460 BELL SYSTEM TECHNICAL JOURNAL 



The ratio of (6) to (7) in amplitude is 



4[1 - JoikR):\ 



(8) 





{kRy 



which is plotted as db loss vs. kR in Fig. 2. 



There are two approximations involved in this analysis, both 

 involving equation (3). One is that for p (which should be constant 

 for a given r), can be taken the average of the actual pressures around 

 the circle of radius r. The other is that the shape of the static deflec- 

 tion curve represents the actual shape up to the highest frequencies 

 of interest in (8). 



APPENDIX II 



Assuming (1) that the air particles, in the plane of the entrance to 

 the cavity, all move in phase with equal velocities Vi which are normal 

 to that plane, and (2) that the impedance per unit area of the micro- 

 phone diaphragm is large compared to pc, where p is the density of 

 air and c is the velocity of sound, the following three relations hold: 

 From the theory of plane wave propagation in a tube, 



(1) p2 = pi cos kl — ipcvi sin kl 



where pi is the pressure in the plane of the entrance to the cavity of 

 depth / and p2 is the pressure at the diaphragm. Also, the input 

 impedance per unit area of this closed cylindrical tube is 



Pi 



(2) ^ = — ipc cot kl. 



Now pi is equal to the pressure P that would exist at the opening if 

 the air particles were held stationary, diminished by the drop in 

 pressure due to their motion and the consequent radiation from the 

 opening. In symbols, 



(3) pi = P - pc{a + ih)vu 



where (a -\- ib) is the radiation impedance coefficient given by Raleigh ^ 

 for the case of a circular piston in an infinite wall : 



, Ji(2kR) 

 ^ = ^--jR~' 



. = ^E(2n+l)(^y(-4.^i^^)-. 



■> "Theory of Sound," Vol. II, § 302. 



