270 DYNAMICAL THEOEY OF SOUND 



by 76 (15). Equating the rate of decay of the energy to W, 

 we are led to the equation 



dA k*a>c A _ 



"3* + 



and the modulus of decay is therefore 



r = 47rZ/A; 2 6>c ...................... (20) 



The ratio of this to the period (2?r/A;c) is 21/kco, or (in the 

 gravest mode) ffi/ira), nearly. Since the moduli of the various 

 normal modes are proportional to the squares of the respective 

 wave-lengths, the decay is the more rapid the higher the order. 

 For a flanged pipe the result (20) would be halved. 



88. Resonator under Influence of External Source. 

 Reaction on the Source. 



The theory of forced vibrations due to an external source 

 of sound, to which we now proceed, involves some rather 

 delicate considerations, and is often misunderstood. That 

 the mass of air contained in a resonator or an organ pipe 

 should be set into vigorous vibration by a source in approxi- 

 mate unison with it is intelligible enough; but it is further 

 desirable to have some estimate of the amplitude of the forced 

 vibration, and in particular to understand why the sound which 

 is apparently emitted by the resonator should under certain 

 conditions enormously exceed that which would be produced 

 by the original source alone. 



For simplicity we will suppose that this source is main- 

 tained at constant amplitude by a suitable supply of energy, 

 so that the vibration of the air is everywhere steady. It is 

 evident at once that under this condition no work is done, 

 on the average of a whole period, at the mouth of a resonator 

 on the contained air, the energy of the latter being constant, 

 and consequently that no work can in turn be done by the 

 reaction of this mass on the external atmosphere. Any 

 increased propagation of sound to a distance must be due to 

 the changed conditions which the action of the resonator has 

 introduced in the neighbourhood of the original source. If 

 this source be not maintained constant, but merely started 

 with an initial fund of energy (as in the case of a tuning 



