266 DYNAMICAL THEOEY OF SOUND 



87. Corrected Theory of the Organ Pipe. 



The same principles can be applied to obtain a correction 

 to the imperfect theory of the open pipe which was given in 

 62. We may begin by a brief examination of the slightly 

 simpler problem of reflection at an open end of an infinitely 

 long pipe ( 61). 



Fig. 83. 



Near the open end there is a certain region, whose dimensions 

 are small compared with the wave-length, within which the 

 transition takes place from plane waves within the tube to 

 diverging spherical waves outside*. We take the origin inside 

 the tube, near the mouth, but in the region of plane waves, and 

 the positive direction of the axis of x along the tube. For the 

 region of plane waves we may write 



<f> = Ae ik * + Be- ik *, ....(1) 



where the first term may be taken to represent a train of waves 

 approaching the end, from the right, whilst the second term 



* The figure, which is based on formulae given by Helmholtz in another 

 connection, relates to the two-dimensional form of the problem. In three 

 dimensions the transition to a state of uniform radial flow outwards from the 

 mouth would be still more rapid. 



