- Sound- Waves of Finite Amplitude. 331 



It is required to write down the differential equation of an 

 infinitesimal spherical disturbance^ which is superposed on a 

 purely radial steady motion. 



Though a steady motion extending inward to the pole 

 would involve a violation of the principle of continuity, we 

 may suppose that throughout a shell of finite thickness the 

 distribution of density and velocity is such as would be con- 

 sistent with steady motion ; the motion within such a shell 

 would then continue steady, provided that its spherical 

 boundaries were constrained to expand or contract in a 

 suitable manner. In the absence of constraints the shell of 

 steady motion would be invaded from without and from 

 within by disturbances emanating from adjoining parts of the 

 fluid, but, at points well within the shell, the character of the 

 steady motion would necessarily be maintained for a finite 

 time. 



Let (f> be the potential of the steady motion. 



Let + yjr be the potential of the actual motion so that 

 i|r and its derivatives are small. 



Let p, p be the pressure and density in the steady motion. 



Let p + Sp, p + Sp be the pressure and density in the actual 

 motion, and assume that the pressure is a function of 

 the density only. From the ordinary equations for the 

 motion of compressible fluids we obtain 



'?-*©' ■ • ••<») 



*\br) -dr-dr-*' ' ' • ( 21 ) 



when small quantities of the second order are neglected. 

 Subtracting (20) from (21) 



& = _*_|*|± (22) 



p r or O r 



Now 

 therefore 



r^ Sn dn ] r\ . £n 



(23) 

 and the equations of continuity for the steady motion and tho 



f 



Sp dp Sp 

 P " dp' p ' 



d Sp _ dp I'd .Sp . 

 ~dt p dp' p ot 



