484 WAVES OF EXPANSION. [CHAP. X 



the positions of the spherical nodes (c&/cfo*=0). Thus in the second mode 

 there is a spherical node whose radius is given by 



r/a = (l'4303)/(2-4590) = -58l7. 



2. In the case of waves propagated outwards into infinite space from a 

 spherical surface, it is more convenient to use the solution of (3), including 

 the time-factor, in the form 



(iv). 



If the motion of the gas be due to a prescribed radial motion 



r=ae i(rt ....................................... (v) 



of a sphere of radius a, C is determined by the condition that r= d(f>/dr for 

 r=a. This gives 



whence, taking the real parts, we have, corresponding to a prescribed normal 

 motion 



r=acosart .................................... (vii), 



^^r (r ~ a}} +ka Sln {<rt ~r (r ~ a} 

 When lea is infinitesimal, this reduces to 



where A = 4ira 2 a. We have here the conception of the 'simple source' of 

 sound, which plays so great a part in the modern treatment of Acoustics. 



The rate of emission of energy may be calculated from the result of Art. 258. 

 At a great distance r from the origin, the waves are approximately plane, of 

 amplitude Aj^ircr. Putting this value of a in the expression p o- 2 a 2 c for the 

 energy transmitted across unit area, and multiplying by 4?jT 2 , we obtain for 

 the energy emitted per second 







267. When the restriction as to symmetry is abandoned, we 

 may suppose the value of <f> over any sphere of radius r, having 

 its centre at the origin, to be expanded in a series of surface- 

 harmonics whose coefficients are functions of r. We therefore 



assume 



