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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 pressure) 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 © 2019-07-30 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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