Digital Waveguide Modeling of a Cylindrical Pipe

• In a previous section, it was found that wave motion in wind instruments with cylindrical bores is adequately represented by plane-wave propagation along the length of the pipe.

• The one-dimensional wave equation for plane waves,
    $$\ensuremath{\frac{\partial^{2} {p}}{\partial {x}^{2}}} = \frac{1}{c^{2}} \ensuremath{\frac{\partial^{2} {p}}{\partial {t}^{2}}} \, ,$$ (34)
  accurately represents lossless wave propagation along the central axis of a cylindrical tube, subject to the boundary conditions at both its ends.

• A discrete-time and -space traveling-wave simulation of this wave propagation is then given by
    $$p\left(t_{n},x_{m}\right) = p^{+}(n - m) + p^{-}(n + m),$$ (35)

  where the spatial sampling distance is given by $X = c T_s$ and the time sampling interval is $T_s = 1 / f_s$.

• At the open end of a pipe, the pressure can be (ideally) approximated as zero for low-frequency wave components.

• Figure 9 represents the digital waveguide implementation of lossless plane-wave pressure propagation in an ideally terminated cylindrical tube.

  Figure 9: Digital waveguide implementation of ideal, lossless plane-wave propagation in a cylindrical tube. The $z^{-1}$ units represent one-sample delays.

  

• The negative one multiplier at $x=L$ implements the low-frequency, open-end approximation for traveling-wave pressure reflection.

• The system of Fig. 9 is a discrete-time and -space implementation of an ideal cylindrical bore reflection function.

• The continuous-time reflectance seen from the entrance of a cylindrical tube of length $L$ is given by
    $$\mathcal{R}(\omega) = e^{-2jkL}\left[\frac{Z_{L}(\omega) - Z_{c}}{Z_{L}(\omega) + Z_{c}}\right],$$ (36)
  where $Z_{c}$ is the real characteristic wave impedance of the cylindrical pipe, and $Z_{L}(\omega)$ is the load impedance at $x=L.$

• For $Z_{L}(\omega) = 0,$ which corresponds to zero pressure at $x=L,$ $R(\omega) = -e^{-2jkL}.$ The resulting continuous-time reflection function $r(t) = -\delta(t - 2L/c)$ is found as the inverse Fourier transform of $\mathcal{R}(\omega)$ when viscothermal losses are ignored.

• This time delay is realized in the digital domain by the delay lines of Fig. 9. The bracketed term in Eq. 36 represents the reflection characteristic at the end of the pipe and must be properly discretized to account for more realistic boundary conditions.