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Physical Audio Signal Processing
    Digital Waveguide Theory
       Scattering at Impedance Changes
          Plane-Wave Scattering
             Scattering Solution

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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 junct... ...f & 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... ...\\ [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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written by Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.