PIPES AND RESONATORS 269 



tion implies that ka. and ka! are small. It is, however, only on 

 this condition that the conductivities at the two ends can, as a 

 rule, be estimated independently of one another. The equation 

 is then equivalent to 



sin &( + + a 7 ) = 0, ............... (14) 



and the frequencies are therefore those which are assigned to 

 a pipe of length I + a + a! by the rudimentary theory. The 

 harmonic relation between the various normal modes is pre- 

 served, but it must be remembered that the approximation is 

 the more precarious, the higher the order of the harmonic. 



The wave-lengths of the proper tones are in all cases fixed 

 by the linear dimensions, but the frequencies, which vary as 

 the velocity of sound, will rise or fall with the temperature. 

 An "open" organ pipe is tuned by means of a contrivance 

 which increases or diminishes the effective aperture at the open 

 end, i.e. the end remote from the " mouth " proper. The pitch 

 of a " closed " pipe is regulated by adjusting the position of a 

 plug which forms the barrier. 



To calculate the rate of decay of the free vibrations it will 

 be sufficient to take the case of the stopped pipe. The kinetic 

 energy corresponding to 



(f) = A cos k (I x) cos nt ............... (15) 



is given with sufficient accuracy by 



T = Ipto dx = pk*c0l . A* cos 2 nt t . . .(16) 



if ka. be small, since cos kl = 0, nearly. A more careful calcula- 

 tion, taking account of the transition region between the plane 

 and the spherical waves, replaces I by I + a, approximately, in 

 this formula, but the correction is not important. The total 

 energy, being equal to the kinetic energy at its maximum, is 



accordingly 



E = \pteu>lA'> ...................... (17) 



If the mouth be unflanged it acts, in relation to the external 

 space, as a simple source of strength kcoA sin kl, or kaA, nearly, 

 and the consequent emission of energy per second is accordingly 

 W = pfrco'cAt/STr, .................. (18) 



