DYNAMICAL THEORY OF SOUND 



The motion is accordingly simple-harmonic, with a period 27r/n, 

 provided 



The nature of the piston is of little importance, provided 

 its mass be sufficiently small. We may even replace it by air, 

 if the length I be small compared with X, for under this 

 condition the column of air in the neck will behave almost as if 

 it were incompressible. We have then p' = p, and 



(3) 



Even in the case of a resonator whose mouth consists of 

 a mere opening in the wall, without a neck, the theory is not 

 very different. It is only a question of obtaining a proper 

 measure of the inertia of the mass of air in the immediate 

 neighbourhood of the mouth, inside and outside, which takes 

 the place of the piston in the above problem. The flow 

 through the aperture at any instant is still regulated, ap- 

 proximately, by the same laws as that of an incompressible 

 fluid, or of electricity in a uniform conductor. There being 

 little motion in the interior, the 

 value of $ there will be sensibly 

 uniform ; we denote it by fa. Out- 

 side, at a short distance beyond the 

 mouth, we shall have <j> = 0, nearly. 

 If q denote the volume of air which 

 has passed through the aperture 

 outwards up to time t, the current, 

 or flux, outwards at this instant will 

 be q, and we have, by the electric analogy, 



........................ (4) 



where K is the " conductivity " ( 82), which depends, of course, 

 on the shape and size of the aperture and the configuration of 

 the wall in its neighbourhood. It is to be observed that this 

 relation (4) is purely kinematical ; from the point of view of the 

 generalized dynamics of a system of one degree of freedom 



