﻿300 On a Quantitative Theory of tlie Telephone. 



of equilibrium. The frequency of the resulting vibration is 



1 /fl0 6 x 7-5-I Kjl1 



With the- aid of a special electric maintenance the plate 

 may be made to speak on its own account. The frequency so 

 found, viz. 896, corresponds undoubtedly to a free vibration, 

 but it does not follow that the vibration is the gravest of 

 which the plate is capable ; and there were indications point- 

 ing to the opposite conclusion. 



As it is almost impossible to form an a priori estimate of 

 the amplitude of vibration (x) when the frequency of the 

 force is in the neighbourhood of any of the free frequencies, 

 I will take for calculation the case of frequency 256, which 

 is presumably much lower than any of them. Under these 

 circumstances an " equilibrium theory " may be employed, 

 the displacement coexisting with any applied force being the 

 same as if the force were permanent. At this pitch the 

 minimum current recorded in the table* is 8*3 x 10 -7 amperes; 

 so that by (12) the maximum excursion corresponding 

 thereto is given by ^ = '080 x 8*3 x 10 _7 = 6*8 x 10 -8 centim. 



The excursion thus found must not be compared with that 

 calculated formerly! for free progressive waves. The proper 

 comparison is rather between the condensations s in the two 

 cases. In a progressive wave the connexion between s and 

 v, the maximum velocity, is v = as, where a is the velocity of 

 propagation. But in the present case the excursion x takes 

 effect upon a very small volume. If A be the effective area 

 of the plate, and S the whole volume included between the 

 plate and the tympanum of the ear, we may take s = Ax/S. 

 This relation assumes that the condensations and rarefactions 

 are uniform throughout the space in question, an assumption 

 justified by the smallness of its dimensions in comparison 

 with the wave-length, and further that the behaviour is the 

 same as if the space were closed air-tight. It would seem 

 that a slight deficiency in the latter respect would not be 

 material. 



For the numerical application I estimate that A=4 sq. 

 centim., S = 20 cub. centim. : so that with the above value 

 of x 



5=1-4 xlO" 8 , (13) 



s being reckoned in atmospheres. 



* Supra, p. 294. 



t Proc. Roy. Soc. vol. xxxvi. p. 248 (1»77). 



