272 DYNAMICAL THEORY OF SOUND 



This must be equal to the mean value of pq, where p is the 

 pressure at the outer face of the disk. Hence by comparison 

 we find that p must have the form 



p = p + D cos nt -r sin nt ............. (5) 



The corresponding pressure on the inner face will be 



p = p Q + D cos nt, ..................... (6) 



simply, since no work is done, on the whole, on the air 

 contained in the resonator. 



We may now invoke the action of the external source. 

 If this be such as would produce the pressure 



P = P<> + ~A~ s ^ n nt .................. CO 



at the mouth of the resonator if the disk were at rest, then 

 in the motion which is compounded of that due to the source 

 and that due to the disk no extraneous force will be required, 

 and the disk may therefore be annihilated without causing 

 any appreciable change in the conditions. If </> 2 be the 

 velocity-potential due to the source alone, at the mouth of 

 the resonator, we must have, in this case, 



since (7) must be identical with p =p Q + p^. 



The hypothesis of a rigid disk vibrating in a cylindrical 

 space was merely introduced for facility of conception, and 

 is in no way essential to the argument. The disk may, if 

 we please, be replaced by a flexible and extensible membrane 

 enclosing the aperture of the resonator, and abutting on the 

 external wall in the region of diverging waves. 



Comparing (1) with (8) we see that to a disturbing potential 

 whose value at the mouth is 



< 2 = Jcos(nt-e) .................. (9) 



will correspond a vibration 





