34 Lord Rayleigh on the Incidence of Aerial 



propagation 



V 2 =m/<r, V' 2 =m7o-' (39) 



The velocity-potential of the undisturbed plane waves is 

 represented bv 



0=^*v*.^fa (40) 



in which k = 2ir[k. The time factor e ikYt , which operates 

 throughout, may be omitted for the sake of brevity. 



The velocity-potential (^) of the disturbance propagated 

 outwards from T may be expanded in spherical harmonic 

 tirms * 



r^n-^So+S^Cttr) +S 1 ^(£tr) +. .}, . (41) 

 where 



^ v ' 2.ikr 2.4. (2^') 2 



1.2.3...2w 

 + + 2.4.d...2n(*'Ar)"' ' ' * ^-) 



At a great distance from the obstacle f n (ikr s ) = l; and the 

 relative importance of the various harmonic terms decreases 

 in going outwards with the order of the harmonic. For the 

 present purpose we shall need to regard only the terms of 

 order and 1. Of these the term of order depends upon 

 the variation of compressibility, and that of order 1 upon the 

 variation of density. 



The relation between the variable part of the pressure 8p, 

 the condensation s, and (j> is 



dt a ' 



so that during the passage of the undisturbed primary waves 

 the rate at which fluid enters the volume T (supposed for 

 the moment to be of the same quality as the surrounding 

 medium) is 



If the obstacle present an unyielding surface, its effect is to 

 prevent the entrance of the fluid (43) ; that is, to superpose 

 upon the plane waves such a disturbance as is caused by the 

 introduction of (43) into the medium. Thus, if the potential 

 of this disturbance be 



p—ikr 



*=A>— , (44) 



* < Theory of Sound,' §§ 323, 324. 



