378 ON THE PROPAGATION OP SOUND IN [402
be taken from 0 to x. As regards the values of the constants of integration (10) may be supposed to identify itself with (7) on the negative side. Thus
yF (x) = e-ik* \A - ~ f Y (A + Be***) das]
( MK Jo J
-f ^ \B + ~ p F(B + Ae-*1™} du\ . .. .(12)
[ AlK Jo J
The integrals disappear when sc is negative, and when x exceeds b they assume constant values.
Let us now further suppose that when as exceeds b there is no negative wave, i.e. no wave travelling in the negative direction. The negative wave on the negative side may then be regarded as the reflexion of the there travelling positive wave. The condition is
1 f5 17 1 } & (
—_ V d.fr\ -J-------
B\l+~ 7d4+^7 Fe-2tock = 0, ............(13)
J ' O „ /„ i O „ /« * \ '
giving the reflected wave (5) in terms of the incident wave (A). There is no reflexion if
o
and then the transmitted wave (so > b} is given by
A a-i
(15)
y
Even when there is reflexion, it is at most of the second order of small-ness, since Y is of that order. For the transmitted wave our equations give (x > 6)
b \
'YaS.ikxJ™
(16)
but if we stop at the second order of smallness the last part is to be omitted, and (16) reduces to (15). It appears that to this order of approximation the intensity of the transmitted sound is equal to that of the incident sound, at least if the tube recovers its original diameter. If the final value of y differs from the initial value, the intensity is changed so as to secure an equal propagation of energy.
The effect of F in (15) is upon the phase of the transmitted wave. It appears, rather unexpectedly, that there is a linear acceleration amounting to
..............................(17)V^-^sm2^' "•'..................( }