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Scattering Solution

Define the junction pressure $ p_j$ and junction velocity $ v_j$ by

p_j &\isdef & p^+_1+p^-_1 = p^+_2\quad\mbox{(pressure at junct...
...f & v^{+}_1+v^{-}_1 = v^{+}_2\quad\mbox{(velocity at junction).}
Then we can write
p^+_1+p^-_1 &=& p^+_2\;=\;p_j\\ [10pt]
...\\ [10pt]
\,\,\Rightarrow\,\,2\,R_1v^{+}_1 - R_1 v_j &=& R_2 v_j

$\displaystyle \,\,\Rightarrow\,\,\zbox {v_j = \frac{2\,R_1}{R_1 + R_2}v^{+}_1.}

Note that $ v_j=v^{+}_2$, so we have found the velocity of the transmitted wave. Since $ v_j = v^{+}_1+v^{-}_1$, the velocity of the reflected wave is simply

$\displaystyle v^{-}_1 = v_j - v^{+}_1 = \left[\frac{2\,R_1}{R_1+R_2} - 1\right]v^{+}_1 = \frac{R_1-R_2}{R_1+R_2} v^{+}_1.

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:
p^+_2 &=& R_2v^{+}_2 = R_2 v_j = \frac{2\,R_2}{R_1+R_2}p^+_1\\ [10pt]
p^-_1 &=& -R_1v^{-}_1 = \frac{R_2-R_1}{R_1+R_2} p^+_1

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Reflection Coefficient
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