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Junctions Between Two Uniform Acoustic Tubes

Consider now a junction between two of the uniform acoustic tubes in the concatenated tube model shown in Figure 1.1(b). The wave speeds in all the tubes are assumed to be constant, and equal to $ \gamma$, so that the discrete-time representation of any single tube will have the form of the pair of digital delay lines shown in Figure 1.2(b). At the junction between the $ i$th and $ (i+1)$th tubes (of cross-sectional areas $ A_{i} = A(i\Delta)$ and $ A_{i+1} = A((i+1)\Delta)$ respectively), for $ i=1,\hdots,N-1$, we will then have a pressure and a velocity on either side; we will write these pressure/velocity pairs as $ (p_{i}, u_{i})$, and $ (p_{i+1}, u_{i+1})$ respectively--see Figure 1.3(a). Continuity arguments (or conservation laws) dictate that these quantities should remain unchanged as we pass through the boundary between the two tubes, and thus

$\displaystyle p_{i} = p_{i+1}\hspace{1.5in}u_{i} = u_{i+1}$ (1.5)

Note that we have dropped the arguments $ t$ and $ x$, since the relationships of (1.5) hold instantaneously, and only at the tube boundaries.

Figure 1.3: (a) The junction between the $ i$th and $ (i+1)$th acoustic tubes in the Kelly-Lochbaum vocal tract model, and (b) the resulting scattering junction for pressure waves.
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\end{picture} \end{center} \vspace{0.3in}

As per (1.2a) and (1.2b), the pressures and velocities can be split into leftward- and rightward-traveling waves as

\begin{subequations}\begin{alignat}{3} p_{i} &= p_{i}^{l} + p_{i}^{r}\hspace{1.0...
...&= Y_{i+1}\left(p_{i+1}^{l} - p_{i+1}^{r}\right) \end{alignat}\end{subequations}

where $ Y_{i}$, the admittance of the $ i$th tube, is defined by

$\displaystyle Y_{i}\triangleq \frac{A_{i}}{\rho\gamma}$ (1.7)

It is then possible, using (1.6) to rewrite (1.5) purely in terms of the wave variables, as
$\displaystyle \begin{eqnarray}p_{i}^{l} &=& \mathcal{R}_{i}p_{i}^{r}+(1-\mathca...
...^{r} &=& (1+\mathcal{R}_{i})p_{i}^{r}-\mathcal{R}_{i}p_{i+1}^{l} \end{eqnarray}$ (1.8a)

where $ \mathcal{R}_{i}$ is defined by

$\displaystyle \mathcal{R}_{i}\triangleq\frac{Y_{i}-Y_{i+1}}{Y_{i}+Y_{i+1}}$    

Here we have written a formula for calculating the pressure waves $ p_{i}^{l}$ and $ p_{i+1}^{r}$ leaving the junction in terms of the waves $ p_{i}^{r}$ and $ p_{i+1}^{l}$ entering the junction--see Figure 1.3(b) for the resulting signal-flow diagram. In particular, (1.8) can be viewed as a scattering operation; incident waves on either side of an interface are reflected and transmitted according to the mismatch in the admittances between the two tubes. The mismatch is characterized by the reflection parameter $ \mathcal{R}_{i}$ which is bounded in magnitude by 1, as long as the admittances of the two tubes are positive. (If $ Y_{i} = Y_{i+1}$, for instance, then $ \mathcal{R}_{i} = 0$, and there is no reflection at the interface.) As we mentioned before, the calculations (1.8) should be viewed as occurring pointwise at the junction interface itself, which does not occupy physical space.

Suppose that we define a set of power-normalized wave variables by

$\displaystyle \underline{p}_{i}^{l} = \sqrt{Y_{i}}p_{i}^{l}\hspace{1.5in}\underline{p}_{i}^{r} = \sqrt{Y_{i}}p_{i}^{r}$ (1.9)

Then the scattering operation (1.8) can be written, in matrix form, as

$\displaystyle \begin{bmatrix}\underline{p}_{i}^{l}\\ \underline{p}_{i+1}^{r}\\ ...
...}\begin{bmatrix}\underline{p}_{i}^{r}\\ \underline{p}_{i+1}^{l}\\ \end{bmatrix}$ (1.10)

Because the $ \mathcal{R}_{i}$ are bounded in magnitude by 1, it is easy to see that scattering, in this case, corresponds to an orthogonal matrix transformation applied to the input wave variables.

next up previous
Next: Power Conservation at Scattering Up: Case Study: The Kelly-Lochbaum Previous: Wave Propagation in a
Stefan Bilbao 2002-01-22