THEORY OF VIBRATION OF THE LARYNX 209 



The capacity of the lungs in an adult man is capable of being varied 

 from about two to five liters. The av'erage in quiet breathing is 

 about 2.6 liters. The average expiration of air in quiet breathing is 

 about .5 liter. The rate of expiration of air in medium loud singing 

 varies from 40 to 200 cm. ^ sec, the lower values obtaining for trained 

 singers. 



The larynx (see Fig. 1) consists of an irregularly shaped cartilaginous 

 box at the top end of a tube, the trachea, about 12 cm. long by 2 cm. 

 in diameter, leading from the lungs. The larynx contains the vocal 

 cords, a pair of fibrous lips which in vibrating vary the width of the 

 slit called the glottis, between them. The length of the glottis in 

 the adult male averages about 1.8 cm. and in the female 1.2 cm. 

 The width of the glottis varies widely with differing sounds. A few 

 tenths of a millimeter may be considered representative. The tension 

 and separation of the vocal cords are controlled by muscles. 



The principal upper vocal cavities are the pharynx, a space just 

 over the larynx, the mouth and the nasal cavities. The first and 

 second may be varied in size and shape at will, but the effect on the 

 last is controlled only by varying the communicating aperture be- 

 tween it and the pharynx. 



Equations of Motiox of the Larynx 



Fig. 2 shows a cross-section of a model which illustrates the 

 essential details of the larynx in so far as it is necessary for this treat- 

 ment, ^o represents the area of the opening to the trachea. The 

 vocal cords are represented by elastically hinged members of com- 

 bined effective area S-^. By effective area is meant the area of aper- 

 ture which displaces the same Aolume of air as the vocal cords when 

 it moves the distance 50 of the tips of the cords. This area is less 

 than that of the vocal cords. The tips of the vocal cords are separated 

 to form a gap, the glottis, of area Su A positive or up and outward 

 displacement q-i of the vocal cords increases S^. It will be assumed 

 that the air is not appreciably compressed in the neighborhood of 

 the glottis, that is, any tendency to compression is relieved by flow 

 into the trachea or pharynx. 



The pressure in the lungs forces a steady current of air through 

 the glottis. Let the velocity in the trachea of this steady flow be /o 

 and in the glottis /]. Small vibrations of the vocal cords superimpose 

 additional small velocities, Zo and /i, in the trachea and glottis re- 

 spectively. If the instantaneous velocity of the vocal cords be u 

 and it be assumed that they are constrained to move in synchronism 



(/o + H)S, = {h + h)S, -f HS2. (1) 



