Consider a plane wave with peak
pressure amplitude
propagating
from wave
impedance into a new
wave impedance , as shown in
Fig.
C.15. (Assume
and
are real and positive.)
The physical constraints on the wave are that
 pressure must be continuous everywhere, and
 velocity in must equal velocity out (the junction has no state).
Since power is pressure times velocity, these constraints imply that
signal power is conserved at the junction.
^{C.5}Expressed mathematically, the physical constraints at the junction
can be written as follows:
As derived in §
C.7.3, we also have the
Ohm's law relations:
These equations determine what happens at the junction.
To obey the physical constraints at the
impedance discontinuity, the
incident planewave must split into a
reflected plane wave
and a
transmitted planewave
such that
pressure is continuous and
signal power is conserved. The physical
pressure on the left of the junction is
, and the
physical pressure on the right of the junction is
, since
according to our setup.
Define the junction
pressure and junction
velocity by
Then we can write
Note that
, so we have found the velocity of the transmitted wave.
Since
, the velocity of the reflected wave is simply
We have solved for the transmitted and reflected velocity waves
given the incident wave and the two
impedances.
Using the
Ohm's law relations, the
pressure waves follow easily:
Define the
reflection coefficient of the
scattering junction as
Then we get the following
scattering relations in terms of
for
pressure waves:
Signal flow graphs for
pressure and
velocity are
given in Fig.
C.16.
It is a simple exercise to verify that
signal power is conserved by
checking that
.
(Leftgoing power is negated to account for its opposite
directionoftravel.)
So far we have only considered a plane wave incident on the left of
the junction. Consider now a plane wave incident from the right. For
that wave, the
impedance steps from
to
, so the reflection
coefficient it ``sees'' is
. By superposition, the signal flow
graph for plane waves incident from either side is given by
Fig.
C.17. Note that the
transmission coefficient is
one plus the reflection coefficient in either direction. This signal
flow graph is often called the ``KellyLochbaum'' scattering junction
[
297].
Figure C.17:
Signal flow graph for plane waves
incident on either the left or right of an impedance discontinuity.
Also shown are delay lines corresponding to sampled traveling
planewave components propagating on either side of the scattering
junction.

There are some simple special cases:

(e.g., rigid wall reflection)

(e.g., openended tube)

(no reflection)
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