WAVE ANALYSIS IN REFERENCE TO VOCAL ACTION. l21 



Owing to the factor of friction in the second element, it rapidly dies 

 away, leaving only the first element. The curve, when this occurs, is a 

 simple sinusoid indefinitely maintained. The case may be illustrated by 

 the vibration of a brass resonator responding to a tuning-fork indefinitely 

 kept in vibration. 



As we have already seen (p. HI), the vibrations 

 produced in the glottis are puff-like and only rarely 

 approach the simple sinusoid form. We can conven- 

 iently represent a puff as a simple sinusoid modified by 

 a large factor of friction, that is, by the equation 



Fio. 111.— Sinusoid with 

 large factor of decrement. 



L 



y = a. e~". sin pt. (11) 



The quantity 6 is analogous to the factor of fric- 



tion £; it may be called the " factor of suddenness" or 



Fig. ii2.-sinusoid with " factor of decrement," as we are not at present con- 

 very large factor of cemed as to how the force acquired this characteristic 

 decrement. ^^^^ Figurcs 111 and 112 show curves with a = 



100, r = 36, and d = 0.050 and 0.100 respectively; the curve without the 

 factor of suddenness has the same period as the curve in figure 67, but 

 an ampUtude ten times as great. 



We have now to consider the effect of a single element (11) of a puff 

 on a single vocal cavit)'. 



The effect of a force of this kind (11) on a vibrating system with 

 one degree of freedom is given by 



m^ = -sy-P'^+ a.e- "' .sin pt 

 dt^ at 



of which the solution— after substituting k and £ as in (6)— is 



V = ^ .e- " .sin {pt - q) + 



^ mi/ (k-' -p' + d'- 2d£y + 4p' {d - ey 



R.e-'^.sin (V k' — e\ t-q'). (12) 



The first element is the impressed vibration; it has the frequency p 

 of the impressed force (that is, the period of the glottal puff) and an 

 amplitude depending on the ampUtude a of the impressing vibration, 

 on the mass m of the system, on the periods p and k, on the suddenness 

 of the impressing vibration, on the friction of the system and on the 

 difference between the last two factors. The amplitude at the start 



