264 DYNAMICAL THEORY OF SOUND 



formula (7) of 7. It was under this form that the theory was 

 presented by Lord Rayleigh. It is to be noticed that the 

 inertia-coefficient is proportional to the " resistance " of the 

 aperture (in the electrical sense), whilst the coefficient of 

 stability, or elasticity, varies inversely as the capacity Q. 



The preceding theory applies only to the gravest mode of the 

 resonator. In the higher modes the internal space is divided 

 into compartments by one or more " loop surfaces " (i.e. surfaces 

 of constant pressure, where <j> = 0), and the frequencies are 

 much greater. The wave-length is then at most comparable 

 with the linear dimensions, as in the problems of 84. 



As already stated ( 82) the calculation of K is usually difficult. 

 For a circular aperture in a thin wall K is equal to the 

 diameter, and for any form differing not too much from a circle 

 we may put K=2 V(w/7r), approximately, where to is the area. 



The frequency, as determined by (9), will then vary as a^/Q . 

 It is remarkable that this law was arrived at empirically by 

 Sondhauss at a date (1850) anterior to the theory. When the 

 aperture is fitted with a cylindrical neck, the conductivity is 

 limited mainly by the neck itself, and we may put K = w/l, 

 approximately, where I is the length. The formula (9) then 

 agrees with (3). It is implied that I is small compared with X, 

 and at the same time large compared with the diameter of the 

 channel. 



We have in the above theory allowed for the inertia of the 

 external atmosphere, but not for its compressibility, and the 

 vibrations as given by (8) are accordingly persistent. In other 

 words, we have neglected the apparent* dissipation of the 

 energy of the resonator due to air-waves diverging outwards 

 from the neighbourhood of the mouth. This will have, in 

 general, no appreciable influence on the period, but will 

 manifest itself by a gradual decay of the amplitude. 



The effect can be estimated with sufficient accuracy in- 

 directly. The flux outwards at the mouth is, by (8), 



q=-nCsm(nt + e) (13) 



* True dissipative influences such as viscosity and thermal conduction are 

 ignored in the present investigation. They probably play as a rule a wholly 

 subordinate part. 



