Rigid Terminations

A rigid termination is the simplest case of a string termination. It imposes the constraint that the string cannot move at the termination. If we terminate a length $L$ ideal string at $x=0$ and $x=L$, we then have the ``boundary conditions''

$$y(t,0) \equiv 0 \qquad y(t,L) \equiv 0 \protect$$ (5.7)

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 is also equal to twice the number of spatial samples along the string.

Applying the traveling-wave decomposition from Eq. (4.2), 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

$$y^{+}(n) = -y^{-}(n)$$ (5.8)

$$y^{-}(n+N/2) = -y^{+}(n-N/2),$$ (5.9)

A digital simulation diagram for the rigidly terminated ideal string is shown in Fig. 4.2. A ``virtual pick-up'' is shown at the arbitrary location $x=\xi$.

Figure 4.2: 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.