108 BELL SYSTEM TECHNICAL JOURNAL 



we may rewrite (1) as 



CO 



iwi2(l + m) + W2] + wi%2' = 0. (la) 



If it were not for p., we could determine from (la) the ratios Ki/Vi and 

 K2/V2 from the known data coi, coo. As will appear later, we can make 

 reasonable assumptions with regard to fx; but it is obvious that even 

 then two further assumptions are required to fix Ki, Ko, Vi, V2 in 

 absolute value. These we supply by assuming a fixed total volume 

 Vi + V2 for the system, and a certain conductivity Ki for the mouth 

 opening, which is the most easily observed element of the system. 



Proceeding in the manner outlined, it will be possible to take the 

 series of the vowel sounds and fit to each sound a doubly resonant 

 system such that the whole series forms a more or less coherent group. 



The following is an outline of the type of calculations required. If 

 we write, from (la), 



^1^(1 + /i) + fli^ = W]- + C02^ 

 «l^W2^ = CO1-CO2'', 



and eliminate n-2~, we have 



(3) 



nr\ _ cor + co-i^ ± Vlcoi + wo-)- — 4(1 + ju)corco2- ... 



nn ~ 2(1 +m) ' ^ ^ 



also, if we eliminate wr, we have 



W9-I _ cor + C02" T -yliwr + co2~)'^ — 4(1 + a^)coiW 



/2 /2s 



(4a) 



In these equations it will be noted that («)-, ni) {fii'^, fii^) each repre- 

 sent possible combinations of simple resonators which will give, on 

 coupling, the observed frequencies coi, coo. In other words, for given 

 (comparable) conductivities Ki, K2, of the two orifices, the outer 

 resonator may be small, and the inner resonator large (Fi < F2), 

 corresponding to the (separate) natural frequencies 



,_ ,Ki _ coi'^ + C02^ + V(coi-^ + C02^)^ - 4(1 + M)a^rW- 

 ''''' ~ ^~ V, 2(1 + m) 



, _ „X'> cor + coo^ — V(cor + coo-)- — 4(1 + m)<^iW (5) 



712 = C" Y^ - 2 ' 



wr > Wo-; 

 or, if V\ > V2, we must apply the other pair of equations 



ill' = c2 -— < W2' = c^jT (5a) 



