DYNAMICAL STUDY OF THE VOWEL SOUNDS 109 



using the lower signs in (4) and (4a). Thus in reconstructing the 

 resonator cavities from the vowel data, we must take care to use, for 

 each particular vowel, that pair of solutions (»i, no, or w/, n^') which 

 places the front and rear cavities in correct order for relative size. 

 From the discussion given above of the data on position of the tongue, 

 sections of the cavities, etc., the application of this principle is a rela- 

 tively easy matter. 



The matter of fixing the coupling factor is not so straightforward. 

 For the loosely coupled systems {oo to ar, the vowels on the left leg of 

 the triangle, Fig. 3), it appears that the maximum allowable coupling 

 factors ;u (that is, the values of p. for which the radicals in (4) and (4a) 

 vanish) are so small that it seems reasonable to adopt them forthwith.^ 

 In these cases we have the single solution 



4c0 1^0)2" Kl 



(6) 



In this situation (since the ratio Vo/Vi is fixed if ncjn^- and Ko/Ki are 

 fixed) all the quantities Vi, F2, Ku K2 are determinate as soon as we fix 

 either K\ or Vi + F2. The practice followed will be to set a value for 

 Ki and check this by noting the value of Vi + F2 to which it leads; 

 thus by trial and error the most reasonable values for the resonator 

 constants for the loosely coupled systems can be found. Incidentally, 

 we shall note in all these cases that the solution requires Vi to be larger 

 than F2. 



The vowel short a marks the transition between the loosely-coupled 

 systems already considered and the closely-coupled systems for the 

 sounds from short e to long e on the right leg of the triangle. Short e 

 is also the first vowel sound of the series to have a high frequency 

 resonance of frequency greater than 1,500 cycles. We might be in a 



^ These values of the coupling factors are not inconsistent with the diagrams of the 

 mouth cavities shown in Fig. 2. Aside from complicating the calculations, the effect 

 of taking still smaller values for f^ (keeping Ki constant) is merely to lower Vo in 

 proportion as K2 is decreased. For example, taking m = M max. for the sound aw, 

 we arrive at the solution Vi = 119 cu. cm., V2 = 22 cu. cm., if Ki = 2.1 cm. as given 

 in Fig. 5. Now if we take n = ^ fj, max., we get Fi = 121, F2 = 10 cu. cm. Thus 

 no great change has been made in the total volume Vi + V2, except that we get a 

 value for V2 which seems unreasonably small. The most satisfactory course, in the 

 case of the loosely coupled systems, is to use the maximum allowable coupling factors. 



