PIPES AND RESONATORS 



259 



Fig. 79. 



85. Vibrations in a Cylindrical Vessel. 



The theory of the purely transversal vibrations of the air 

 enclosed by a circular cylinder is very similar. As in 54, the 

 equation 



v (p (Z> j i f\ /-i \ 



-^- -j- fcP(p = ( 1 ) 



03j Vll 



where x, y are Cartesian coordinates in the plane of a cross- 

 section, becomes in polar coordinates 



and the typical solution, when there is no source at the 

 origin, is 



x 

 GJ m (kr) cos W0.+<k ) ............... (3) 



The admissible values of k are determined by the condition that 

 d<f>/dr = for r = a, or 



J m '(ka)=0 ......................... (4) 



For the radial vibrations (ra = 0) the earlier roots are given by 



ka/ir = 1*2179, 2-2330, 3-2383,..., ......... (5) 



172 



j 



