THEORY OF VIBRATION OF THE LARYNX 223 



Now the condition for sustained oscillation is that A = and if the 

 value of o) when this obtains be wo then 



D C 



coo- = -g and '^°^""J' (^^) 



or the condition for sustained vibration in terms of the constants is 



AD = BC. (42) 



In addition to this if use be made of the fact that in an algebraic 

 equation such as (36) the coefficient of p- is the negative sum of all 

 the roots then this coefficient is the real root. Let this be Ao and 

 then (36) may be written 



p^ + p'-^, + ;^coo- + Aocoo- = 0. (43) 



The coefficient of p in (36) is therefore the square of radial frequency 

 at which sustained oscillation will take place and this is seen to be 

 higher than the natural frequency wa of the vocal cords, the difference 

 being increased when the damping of either mesh is greater or when 

 the coupling mutual G is greater. 



It might be noted in passing that (43) is the free oscillation equation 

 for any system which may be represented by a cubic equation and 

 is not confined to the simple larynx. Such an equation always results 

 when there is only one kind of reactive element in one of the meshes. 

 It holds also for the tuned grid circuit. 



The condition for sustained oscillation to be fulfilled for the con- 

 stants may from (42) be reduced to: 



R,R, = G'[^- ij- (44) 



It is rather difficult to place a simple physical interpretation on 

 this formula. The qualitative import of it may however be seen by 

 substituting the values of G and Ku from (13) and (10): 



R\R'i = Li' I]' 



doR/dqo doLi/dqo 



doLi/dqo ,.. 



The first term in brackets is in the nature of a resistance modula- 

 tion constant, a fractional change in glottis resistance per unit dis- 

 placement of the cords, to be designated by r and the second term 

 similarly a glottis mass modulation constant, /. The quantity Lil^ 

 is the momentum of the air in the glottis. This equation is then 



i?ii?2 = {UU)\r - /)/. (46) 



