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Rigid Terminations

A rigid termination is the simplest case of a string (or tube) termination. It imposes the constraint that the string (or air) cannot move at the termination. (We'll look at the more practical case of a yielding termination in §9.2.1.) If we terminate a length $ L$ ideal string at $ x=0$ and $ x=L$, we then have the ``boundary conditions''

$\displaystyle y(t,0) \equiv 0 \qquad y(t,L) \equiv 0 \protect$ (7.9)

where ``$ \equiv$'' means ``identically equal to,'' i.e., equal for all $ t$. Let $ N\isdef 2L/X$ denote the time in samples to propagate from one end of the string to the other and back, or the total ``string loop'' delay. The loop delay $ N$ is also equal to twice the number of spatial samples along the string.

Applying the traveling-wave decomposition from Eq.$ \,$(6.2), we have

\begin{eqnarray*}
y(nT,0) &=& y^{+}(n) + y^{-}(n) \;\equiv\; 0\\
y(nT,NX/2) &=& y^{+}(n-N/2) + y^{-}(n+N/2) \;\equiv\; 0.
\end{eqnarray*}

Therefore, solving for the reflected waves gives

$\displaystyle y^{+}(n)$ $\displaystyle =$ $\displaystyle -y^{-}(n)$ (7.10)
$\displaystyle y^{-}(n+N/2)$ $\displaystyle =$ $\displaystyle -y^{+}(n-N/2).$ (7.11)

A digital simulation diagram for the rigidly terminated ideal string is shown in Fig.6.3. A ``virtual pickup'' is shown at the arbitrary location $ x=\xi $.

Figure 6.3: The rigidly terminated ideal string, with a displacement output indicated at position $ x=\xi $. Rigid terminations reflect traveling displacement, velocity, or acceleration waves with a sign inversion. Slope or force waves reflect with no sign inversion.
\includegraphics[width=\twidth]{eps/fterminatedstring}



Subsections
Previous: Ideal Acoustic Tube
Next: Velocity Waves at a Rigid Termination

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About the Author: 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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