THE 

 LONDON, EDINBURGH, and DUBLIN 



PHILOSOPHICAL MAGAZINE 



AND 



JOURNAL OF SCIENCE. 



[SIXTH SERIES] 



*\* V 



FEBRUARY 1916. 



XII. On the Propagation of Sound in narrow%Tubes of 

 variable section. By Lord Rayleigh, O.M., F.R.S.* 



UNDER this head there are two opposite extreme cases 

 fairly amenable to analytical treatment, (i.) when 

 the changes of: section are so slow that but little alteration 

 occurs within a wave-length of the sound propagated and 

 (ii.) when any change that may occur is complete within a 

 distance small in comparison with a wave-length. 



In the first case we suppose the tube to be of revolution. 

 A very similar analysis would Jipply to the corresponding- 

 problem in two dimensions, but this is of less interest. 

 If the velocity-potential </> of the simple sound be pro- 

 portional to e ikat , the equation governing (f> is 



2+g+2+"*-«. • • • • « 



where a; is measured along the axis of symmetry and 

 r perpendicular to it. Since there are no sources of sound 

 along the axis, the appropriate solution is f 



* = j,{,V(«PAfr , + -WW, ■'••(*) 



in which F, a function of x only, is the value of <£ when 

 r = 0. 



* Communicated by the Author. 



t Compare Proc, Lond. Math. Soc. vol. vii. p. 70 (1876) ; Scientific 

 Papers, vol. i. p. 275. 



Phil. Mag. S. 6. Vol. 31. No. 182. Feb. 1916. H 



