PIPES AND RESONATORS 261 



the least diameter considerably exceeds the dimensions of 

 the aperture. In his synthetic work on the vowel sounds 

 Helmholtz used cylindrical resonators having a circular opening 

 at the centre of one end. When the object was to detect and 

 to isolate a particular overtone in a complex sound, he used 

 the more convenient form shewn in Fig. 80. The small open 

 nipple opposite the mouth is inserted into the ear cavity, so 

 that the tympanic membrane becomes part of the internal wall 

 of the resonator. 



Fig. 80. Fig. 81. 



The theory of resonators was treated mathematically for the 

 first time by Helmholtz in 1860, and was afterwards greatly 

 simplified by Lord Rayleigh (1871). Suppose in the first place 

 that we have a vessel with a narrow cylindrical neck which is 

 occupied by a plug or piston freely movable to and fro (Fig. 

 81). Let Q denote the capacity of the vessel, I the length of 

 the neck, o> its sectional area, p the density of the piston. We 

 will assume that the period of vibration is so long that the 

 corresponding wave-length (X) in air is large compared with 

 the diameter of the vessel. Under this condition the con- 

 densation s will at any instant be almost uniform throughout 

 the interior, and we may put s = cox/Q, where x denotes the 

 small displacement of the piston outwards from its mean 

 position. The resulting excess of pressure on the base of the 

 piston is /5C 2 so), or pc 2 (0 2 x/Q, and the equation of motion of 

 the system is, approximately, 



(1) 



