``Traveling Waves'' in Lumped Systems

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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 [34]. 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 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

$\begin{eqnarray*} f^{{+}}&=& R_0v^{+}\\ f^{{-}}&=& -R_0v^{-} \end{eqnarray*}$

and

$\begin{eqnarray*} f &=& f^{{+}}+ f^{{-}}\\ v &=& v^{+}+ v^{-} \end{eqnarray*}$

These are the basic relations for traveling waves in an ideal medium such as an ideal vibrating string [325]. Replacing force by pressure, we obtain the traveling-wave relations for acoustic tubes [301]. Using voltage and current gives elementary transmission line theory.

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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.

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Physical Audio Signal Processing
Introduction to Lumped Models
``Traveling Waves'' in Lumped Systems