[ G55 ] 



LXIII. On the Emission of Sound by a Source on the Axis of 

 a Cylindrical Tube. By Mary Taylor,, M.Sc. (Manchr.)*. 



^|^HE following problem was suggested by Lord Rayleigh 

 JL in his ' Theory of Sound/ Art. 301. It is here worked 

 out; and attention is drawn to a point of some interest. It 

 appears that the emission of energy by the source is a dis- 

 continuous function of the frequency. If the frequency be 

 supposed to be gradually increased, the emission tends to 

 become infinite as the frequency approaches any one of a 

 series of critical values, but becomes finite again immediately 

 on passing these. This is on the hypothesis of no dissipative 

 forces. An attempt is also made to show the manner in 

 which such forces will affect the above results. 



First consider the case of an incompressible fluid. The 

 motion is evidently symmetrical about the axis of the 

 cylinder. Hence, using cylindrical coordinates r, 0, z, the 

 origin being at the position of the source, and the axis of z 

 along the axis of the cylinder, the equation of motion, 

 which is satisfied at all points except the origin, is 



w + i.M + y* =0 (i) 



3r 2 + r -dr H dc 2 U ' • • • • W 



cf> being the velocity potential. 



Considering only that part of the cylinder which lies on the 

 positive side of the plane z = 0, assume that <j) varies as e~ kz , 

 the minus sign being taken to insure finiteness at infinity. 

 The equation (1) then becomes 



|^+i.|*+^=0 (2) 



or 2, r or 



The most general solution of (2) which is finite for r = 0, 

 is = AJ o (At) . e~ ks , in the notation of Bessel's functions. 

 The condition of zero normal velocity over the surface of the 

 cylinder gives 



Jo'(fo)=0, (3) 



where a is the radius of the cylinder. Thus the complete 

 solution of (1) applicable to the present case is 



cj> = Az + 2A s J (k s r)e-h*, .... (4) 



where the summation extends over all values of k s , other 

 than zero, which satisfy (3). 



* Communicated by Prof. II. Lamb, F.R.S. 



