4 BELL SYSTEM TECHNICAL JOURNAL 



For the absolute value of the pressure at any point in the tube we 

 get from equation (2) 



\P\ = 



i?2- + X2- -\- R- -\- {R2- + X22 - i?2) cos 2l3y 



. + 2X2R sin 2(3y 



Rr + X2- + R' - (i?2'' + ^2' - R^) cos 2/3/ 



- 2X2^? sin 2/3/ 



RL (6) 



where y = I — x. 



\p\ has maximum or minimum values when 



tan 2l3y 



2X2R 



X2' + i?2' - R' ' 



(7) 



for the maximum value 2j8v lies in the first and for the minimum, in the 

 third quadrant. We therefore get 



=[ 



X2^ + i^2-+i^- + \(X2^ + i?2''-i^-)- + 4X2'^i?'^ 



X2'-\-R2--\-R''-^f{X^TR^^^^y^^^x7R' - 



^A. (8) 



Let 3'i be the value of y for which the pressure is a maximum; we then 

 have from (7) and (8) and (4) 



2AR 



R2 = 



"' ~ (^- + 1) - {A- - 1) cos 2^y ' 



Xo = 



{A- + 1) - {A- - 1) cos 2/33/1 ' 

 R{A^ - 1) sin 20yi 



G = 



^ + 1 ' 



(9) 

 (10) 



(11) 



^ = 2^3'i. 



The relation (11) can be derived more simply on the classical theory, 

 as it was done by H. O. Taylor. A derivation of (11) is given by 

 Eckhardt and Chrisler,^ which differs from that of H. O. Taylor. From 

 their derivation it would appear that for (11) to be valid the length 

 of the tube should be adjusted for resonance and that the change in 

 phase at the reflecting surface should be small. The derivation here 

 given shows that (11) is general; it implies only that the waves be 

 plane and that there be no dissipation of power along the tube. 



* Scientific Paper of the Bureau of Standards, No. 526, page 56. 



