooth curves drawn through harmonic plots 

 have led two investigators to discuss a "curve of resonance" for each 

 vowel, whereas such a concept is absolute nonsense. To famiharize the 

 mind with the practice of harmonic analysis I will give some elementary 

 illustrations, using only 12 ordinates and one decimal place. By consult- 

 ing the patterns at the back of the book the reader can readily perform 

 the simple calculations mentally. Actual work 

 with speech curves is, of course, much more 

 laborious, requiring usually 36 or 72 ordinates 

 and two to four decimal places; such an anal- 

 ysis of a single wave usually requires the work 

 of one person for two or three days. 



A curve with the ordinates 0.0, 5.0, 8.7, 

 10.0, 8.7, 5.0, 0.0, 5.0,-8.7,-10.0,-8.7, -5.0, 

 0.0 yields the neighboring table of products. 

 The analysis gives ai = 0, 6, = 10.1, with the 

 other coefficients = 0. The curve is therefore present only as the first 

 harmonic in the sine series with a phase factor = 0. Assuming 36 as the 

 length of the period and neglecting the tenths of a unit, we have as the 



which is the equation according to 

 The curve itself is given in figure 67. 



equation of the curve y= 10. sin t, 



oO 



which the ordinates were calculated. 



A curve yields ordinates and products as 

 in the adjacent table. The analysis gives C = 

 12, Oi = 0, 6i = 10, and all other coefficients 

 = 0. The curve is therefore the same as the 

 preceding one except in having its axis at 12 

 above the X-axis. This illustrates the prin- 

 ciple that the addition of a constant to the 

 ordinates of the curve does not affect the coef- 

 ficients for the sinusoids. 



The curve in figure 69 yields a table of 

 products like the first one given above, except that three values are 

 removed from the beginning and placed at the end. The analysis gives 

 Oi= 10 and all other coefficients =0. The curve is therefore 



,-10.cos|<-10.sin(|<+^). 



This case illustrates the fact that values can be transferred from one end 

 of the table to the other without altering anything except the phase. It 



