656 Miss M. Taylor on the Emission of Sound by a 



The constants A and A s are to be determined from con- 

 sideration of the flux across the section z = 0. Over this 

 section ~d(j)/^z will clearly vanish except for r = 0, but the 

 total flux across it must be J-, the strength of the source being 

 taken to be unity. Hence we must have 



f( 



s ' c— n 



(5) 



Now, from (4), by the conjugate property of BesseFs 

 functions. 



\ X ~^) Jo(hr)2irrctr =Jc s A S C{ J (k s r)\ 2 2irrdr 

 J Q \ OZ J Z=Q Jo 



= 7ra 2 k s A s {J Q (k s a)}\ . (6) 



in virtue of the equation (3)*. 

 But 



j;(-m) 2 _jo(^,)2.,^=£(-M)^ o2 .,^=x 



since ( — B^>/9^=o vanishes except for infinitesimal values 

 of r> Hence 



A, 



27ra 2 k s {J (k s a)} 2 ' 



and similarly 

 Hence 



A " " as* 



(7) 

 (8) 



4> 





{J (k s a)y 



For large values of s the first term in this solution is 

 preponderant, and —'dtyfoz assumes approximately the 

 constant value 1/2 ira 2 . 



Next consider the case when the fluid is gaseous. The 

 equation of motion now is 



|f=c 2 vv, (io) 



where c is the velocity of sound in the fluid. If <f> varies as 

 * Eayleigh, ' Theory of Sound,' § 203. 



