DYNAMICAL STUDY OF THE VOWEL SOUNDS 107 



In Rayleigh (II, p. 191, eq. 12) it is shown that the natural frequen- 

 cies of a double resonator of the type described are the roots coi, w^, of 



aj"* — (jr{ni~ -\- n^} + «i2^) + Mr«2" = 0, (1) 



= (^ A/TT > the natural frequency of the outer resonator, with inner 

 orifice closed ; 



ni = <" \ /TT , the natural frequency of the inner resonator alone; 



in which 



Wl 



Wl2 = c 





and c is the velocity of sound. Equation (1) is easily obtained by 

 writing the equations of motion of the system, for zero applied forces 

 and zero damping, and placing the determinant of the coefificients of 

 the amplitudes or velocities equal to zero.^ (This is equivalent to 

 placing the driving point impedance, as viewed from the front orifice, 

 equal to zero.) If nn = (the case of a very constricted inner orifice), 

 the roots of (1) are simply ni, no- 



We neglect damping in the system in order to get an easily-managed 

 solution for the natural frequencies. Damping arises in two ways : (1) 

 from sound absorption by the soft (tissue) lining of the cavities, and 

 (2) by radiation from the mouth. Both are very variable, that due to 

 radiation particularly so because of the considerable change in size 

 of the mouth opening from one vowel sound to another. A great deal 

 can be learned of the mechanism of the system by studying only the 

 natural frequencies, and although it is not entirely impracticable to 

 solve the problem with an allowance for radiation damping, we shall 

 ignore this here. 



The general procedure in this study will be to take as known from 

 the vowel spectra the actual natural frequencies coi, co2 of the system, 

 and to find the most reasonable values for the four quantities Ki, K2, 

 Vi, F2, in order that these natural frequencies may result. We thus 

 reconstruct the hypothetical resonator, or throat-mouth system which 

 produces the vowel sounds. If we take 



W12 = c Jy = wi Vm, ('^ ^ '^i ' ^^^ 



" A typical solution of a double resonator problem is given in the author's "Theory 

 of Vibrating Systems and Sound," Van Nostrand (1926), pp. 59-64. The double 

 resonator as a sound amplifier is discussed by E. T. Paris, Science Progress, XX, No. 

 77 (1925), p. 68. 



