Mr. R. H. M. Bosaiiquet on the Theory of Sound. 29 



Symmetrical Splieincal Dwergence. 



The simplest case of the divergence of sound is where it 

 issues from a spherical source, and spreads symmetrically in 

 all directions. This may be realized by supposing a small 

 sphere, pierced with holes like the rose of a watering-pot, to 

 be connected through a long tube with a source from which 

 air alternately issues and is abstracted, the tube being thin 

 enough not to interfere sensibly with the spherical divergence. 

 Of course also the conditions are the same in hemispherical or 

 sectorical divergence, the sound diverging always from a cor- 

 responding portion of a spherical surface. In these cases the 

 assumption that R, the velocity-potential, is a function of r 

 and t only, is legitimate, but in no other case. 



If we form the equations of motion in three dimensions in 

 the ordinary manner, making the above assumption, we have 

 (Airy, ' On Sound,' p. 92) 



1 cZ'R _d?n 2 clR 

 v^ dt^ ~~ rZr'^ r dr 



There are three principal forms of integral of the above equa- 

 tion with which we are concerned : 



-P, Csin k(vt — r) ... 



^= r ' ^'-^ 



-r, C sin k(vt -\- r) .... 



^= ^ ^ (") 



K=p • • (iii) 



The corresponding velocities are, 



^^^ -^=-^{^^cosy(:(v«^-r)+sin^(t'^-r)}, 



(ii) -7- = 2 { ^^ ^^^ ^(y^ + *0 ~ ^i^ ^(y^ + ^0 } ; 



(iii) 



dr r^ 



In (i), (ii) of the values of -7— the coefficient 'kr= ; if 



^ ^^ ^ ■' dr \ 



X, be regarded as indefinitely great with respect to r, the last 

 terms alone survive, and the velocities reduce to 



