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``Traveling Waves'' in Lumped Systems

One of the topics in classical network theory is the reflection and transmission, or scattering formulation for lumped networks [35]. Lumped scattering theory also serves as the starting point for deriving wave digital filters (the subject of Appendix F). In this formulation, forces (voltages) and velocities (currents) are replaced by so-called wave variables

\begin{eqnarray*}
f^{{+}}(t) &\isdef & \frac{f(t) + R_0v(t)}{2} \\
f^{{-}}(t) &\isdef & \frac{f(t) - R_0v(t)}{2}
\end{eqnarray*}

where $ R_0$ is an arbitrary reference impedance. Since the above wave variables have dimensions of force, they are specifically force waves. The corresponding velocity waves are

\begin{eqnarray*}
v^{+}(t) &\isdef & \frac{1}{2}[v(t) + f(t)/R_0], \\
v^{-}(t) &\isdef & \frac{1}{2}[v(t) - f(t)/R_0].
\end{eqnarray*}

Dropping the time argument, since it is always `(t)', we see that

$\displaystyle f^{{+}}$ $\displaystyle =$ $\displaystyle R_0v^{+}$  
$\displaystyle f^{{-}}$ $\displaystyle =$ $\displaystyle -R_0v^{-}\protect$ (C.73)

and
$\displaystyle f$ $\displaystyle =$ $\displaystyle f^{{+}}+ f^{{-}}$  
$\displaystyle v$ $\displaystyle =$ $\displaystyle v^{+}+ v^{-}\protect$ (C.74)

These are the basic relations for traveling waves in an ideal medium such as an ideal vibrating string or acoustic tube. Using voltage and current gives elementary transmission line theory.



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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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