258 



DYNAMICAL THEOEY OF SOUND 



tained in a spherical cavity. The condition 3</>/3r = is satisfied 

 for r = a, provided 



tan ka = 



2 - 



,(16) 



The solution can be carried out as in the case of (7). The 

 annexed diagram of the curves y = cot x, y = (2 # 2 )/2#, shews 

 that the roots tend after a time to the form mnr. Approximate 

 values of the first few roots are 



5, 1-891, 2'930, 3'948, 4*959, ...(17) 



Fig. 78. 



the first of which alone gives any trouble. This root corresponds 

 to the gravest of all the normal modes of the cavity. The air 

 swings from side to side, much as in the case of a doubly closed 

 pipe, and the wave-length is X = 2-7T/A; = T509 x 2a. The 

 forms of the equipotential surfaces, to which the directions of 

 vibration of the air-particles are orthogonal, are shewn in Fig. 79. 

 In the next mode the radial velocity vanishes over the sphere 

 r/a= '6625/1-891 ='350. 



The study of the more complicated normal modes of vibra- 

 tion in a spherical vessel would lead us too far. The problem 

 is fully discussed in Lord Rayleigh's treatise. 



