### Plane-WaveScattering 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.5Expressed 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 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  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: #### Reflection Coefficient

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 . (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 . There are some simple special cases:

• (e.g., rigid wall reflection)
• (e.g., open-ended tube)
• (no reflection)

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Total Energy in a Rigidly Terminated String