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Force or Pressure Waves at a Rigid Termination

To find out how force or pressure waves recoil from a rigid termination, we may convert velocity waves to force or velocity waves by means of the Ohm's law relations of Eq.$ \,$ (6.6) for strings (or Eq.$ \,$ (6.7) for acoustic tubes), and then use Eq.$ \,$ (6.12), and then Eq.$ \,$ (6.6) again:

\begin{eqnarray*}
f^{{+}}(n) &=&Rv^{+}(n) \eqsp -Rv^{-}(n) \eqsp f^{{-}}(n) \\
f^{{-}}(n+N/2) &=&-Rv^{-}(n+N/2) \eqsp Rv^{+}(n-N/2) \eqsp f^{{+}}(n-N/2)
\end{eqnarray*}

Thus, force (and pressure) waves reflect from a rigid termination with no sign inversion:7.3

\begin{eqnarray*}
f^{{+}}(n) &=& f^{{-}}(n) \\
f^{{-}}(n+N/2) &=& f^{{+}}(n-N/2)
\end{eqnarray*}

The reflections from a rigid termination in a digital-waveguide acoustic-tube simulation are exactly analogous:

\begin{eqnarray*}
p^+(n) &=& p^-(n) \\
p^-(n+N/2) &=& p^+(n-N/2)
\end{eqnarray*}

Waveguide terminations in acoustic stringed and wind instruments are never perfectly rigid. However, they are typically passive, which means that waves at each frequency see a reflection coefficient not exceeding 1 in magnitude. Aspects of passive ``yielding'' terminations are discussed in §C.11.


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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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