PIPES AND RESONATORS 271 



fork), this fund will under the influence of the resonator be 

 more rapidly consumed. 



In order to treat the question in a form free from unessential 

 details, which may vary from one case to another, we take the 

 case of a resonator of the type considered in 86, whose 

 dimensions are small compared with the wave-length. 



The theory is simplest when the frequency of the source 

 is very nearly equal to the natural frequency of the resonator, 

 as determined by 86 (9), so that the forced vibration in the 

 latter is at its strongest. It will perhaps make the matter 

 clearer if we imagine in the first instance that the resonator 

 has a short cylindrical neck in which a thin massless disk, 

 almost exactly fitting it, can be made to move to and fro by 

 a suitable application of force. Suppose then that the disk 

 is made to execute a vibration such that the volume swept 

 over by it outwards up to time t is 



q = C cos nt ; (1) 



and let the extraneous force which must be applied to the 

 disk to compensate the difference of the air-pressures on the 

 two sides be denoted by 



A cos nt + B sin nt, (2) 



this expression being (say) positive when the force is outwards. 

 The component Acosnt which keeps step with the displace- 

 ment is required to control the inertia of the air. From the 

 general theory of forced vibrations ( 9, 12) it appears that the 

 coefficient A can be made to have one sign or the other 

 by adjusting the value of n, the sign being the same as that 

 of C when the imposed vibration is relatively slow, and the 

 opposite when it is relatively rapid. We may therefore 

 suppose n to be so adjusted that A = 0. The circumstances 

 are then very nearly those of a free vibration, and the required 

 value of n is given by 



n* = Kc*/Q, (3) 



very approximately. The second component of the force (2), 

 which keeps step with the velocity (q), is required to maintain 

 the emission of energy outwards, which is, by 76 (15), 



(4) 



