WAVE ANALYSIS IN REFERENCE TO VOCAL ACTION. 119 



Here the amplitude a depends on the energy received by the system; 

 the period 



depends exchisively on the elasticity and the mass of the system itself; 

 the factor of phase q can be made zero by proper choice of coordinates. 

 Putting _ 



k = Ji. and g = 0, 



we have 



y = a. sin kt. (3) 



This is the simple sinusoid discussed above (p. 84). 



In speech vibrations the friction can not be neglected. For small 

 movements the friction may be considered as proportional to the velocity 

 and opposed to the movement. Adding the term for friction the equation 

 of motion becomes 



of which the solution is 



The period 



is thus dependent not only on the elasticity and the mass, but also on 

 the friction; it is longer than the period without friction. The amplitude 

 a depends on the energy imparted to the system. The frictional element 



e *"• acts to steadily decrease the amplitude. 

 For ' _ 



k = J^, ,= ^, and5' = 0, (6) 



we have 



y:= a.e- ". sin (i/F^T^O- C^) 



This is the frictional sinusoid considered above (p. 104). 



