Force Waves at a Rigid Termination

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Force Waves at a Rigid Termination

To find out how force waves recoil from a rigid termination, we may convert velocity waves to force waves by means of the Ohm's law relations of Eq. $\,$ (4.6), and then use Eq. $\,$ (4.10), and then Eq. $\,$ (4.6) again:

$\begin{eqnarray*} f^{{+}}(n) &=&Rv^{+}(n) \,\mathrel{\mathop=}\,-Rv^{-}(n) \,\ma... ...el{\mathop=}\,Rv^{+}(n-N/2) \,\mathrel{\mathop=}\,f^{{+}}(n-N/2) \end{eqnarray*}$

Thus, force waves reflect from a rigid termination with no sign inversion:^5.2

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

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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
    Elementary String Instruments
       Rigid Terminations
          Force Waves at a Rigid Termination