268 DYNAMICAL THEOEY OF SOUND 



an unflanged cylindrical tube experiment seems to indicate a 

 value of about '6 of the radius. 



We will next suppose the pipe to be of finite length, and to 

 be closed at x = I, the origin being chosen as before, near the 

 mouth, in the region of plane waves. For this latter region we 



may assume 



<f> = A cos k(l x), .................. (9) 



since d<f>/dx must vanish for x = l. The flux outwards at 

 the mouth is therefore 



q = codcj>/dx = kco A sin kl, .................. (10) 



and the potential at is A cos kl. Hence with the same 

 meaning of a as before we have 



A cos kl-x kco A sin kl, 



CO 



or cotkl = ka ...................... (11) 



This equation determines the wave-lengths (^Tr/k) of the 

 various normal modes. Usually, ka. is small, and the solution 

 of (11) is then 



kl = (m + J) TT ka, 



or fc(/ + ) = (w + 4)9r, ............... (12) 



where m is integral. The character of the normal modes is there- 

 fore the same as on the rudimentary theory ( 62), provided we 

 imagine the length of the pipe to be increased by the quantity 

 a. In particular, the frequencies are as the odd integers 

 1, 3, 5, ... , so long as the wave-length remains large compared 

 with the diameter. 



If the aperture be contracted the value of a is increased, 

 and the result tends to become less simple. In particular, the 

 harmonic relation of the successive frequencies is violated, as 

 may easily be seen from a graphical discussion of the equation 

 (11). When the pipe is almost closed, a is relatively great, and 

 the solution of (11) is kl = 1/ka, or k* = 1 /la. This agrees with 

 the formula (10) of 86, if we put col = Q, co/a = K. 



In the case of a pipe open at both ends the period equation 

 is found to be 



tan &Z = -( + '), ............... (13) 



where a, a' are the corrections for the two ends, but the calcula- 



