380
ON THE PROPAGATION OF SOUND IN
[4
so that at xl and aaa we have
dfa/dx = - i
(23'
. . .(24
dfa/dx = ik (- A e-'**-
When X is sufficiently great we may ignore altogether the space betw( %i and x2, that is we may suppose that the pressures are the same at th two places and that the total flow is also the same, as if the fluid w incompressible. As there is now no need to distinguish between os^ and we may as well suppose both to be zero. The condition fa = <p2 gives
A+B = 0, ................................. (25;
and the condition a-ldfa/dx = a-2dfa/dx gives
<r1(-j. + 5) = -<72a ........................ (26;
Thus
CT2
A
+ cr2
C
A
,.(27;
These are Poisson's formulas*. If o~l and o-2 are equal, we have of cou J3 = 0, C — A. Our task is now to proceed to a closer approximation, s supposing that the region of irregularity is small
For this purpose both of the conditions just now employed need c rection. Since the volume V of the irregular region is to be regarded sensible and the fluid is really susceptible of condensation (s), we have
fi"f" (1 IT " ujT
and since in general s = — a~2d^/dt, we may take ds d-
dt
_ _ ~~a
or -
the distinction being negligible in this approximation in virtue of smalj.ness of V. Thus
dfa dfa
a? dtf
.(2S
In like manner, assimilating the flow to that of an incompressible fli we have for the second condition
dfa ' dccz'
.(29
where R may be defined in electrical language as the resistance betweer and x2, when the material supposed to be bounded by non-conducting wi coincident with the walls of the tube is of unit specific resistance.
Compare Theory of Sound, § 264.cident wave.