PIPES AND RESONATORS 267 



represents the reflected waves. The outward velocity at is 

 therefore represented by ik (A - B), and the flux is 



q = ika>(A-B\ ..................... (2) 



where a> is the sectional area. The velocity-potential at is 

 A + B. The " resistance " between the section x = and the 

 external region to the left may be specified as equivalent to 

 that of a certain length a of the pipe, and is accordingly 

 denoted by a/to. Hence, by the electrical analogy, 



to 



(3) 



u ^ lika , 



whence = - - ...................... (4) 



A I+ika. 



If we put ka tan & fi, ..................... (5) 



this may be written B/A = - e -* k t ...................... (6) 



Hence < = A {e ikx - e~* ***> ) ................ (7) 



The reflected train is therefore equal in amplitude to the 

 incident one, as was to be expected, since the inertia only of the 

 external air is so far taken into account ; but there is a difference 

 of phase. In the theory of 61 the condition to be satisfied at 

 an open end was s = 0, or <f) = 0. Hence if we write (7) in the 

 form 



= 4*^ {***> -e-*^} ............ (8) 



we recognize that the circumstances are the same as if the pipe 

 were prolonged to the left for a length ft, and the reflection at the 

 mouth were to take place according to the rudimentary theory. 

 The wave-length being assumed to be large compared with the 

 diameter of the pipe, ka. will usually be small, so that ft = a, 

 nearly. But if the pipe be very much contracted or obstructed 

 at the mouth, ka may be considerable, and k/3 will in that case 

 approach \ir. We then have B = A, nearly, and the circum- 

 stances approximate to those of reflection at a closed end. 



The actual determination of a is a problem in electric 

 conduction which has at present only been solved, even 

 approximately, in a very few cases. Lord Rayleigh estimates 

 that for an accurately cylindrical tube fitted with an infinite 

 flange the value of a is about "82 of the sectional radius. For 



