ANALYSIS INTO FRICTIONAL SINUSOIDS. 



103 



eo 



sa 



■»o 



10 



60 



so 



40 



30 



20 



10 



_L_L 



To get some idea of the degree of accuracy of the results of harmonic 

 analysis applied to a frictional sinusoid, and also of the error introduced 

 by using the wrong factor of friction, the curve in figure 94 was submitted 



to harmonic an- 

 alysis into simple 

 sinusoids (£ = 0) 

 and into various 

 systems of fric- 

 tional sinusoids 

 with £= 0.002, 

 0.005, 0.010. The 

 results are given in 

 detail in the table 

 on page 102. The 

 numbers 12, 24, 36 

 give the number 

 of oidicates used. Frcm these results several conclusions 

 can be drawn. The curve in figure 94 can be analyzed 

 into simple sinusoids with the amplitudes given in the 

 columns headed £ = 0; or it can also be analyzed into 

 frictional sinusoids with various factors of friction. If 

 we know nothing whatever about the original composition 

 of the curve, one mode of analysis will be just as good as 

 another. The equation according to which the cuive was 

 obtained was 



-o.oo8( 2n 

 y = l0.e .sin —.t. 



1 2 3 4 5 6 7 8 9 10 II 12 



Fig. 95. — Harmonic plot to 

 figure 94 with £ = 0. 



t 2 3 4 5 6 7 8 9 10 II a 



Fio. 96.— Harmonic plot to figure 94 

 with £ = 0.002. 



100 

 90 



SO 



70 

 60 

 DO 

 40 



cs 



20 



36 



I 2 3 d 5 6 7 8 a 10 II 12 



Fig. 97.— Harmonic plot to figure 94 

 with 6= 0.005. 



We find that analysis with any other 

 factor than £ = 0.005 gives results that 

 differ widely from the original equation. The 

 difference increases with the divergence in 

 the factor of friction. We therefore con- 

 clude that the analysis must be made with the appropriate factor of 

 friction if it is to give the way in which the curve was obtamed^ 



These considerations are of great practical importance. As will 

 be shown in the following sections, the vibrations in speech always have 

 large factors of friction. The treatment of the curves by simple har- 

 monic analysis-the only method that has hitherto been tried-furnishes 

 results that are so wrong as to be utterly misleading when used to indi- 

 cate the manner in which the vibrations were produced. In such a case the 



