104 THE STUDY OF SPEECH CURVES. 



use of a large number of members — as the table clearly shows — involves 

 merely an increase of labor without any essential increase in correctness. 

 The only advantage of a large number of members occurs when the correct 

 factor is used ; then the number of members is settled by the number of 

 tones to be found. 



To show the application to a compound curve, the wave in figure 94 

 was analyzed with the assumptions e = 0.000, 0.002, 0.005. The results 

 are given in the harmonic plots in figures 95, 96, 97 with the ordinates ten 

 times enlarged. It is clear that the curve is most simply represented by 

 the results of the analysis with £ = 0.005, according to which its equation is 



-0.006< 9 -0.005/ 9 -0.005» 9 



t/=10.e .s'm—t+b.e .sm—t+b.e .sin_/. 



3(j 18 12 



This is the equation according to which the curve was obtained. The 

 reader who does not wish to go into the detailed treatment may pass to 

 the following chapter. 



In a frictional sinusoid 



y = a.e-".sm{—t-q), (1) 



y is the elongation at the moment /, T' is the period, a the ampUtude 

 which would be present in the case of no friction, and £ is a factor depend- 

 ing on friction. The actual amplitude a.e~" = a/e" steadily decreases. The 

 period T" is longer than T for the same curve without friction; the rela- 

 tion is 



The difference is so small that in practical work T may be used for 

 T'. Using the form of equation (6) of Chapter V the frictional sinusoid is 



y=a.e~".sm{]/p-s\t). (3) 



Let the function y be built up of a harmonic series of frictional sinu- 

 soids with the same factor of friction, thus, 



9_ 2;r 



y=ai.e-".cos-^^t + aie-".cosj^,t= ... (4) 



+ 6i.e-".sin —t + 62.e-".sin-— -/ + . . . 



