Plane-Wave Scattering
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).
As derived in §C.7.3, we also have the Ohm's law relations:
To obey the physical constraints at the impedance discontinuity, the incident plane-wave must split into a reflected plane wave and a transmitted plane-wave 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 set-up.
Scattering Solution
Define the junction pressure and junction velocity by
Then we can write
Using the Ohm's law relations, the pressure waves follow easily:
Reflection Coefficient
Define the reflection coefficient of the scattering junction as
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 . (Left-going power is negated to account for its opposite direction-of-travel.)
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 ``Kelly-Lochbaum'' scattering junction [297].
There are some simple special cases:
- (e.g., rigid wall reflection)
- (e.g., open-ended tube)
- (no reflection)
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Plane-Wave Scattering at an Angle
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Total Energy in a Rigidly Terminated String