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Acoustic Tubes

In acoustic tubes, we again work with
Pressure Plane Waves:

\begin{eqnarray*}
p^+(n) &\mathrel{\stackrel{\mathrm{\Delta}}{=}}& R_{\hbox{\tiny T}}\,U^{+}(n)\\
p^-(n) &\mathrel{\stackrel{\mathrm{\Delta}}{=}}& -R_{\hbox{\tiny T}}\,U^{-}(n)
\end{eqnarray*}

However, now $ U^{+},U^{-}$ are
Longitudinal Volume-Velocity Waves:

\begin{eqnarray*}
U^{+}(n) &\mathrel{\stackrel{\mathrm{\Delta}}{=}}& A\,u^{+}(n)\\
U^{-}(n) &\mathrel{\stackrel{\mathrm{\Delta}}{=}}& A\,u^{-}(n)
\end{eqnarray*}

where $ A$ is the cross-sectional area of the tube. In an acoustic tube, it is volume velocity that is conserved from one tube section to the next.

Ohm's Law for Traveling Plane Waves in an Acoustic Tube:

\fbox{%
\begin{minipage}[c]{3in}%
\begin{displaymath}\begin{array}{rcrl}%
p^+(n)&=&&R_{\hbox{\tiny T}}\,U^{+}(n) \\
p^-(n)&=&-&R_{\hbox{\tiny T}}\,U^{-}(n)
\end{array}\end{displaymath}\end{minipage}}
where

$\displaystyle \zbox{R_{\hbox{\tiny T}}= \frac{\rho c}{A}}
$

is the wave impedance of air in terms of mass density $ \rho$ , sound speed $ c$ , and tube cross-section area $ A$ .


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``Digitizing Strings Waves in Vibrating Strings'', by Julius O. Smith III, (From Lecture Overheads, Music 420).
Copyright © 2014-03-24 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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