Sound in narrow Tubes oj variable section. 95 



In substituting the values of <f) and d<\>]dx from (23), (24) 

 it will shorten our expressions if for the time we merge the 

 exponentials in the constants, writing 



A' = Ae~^\ B' = Be ik *, C = Qe~ ik ^. . (30) 



Thus a- 1 (-A'+B').+ ^0 / =^-i*VO , 1 . . . (31) 



A' + B'-C^ttogRC. . . . (32) 



We may check these equations by applying them to the 

 case where there is really no break in the regularity of 

 the tube, so that 



°1 = &2, V = (#2 — #i)0"> R = (il'2 — X\)/<T. 



Then (31), (32) give B' = 0, or B=0, and 



A' l+i*(j? g — %) ' 



with sufficient approximation. Thus 



CV A '*2 = AV** 1 , or C = A. 



The undisturbed propagation of the waves is thus verified. 

 In general, 



B' a 1 - a 2 + ik(a 1 <T 2 'R — V) 



A' Oi + ffj + ^CoiOsR + V)' • • * * y™> 



U ~~ o- 1 + o- 2 + z'A:(a 1 o- 2 R+ V) t 34 ) 



When cr 1 — a 2 is finite, the effect of the new terms is only 

 upon the phases of the reflected and transmitted waves. 

 In order to investigate changes of intensity we should need 

 to consider terms of still higher order. 



When cr 1 = cr 2 , we have 



C = A' J 1« (o- 2 R + V) \ = A^-^ 2R + y )/^, 



C = A^^"^-^ 11 -^ 2 ^, (35) 



making, as before, 0= A, if there be no interruption. Also, 

 when cti = cr 2 absolutely, 



F = 2^ ' (36) 



indicating a change of phase of 90°, and an intensity referred 

 to that of the incident waves equal to 



gggfS! (37) 



