PIPES AND RESONATORS 



255 



where p, q, r are any integers; and the equation (1) is also 

 satisfied provided 



(4) 



If we put <? = 0, r = 0, the case degenerates into that of the 

 doubly closed pipe ( 62). 



A more interesting case is that of a spherical cavity. The 

 symmetrical radial vibrations come under the methods of 71, 

 76. The formula (15) of 71, which implies that there is no 

 source at the origin, gives, in the case of simple-harmonic 

 vibrations, 



^Ller ttr 



(5) 



...(6) 



_ 



or, say, 



sin kr 



