CHAPTER IX 



PIPES AND RESONATORS 



84. Normal Modes of Rectangular and Spherical 

 Vessels. 



The main object in this chapter is to develop the laws of 

 vibration of air contained in cavities, such as those of resonators 

 and organ pipes, which are in communication with the external 

 atmosphere. A little space may however be devoted in the 

 first instance to some problems relating to the vibrations of air 

 in spaces which are completely enclosed by rigid walls. These 

 will at all events supply some interesting examples of the 

 general theory of normal modes ( 16). 



The analytical process consists in finding solutions of the 

 equation 



V 2 </> + #ty = ........................ (1) 



consistent with the condition 



which expresses that the component of the fluid velocity in 

 the direction of the normal (n) vanishes at the boundary. It 

 appears that, as in former analogous problems, this is only 

 possible for a certain sequence of values of k, which determine 

 the nature and the frequency of the respective normal modes. 

 In the case of a rectangular cavity we take the origin at a 

 corner, and the coordinate axes along the edges which meet 

 there. If the lengths of these edges be a, b, c, the condition 

 (2) is fulfilled by 



TTTZ 

 </>= (7 cos- cos -j. cos -- 



d G 



