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Scattering Solution

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

\begin{eqnarray*}
p_j &\isdef & p^+_1+p^-_1 = p^+_2\quad\mbox{(pressure at junction)}\\
v_j &\isdef & v^{+}_1+v^{-}_1 = v^{+}_2\quad\mbox{(velocity at junction).}
\end{eqnarray*}

Then we can write

\begin{eqnarray*}
p^+_1+p^-_1 &=& p^+_2\;=\;p_j\\ [10pt]
\,\,\Rightarrow\,\,R_1v^{+}_1 - R_1v^{-}_1 &=& R_2 v^{+}_2 \;=\; R_2 v_j\\ [10pt]
\,\,\Rightarrow\,\,R_1v^{+}_1 - R_1(v_j-v^{+}_1) &=& R_2 v_j\\ [10pt]
\,\,\Rightarrow\,\,2\,R_1v^{+}_1 - R_1 v_j &=& R_2 v_j
\end{eqnarray*}

$\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:

\begin{eqnarray*}
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
\end{eqnarray*}


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``Physical Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2010, ISBN 978-0-9745607-2-4.
Copyright © 2014-03-23 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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