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Scattering Solution

Let

\begin{eqnarray*}
p_j &\mathrel{\stackrel{\mathrm{\Delta}}{=}}& p^+_1+p^-_1 = p^+_2\quad\mbox{(pressure at junction)}\\
v_j &\mathrel{\stackrel{\mathrm{\Delta}}{=}}& 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}
$

We have solved for the junction velocity $ v_j=v^{+}_2$ . The transmitted pressure is then $ p^+_2 = R_2v^{+}_2 = R_2 v_j$ .

Since $ v_j = v^{+}_1+v^{-}_1$ , the reflected velocity 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
$

Thus, 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:

\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*}

Define

$\displaystyle \zbox{\rho = \frac{R_2-R_1}{R_1+R_2} =
\frac{\mbox{Impedance Step}}{\mbox{Impedance Sum}}}
$

Then we get the following scattering relations in terms of $ \rho$ for pressure waves:
\fbox{\begin{minipage}{3in}{\vspace{-0.15in}%
\par\begin{center}\begin{eqnarray*}
p^+_2 &=& (1+\rho)p^+_1\\ [5pt]
p^-_1 &=& \rho\,p^+_1
\end{eqnarray*}\end{center}\par
}\end{minipage} }
Signal Flow Graph:
\epsfig{file=eps/planewavescatdm2.eps}

Signal power conserved (left-going power negated):

$\displaystyle p^+_1v^{+}_1 = p^+_2v^{+}_2 + ( - p^-_1v^{-}_1)
$


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``Scattering at an Impedance Discontinuity'', by Julius O. Smith III, (From Lecture Overheads, Music 420).
Copyright © 2015-03-30 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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