222 BELL SYSTEM TECHNICAL JOURNAL 



vibration considered as the free oscillation of the system. The de- 

 terminant (31) (or 30) is then used to determine the free frequencies 

 and decrements of the system. The method is as usual to solve for p 

 in the equation 



D = {). {32,) 



To simplify the demonstration the simple larynx without the load of 

 the air chambers will be considered. Taking D of (31) then and 

 expanding: 



^^L,Lo + p\L,R, + L.Ri) + p\UK. + R,R. + G^) 



+ p{R,K, + GK^,) = 0. (34) 



If this be divided by Z1L2 and the uncoupled decrements and natural 

 frequency defined : 



-^1; ;77- = Ao; -7- = coo-, (35) 



= 0. (36) 



2.Li\ 2.L11 Li 



then 



p"- + p\l^, + 2A2) ^ pi coo- + 4A,A2 + -^ ) 



One of the roots of this equation is zero and another is negative 

 real since all coefficients are positive. This root is therefore the decre- 

 ment of a mode of non-vibratory motion. The remaining two roots 

 may be real, imaginary or generally complex, of the form 



Aija,. (37) 



If it is found that A = 0, then an oscillation once started will be 

 sustained. If A be negative then any existing oscillation must sub- 

 side or if A be found positive then an impulse will start an oscillation 

 which of itself increases in amplitude to a point where its violence 

 modifies the constants to such an extent as to make A vanish, leaving 

 a sustained oscillation, or negative leaving the oscillation to subside 

 to a lower amplitude or completely if sufficient permanent changes 

 have been made. 



If now (36) be written 



Ap-' + Bp~ + Cp + D ^ Q (38) 



and the tirst root (37) be substituted, two equations result, one from 

 the real and the other from the imaginary terms, as follows: 



^A(A- - 3co-) + 5(A- - CO-) + CA + /^ = 0, (39) 



^(3A- - CO-) + B{2^) + C = 0. (40) 



