260 DYNAMICAL THEOEY OF SOUND 



the limiting form being integer + J. In the case m = 1, which 

 includes the gravest mode, 



&a/7r=-586, 1-697, 2717,..., ............ (6) 



the limiting form being integer J . 



The purely longitudinal modes of a closed circular cylinder 

 come under 62. There remain the vibrations of mixed type. 

 The equation (2) has now to be modified by the inclusion of 

 a term 9 2 </>/d 2 , where z is the longitudinal coordinate. It is 

 found that the equation is satisfied by 



(7) 



provided k* = /3 2 + m'V 2 /^, .................. (8) 



the origin being taken at the centre of one end. The condition 

 of zero normal velocity (d(f>/dz) at the other end (z = I) is 

 satisfied if m' be integral. The corresponding condition at the 

 cylindrical surface requires that /3 should be a root of 



J m '(l3a) = ...................... (9) 



86. Free Vibrations of a Resonator. Dissipation. 



The foregoing examples are of theoretical rather than 

 practical interest, since the vibrations of a mass of air enclosed 

 by rigid walls would be completely isolated. For acoustical 

 purposes the vibrating mass must have some communication 

 with the external atmosphere ; on the other hand it is essential 

 that the communication should be so restricted that the frac- 

 tion of the energy which is used up in a single period in 

 the generation of diverging waves shall still be very small. 

 Otherwise the free vibrations could hardly be regarded as 

 approximately simple-harmonic, and might even resemble 

 the "dead-beat" type (11). 



The theory is simplest in the case of " resonators " such as 

 were employed by Helmholtz in his researches on the quality 

 of musical notes. These are nearly closed vessels, with an 

 aperture, and are used to intensify, by sympathetic vibration of 

 the enclosed air, the effect of a simple tone produced in the 

 neighbourhood. The precise form is not important; it may 

 be spherical or cylindrical, or almost any shape, so long as 



