MINIATURE CONDENSER MICROPHONE SYSTEM 459 



and hence that 



p = P cos [(co/ - kR) + kr cos dj 



= P[cos (co/ - kR) cos {kr cos 9) - sin (co/ - ^i?) sin {kr cos 0)]. 



The average pressure is 



1 r" 



P COS (co/ - /^i?) 



I COS {kr COS 0)rf^ 



P sin (co/ - /^i?) 



(sin {k 



r cos 



Here the seccjnd integral is zero, while the first may be written 



/)av = ^^ COS (co/ — kR) I COS (^r COS 6)dd 

 ^ Jo 



p /^'r/2 



= ±11 COS (co/ - y^i?) cos (^r sin e)dd 



^ Jo 



and is readily identified as a Bessel function. 



(4) ^av = P COS (co/ - kR)Jo{kr). 



Substitution of (4) in (3) gives an approximation to the total effect 

 (on the central displacement) of these average pressures acting over 

 the whole diaphragm. 



(5) 

 Now 



and 



so that 

 (6) 



^c 



P COS (co/ — 



r J„{kr)rdr = -yJiikr) 

 Jo 



r^Mkr)dr = ^,li -J,{kR)2 



P cos (co/ — kR) p. J /, D\-i 

 lo = -^ [1 - /o(^^)]. 



The displacement when R is small compared to the wave-length, i.e. 

 y^ = Ois 



c / PR^ cos co/ 



(7) {.' = -t; 



