120 THE STUDY OF SPEECH CURVES. 



When vibrations are aroused in the vocal cavities by a sudden blow, 

 they can be treated as if aroused by a momentary impulse and then left 

 to themselves. The equation of motion is then for each cavity the one 

 just considered. For a system of cavities struck by a sudden blow we 

 may treat the effect as the sum of the effects on each element of the 

 system, whereby the constants for each element are determined while it 

 is in connection with the other elements and not alone. The result is 

 then a vibration of the form 



y^^a.e-'.sin{V^'^-s\t~q,) (8) 



The ampHtudes a,, Oj, Os, . . . , the factors of friction £„ £■,, 

 s,, . . . , the factors of system ki, k^, k^, . . . , and the phases 51, 

 q^, Qi, . . . , have to be determined for each cavity, not as an isolated 

 one but in connection with the rest of the system. 



Let us suppose, now, the glottis to produce sinusoidal vibrations in 

 the air current. When the resonating cavity is acted upon by a main- 

 tained force of the character a. sin pt, the equation of motion becomes 



m-^ = —sy — b^-+ a.sin pt. 

 de ^ dt ^ 



of which the solution — after substituting e and k as in (6) — becomes 

 y = ^ — sin {pt-q)+R.e-" .^m {V k"" - e" t — q'). (9) 



The first element is the "forced vibration"; it has the frequency of 

 the glottal vibration impressed on the resonating cavity and the amplitude 



^ (10) 



m 



V {k-^ - py + 4p' £* 



which depends on the amplitude a of the glottal vibration, on the mass 

 m of the air in the cavity, on the friction £, on the frequency p of the 

 glottal vibration, and on the difference between the frequency k of the 

 cavity and that of the glottal vibration. 



The second element is the vibration of the air in the cavity aroused 

 and left to itself; its frequency has no relation to that of the glottal vibra- 

 tion, but depends solely on the cavity and the friction. 



