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''

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

$$y^{+}(n) = -y^{-}(n)$$ (7.10)

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