MINIMUM AUDIBLE INTENSITY OF SOUND. 



431 



If the force is distributed over a solid plane of infinite extent, double 

 the energy is emitted, and 



dp c~ lkr 



V" 



-US 



dn r 



dS 



If we consider the reaction of the air upon the vibrating plane, the rate 

 of emission of energy will be 



dw r r r 



-(U=JJt- 



dip 



dn 



where 8p is the pressure variation. If a is the density of the air, 



dip 



V C C dip dp 



~ = - a J J ^7f f/lS 

 ., fa C C 



= — ikva ~r I I <p do 

 dn *J J 



dip 

 if -7- is constant over the plane. 

 an 



Then 



dW 



dt 



II 



-i'A-r 



dS-dS' 



This is to be solved for the case where a circular rigid portion of the 

 plane is vibrating with simple harmonic motion, while the remainder 

 of the plane is fixed. 



The evaluation of this expression for the case in question has been 

 shown by Rayleigh 20 to be 



c/ir ikva 



dt 



IT 



dp 

 dn 



tir-a- 



2^*i(2*a) -—, 



1 



Ji(2ka) 

 ka 



where a is the radius of the disc, and J \ and K\ are Bessel's functions 

 of the first and second kinds respectively. 



If .r is the displacement of the disc at time /, and a is its amplitude 

 of vibration, 



20 Theory of Sound, Vol. II, Art, 302. 



