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Ideal Acoustic Tube

As discussed in §C.7.3, the most commonly used digital waveguide variables (``wave variables'') for acoustic tube simulation are traveling pressure and volume-velocity samples. These variables are exactly analogous to the traveling force and transverse-velocity waves used for vibrating string models.

The Ohm's law relations for acoustic-tube wave variables may be written as follows (cf. Eq.$ \,$(6.6)):

\begin{displaymath}\begin{array}{rcrl} p^+(n) &=& &R_{\hbox{\tiny T}}\, u^{+}(n)... ...p^-(n) &=& -&R_{\hbox{\tiny T}}\, u^{-}(n) \end{array} \protect\end{displaymath} (7.7)

Here $ p^+(n)$ is the right-going traveling longitudinal pressure wave component, $ p^-(n)$ is the left-going pressure wave, and $ u^\pm (n)$ are the left- and right-going volume velocity waves. For acoustic tubes, the wave impedance $ R_{\hbox{\tiny T}}$ is given by

$\displaystyle R_{\hbox{\tiny T}}= \frac{\rho c}{A}$   (Acoustic-Tube Wave Impedance) (7.8)

where $ \rho$ denotes the density (mass per unit volume) of air, $ c$ is sound speed in air, and $ A$ is the cross-sectional area of the tube. In this formulation, the acoustic tube is assumed to contain only traveling plane waves to the left and right. This is a reasonable assumption for wavelengths $ \lambda$ much larger than the tube diameter ( $ \approx \sqrt{A}$). In this case, a change in the tube cross-sectional area $ A$ along the tube axis will cause lossless scattering of incident plane waves. That is, the plane wave splits into a transmitted and reflected component such that wave energy is conserved (see Appendix C for a detailed derivation).
Figure 6.2: Kelly-Lochbaum piecewise cylindrical tube model of the vocal tract (top) and sampled traveling-wave formulation (bottom).
\includegraphics[width=\twidth]{eps/KellyLochbaum}
Figure 6.2 shows a piecewise cylindrical tube model of the vocal tract and a corresponding digital simulation [245,297]. In the figure, $ k_1$ denotes the reflection coefficient associated with the first tube junction (where the cross-sectional area changes), and $ 1+k_1$ is the corresponding transmission coefficient for traveling pressure plane waves. The corresponding reflection and transmission coefficients for volume velocity are $ -k_1$ and $ 1-k_1$, respectively. Again, see Appendix C for a complete derivation. At higher frequencies, those for which $ \lambda\ll\sqrt{A}$, changes in the tube cross-sectional area $ A$ give rise to mode conversion (which we will neglect in this chapter). Mode conversion means that an incident plane wave (the simplest mode of propagation in the tube) generally scatters into waves traveling in many directions, not just the two directions along the tube axis. Furthermore, even along the tube axis, there are higher orders of mode propagation associated with ``node lines'' in the transverse plane (such as Bessel functions of integer order [541]). When mode conversion occurs, it is necessary to keep track of many components in a more general modal expansion of the acoustic field [336,13,50]. We may say that when a plane wave encounters a change in the cross-sectional tube area, it is ``converted'' into a sum of propagation modes. The coefficients (amplitude and phase) of the new modes are typically found by matching boundary conditions. (Pressure and volume-velocity must be continuous throughout the tube.) As mentioned above, in acoustic tubes we work with volume velocity, because it is volume velocity that is conserved when a wave propagates from one tube section to another. For plane waves in open air, on the other hand, we use particle velocity, and in this case, the wave impedance of open air is $ R_0=\rho c$ instead. That is, the appropriate wave impedance in open air (not inside an acoustic tube) is pressure divided by particle-velocity for any traveling plane wave. If $ u^{+}(n)$ denotes a sample of the volume-velocity plane-wave traveling to the right in an acoustic tube of cross-sectional area $ A$, and if $ v^{+}(n)$ denotes the corresponding particle velocity, then we have

$\displaystyle u^{+}(n) \eqsp v^{+}(n)\, A. $

Note that particle velocity is in units of meters per second, while volume velocity is in units of meters-cubed per second (literally a volume of flow per unit time--hence the name). In summary, particle velocity is the appropriate velocity for simulations of waves in open air, while volume velocity is the right choice for acoustic tubes, or ``ducts,'' of varying cross-sectional area. See §B.7.2 and Appendix C for further discussion.
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Ideal Vibrating String