CHAPTER VII. 

 ANALYSIS INTO FRICTIONAL SINUSOIDS. 



When the finger is laid softly against the side of the prong of a 

 vibrating fork the movement dies away with a rapidity depending on the 

 amount of friction of the prong against the finger. The curve traced in 

 such a way may be called a " frictional sinusoid"; curves with the same 

 period, ampUtude, and phase, but with different degrees of friction, are 

 shown in figure 92. 



Fig. 93. — Frictional sinusoid 

 showing logarithmic decre- 

 ment. 



Fig. 94.— Curve compounded of 

 frictional sinusoids. 



The vibrations of the voice in speech are 

 — as will be conclusively shown in the follow- 

 ing chapters — composed exclusively of frictional 

 sinusoids and not of simple sinusoids, as has 

 hitherto been assumed. Can a method of anal- 

 ysis into frictional sinusoids be found? Does an 

 analysis into simple sinusoids give false results for 

 the vowel curves? The purpose of this chapter is 

 to answer these two questions. 



When a line is drawn along the tops of the 

 vibrations of such a tuning-fork curve (figure 93) 

 it is found to bend downward with greater or 

 less rapidity and to gradually approach the zero 

 line. This curve — which we may call the "curve of friction" — has a 

 definite form indicated by the expression a.e~" where a is the amphtude 

 of the vibration without friction, e = 2.71828, s is a figure indicating 

 the amount of friction, and t is the time from the start. The number 

 £ may be called the factor of friction (it is not the same as the coeffi- 

 cient of friction used in physics, but depends directly on it). The entire 

 expression e'", which modifies the amplitude, may be termed the frictional 

 element of the curve. 



101 



Fig. 92.— Sinusoid with dif- 

 ferent factors of friction. 



