Reflection at a Rigid Boundary

In an earlier section, it was observed that pressure and volume flow traveling-wave components grow increasingly out of phase as they approach the cone apex. This behavior is defined by the complex characteristic impedance of conical air columns.

The boundary condition imposed by a rigid termination requires that volume flow normal to the boundary equal zero.

Where spherical volume velocity traveling-wave components reflect with an inversion, so that their sum is zero, spherical pressure traveling-wave components reflect with a phase angle $2 \, \angle Z_{c}(x,\Omega).$

In the case of a complete cone, $\angle Z_{c}(0,\Omega) = 90^{\circ},$ and pressure traveling-wave components reflect from the cone apex with a $180^{\circ}$ phase shift (or an inversion).

**Figure 5:** Divergent conical section rigidly terminated by a spherical cap at $x_{0}$ .
$\begin{figure}\begin{center} % \begin{picture}(4,2.25) \dashline{0.02}(0.5,1)(... ....25){$x_{0}$} \put(2.43,1.63){$\gamma$} \end{picture}\end{center} \end{figure}$

Figure 5 illustrates a divergent conical bore truncated and rigidly terminated at a distance $x_{0}$ from its apex.

The pressure reflectance at $x_{0}$ is found from the boundary condition $U(x_{0}) = 0,$ such that

$\displaystyle U^{+}(x_{0}) + U^{-}(x_{0})$	$\textstyle =$	$\displaystyle 0$
$\displaystyle \frac{P^{+}(x_{0})}{Z_{c}(x_{0})} - \frac{P^{-}(x_{0})}{Z_{c}^{*}(x_{0})}$	$\textstyle =$	$\displaystyle 0$

$\displaystyle \Rightarrow \frac{P^{+}(x_{0}, s)}{P^{-}(x_{0}, s)}$	$\textstyle =$	$\displaystyle \frac{Z_{c}(x_{0}, s)}{Z_{c}^{*}(x_{0}, s)} = \frac{j k x_{0} - 1}{j k x_{0} + 1}$
	$\textstyle =$	$\displaystyle \left. \frac{x_{0} s - c}{x_{0} s + c} \right\vert _{s=jkc},$	(23)

where

is the Laplace transform variable and $c$

is the speed of sound in air.

As $x_{0} \rightarrow 0,$ the reflectance approaches negative one as observed above.

Using the bilinear transform, this expression can be discretized as follows

$\displaystyle \frac{P^{+}(x_{0}, z)}{P^{-}(x_{0}, z)} = \frac{-a_{1} - z^{-1}}{1 + a_{1} z^{-1}},$ (24)

where

$\displaystyle a_{1} = \frac{c - \alpha x_{0}}{c + \alpha x_{0}}$ (25)

and $\alpha$ is the bilinear transform constant that controls frequency warping.

Equation (24) is a first-order discrete-time allpass filter. This filter structure implements the $2 \, \angle Z_{c}(x,\Omega)$ frequency-dependent phase delay experienced by pressure traveling-wave components reflecting from a rigid termination in a conical waveguide.

The upper plot of Fig. 6 illustrates the continuous-time phase response of the reflectance given by Eq. (23) and the discrete-time phase response of the digital filter of Eq. (24).

The lower plot of Fig. 6 indicates the allpass filter coefficient value as a function of cone truncation.

As $x_{0} \rightarrow 0,$ the filter becomes unstable because $a_{1} \rightarrow 1$ and the allpass pole falls on the unit circle. Thus, the truncation filter cannot be used to simulate a complete cone, but in this case the reflectance is simply negative one anyway.

**Figure 6:** The conical truncation reflectance: (top) Continuous-time and discrete-time filter phase responses; (bottom) Digital allpass filter coefficient value versus $x_{0}$ .
$\begin{figure}\begin{center} \epsfig{file = figures/truncflt.eps,width=4in} \end{center} \end{figure}$

The digital waveguide implementation of a closed-open conic section is shown in Fig. 7.

**Figure 7:** Digital waveguide implementation of a closed-open truncated conical section excited at $x=x_{0}.$
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The cone truncation reflectance filter is represented by $\mathcal{R}_{x_{0}}(z).$