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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

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

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.

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``Physical Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2010, ISBN 978-0-9745607-2-4
Copyright © 2023-08-20 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University