1916] NARROW TUBES OP VARIABLE SECTION 381
Iii substituting the values of $ and d^jdos from (23), (24) it will shorten our expressions if for the time we merge the exponentials in the constants, writing
A' = Ae~^, B'^Be^, C' = Ce-ikx* ................ (30)
Thus . ff1(-A' + B') + <raC' = -ikVG', .. ................ (31)
A' + B' - C' = ik<rzRC' ......................... (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
cr1=cr2, V=(x« — xl}cr, R = (oGz~xl}la. Then (31), (32) give £' = 0, or # = 0, and C' 1
_^_ __ ____ __ _ . _ _. Q— ik (#.2— ajj)
A' I + ik (#2 - #1) '
with sufficient approximation. Thus
Q'eikxts-A'eik»li or c=*A.
The undisturbed propagation of the waves is thus verified.
In general,
& = Q-i - Pit + ik (O^Q-B R—V)
A' <r+o- + i-- J ..................... ( '
+ V}
When o-a — cra 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 o-j = cr2, we have
(J -
making, as before, C = A, if there be no interruption. Also, when or1 = cr2 absolutely,
B' _ik(a*R-V)
_ -- __ - , ........................... (36)
indicating a change of phase of 90°, and an intensity referred to that of the incident waves equal to
*--Q .............................. (37)be bounded by non-conducting wi coincident with the walls of the tube is of unit specific resistance.