122 THE STUDY OF SPEECH CURVES. 



decreasas more or less rapidly according to d, the factor of suddenness. 

 When the puff is very sharp (H large) this element will not be prominent 

 in the result and the impressing vibration will appear as a momentary 

 modification of the second element at the start. 



The second element is the vibration of the cavity aroused by the puff 

 and left to itself; its period does not depend on that of the puff. When 

 the first element fades, this alone is left. 



The two extremes of the predominance of the impressed (or glottal) 

 period or of the natural period of the cavity are found in the curves of 

 some musical instruments and of some vowels. In musical instruments 

 whose source of tone consists of smooth puffs (3) continued for some time, 

 the vibrations are given by equation (9) ; but the second element is present 

 only at the very beginning of each tone and generally can not be detected 

 in the curve; the tone then consists exclusively of vibrations represented 

 by the first element. Such are the curves for the clarionet, cornet, and 

 saxophone given in figures 89, 90, 91. The other extreme appears strik- 

 ingly at the beginnings of some vowels where the puffs are sudden (11) 

 and so far apart that the effect of one dies away before the next occurs, 

 as in the curve for [ai] "I," Depew, line 110; here the solution is given by 

 equation (12), but the first member hardly appears, and the curve seems to 

 consist solely of the second element with the period of the resonance cavity. 



The equation (12) is sufficient to express the action of a single glottal 

 puff of greater or less sharpness on a single vocal cavity; for the action of 

 the puff on a system of cavities we would have to take the svim of a set 

 of such equations where the factor of system (period), the mass, and the 

 friction for each one is considered. When the complicated structure of 

 the glottal lips is considered, we must admit that the puff may be of such 

 a form that 6, the factor of suddenness, has to be replaced by a more 

 complicated expression, or that the puff is to be expressed by a sum of 

 frictional sinusoids with various values for a, 0, p, and q. In the absence 

 of any definite information we may, however, for the following remarks 

 use d as in (12). The complete expression for the action of a single puff 

 on the set of vocal cavities then becomes 



V = ^> ====^=^=^=^^^^^^=. .e" ^' .sin (pt — q) 



^ZL. m \/ {k^ - p^ + e"" - 2dsy + ^p%0 - eY 



+ ^ R.e-'* .sin (i ' P - £^.< - q') (13) 



where the sum is to be taken over as many elements as required. This 

 is the fundamental equation for each wave of a vowel curve. The various 

 waves differ in the values given to the different letters. 



