Diameter and Taper Discontinuities

A frequency-domain, transfer-matrix method for modeling piece-wise conical sections, as previously presented for cylindrical sections, is possible. The matrix elements are a bit more complicated, but the approach is well documented and validated and can include accurate characterizations of thermoviscous losses. In this section, however, we focus on the time-domain, traveling-wave approach for modeling combinations of conical sections.

**Figure 10:** Junction of two conical tube sections.
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At the boundary of two discontinuous and lossless conical sections, Fig. 10, the abrupt change in diameter and rate of taper will cause scattering of traveling wave components.

Assuming continuity of pressure and conservation of volume flow at the boundary,

$\displaystyle P_{1}^{+}(\Omega) + P_{1}^{-}(\Omega) = P_{2}^{+}(\Omega) + P_{2}^{-}(\Omega)$ (29)

and

$\displaystyle P_{1}^{+}(\Omega) Y_{c1}(\Omega) - P_{1}^{-}(\Omega) Y_{c1}^{\ast... ... = P_{2}^{+}(\Omega) Y_{c2}(\Omega) - P_{2}^{-}(\Omega) Y_{c2}^{\ast}(\Omega),$ (30)

where $Y_{c1}$ is the characteristic admittance for section $1$

at the boundary looking in the positive $x$

direction.

The characteristic admittance for spherical waves in a cone is given by

$\displaystyle Y_{c}(x,\Omega) = \frac{1}{Z_{c}(x,\Omega)} = \frac{A(x)}{\rho c}... ...}{Z_{c}^{\ast}(x,\Omega)} = \frac{A(x)}{\rho c}\left(1 - \frac{1}{jkx}\right),$ (31)

where $Y_{c}(x,\Omega)$ applies to traveling-wave components propagating away from the cone apex in the positive $x$

direction and $Y_{c}^{\ast}(x,\Omega)$ to traveling-wave components propagating toward the cone apex in the negative $x$

direction.

Solving for $P_{1}^{-}(\Omega)$ at the junction,

$\displaystyle P_{1}^{-}(\Omega) = \left[\frac{Y_{c1}(\Omega) - Y_{c2}(\Omega)}{... ...st}(\Omega)}{Y_{c1}^{\ast}(\Omega) + Y_{c2}(\Omega)}\right] P_{2}^{-}(\Omega).$ (32)

The frequency-dependent scattering coefficient that relates $P_{1}^{-}(\Omega)$ to $P_{1}^{+}(\Omega)$ is

$\displaystyle \mathcal{R}^{-}(\Omega)$	$\textstyle =$	$\displaystyle \frac{Y_{c1}(\Omega) - Y_{c2}(\Omega)}{Y_{c1}^{\ast}(\Omega) + Y_{c2}(\Omega)}$
	$\textstyle =$	$\displaystyle \frac{B-1}{B+1} - \frac{2 B \gamma}{(B+1)(j \Omega + \gamma)},$	(33)

where

is the ratio of wave front surface areas $A_{1}/A_{2}$ at the boundary and $\gamma$ is given by

$\displaystyle \gamma = - \frac{c}{A_{1} + A_{2}}\left(\frac{A_{1}}{x_{1}} - \frac{A_{2}}{x_{2}}\right)$ (34)

(Martínez and Agulló, 1988; Gilbert et al., 1990).

This reflectance is given a negative superscript to indicate scattering in the negative $x$

direction.

The parameters $x_{1}$ and $x_{2}$ are measured from the (imaginary) apices of cones $1$

and

, respectively, to the discontinuity.

Similarly, the expression for $P_{2}^{+}(\Omega)$ at the junction is

$\displaystyle P_{2}^{+}(\Omega) = \left[\frac{Y_{c1}(\Omega) + Y_{c1}^{\ast}(\O... ...st}(\Omega)}{Y_{c2}(\Omega) + Y_{c1}^{\ast}(\Omega)}\right] P_{2}^{-}(\Omega),$ (35)

and the reflectance that relates $P_{2}^{+}(\Omega)$ to $P_{2}^{-}(\Omega)$ is

$\displaystyle \mathcal{R}^{+}(\Omega)$	$\textstyle =$	$\displaystyle \frac{Y_{c2}^{\ast}(\Omega) - Y_{c1}^{\ast}(\Omega)}{Y_{c2}(\Omega) + Y_{c1}^{\ast}(\Omega)}$
	$\textstyle =$	$\displaystyle \frac{1-B}{1+B} - \frac{2 \gamma}{(1+B)(j \Omega + \gamma)}.$	(36)

This reflectance is given a positive superscript to indicate scattering in the positive $x$

direction.

The scattering equations can then be expressed in terms of $\mathcal{R}^{-}(\Omega)$ and $\mathcal{R}^{+}(\Omega)$ as

$\displaystyle P_{1}^{-}(\Omega)$	$\textstyle =$	$\displaystyle \mathcal{R}^{-}(\Omega) P_{1}^{+}(\Omega) + \left[1 + \mathcal{R}^{+}(\Omega) \right] P_{2}^{-}(\Omega)$	(37)
$\displaystyle P_{2}^{+}(\Omega)$	$\textstyle =$	$\displaystyle \left[1 + \mathcal{R}^{-}(\Omega) \right] P_{1}^{+}(\Omega) + \mathcal{R}^{+}(\Omega) P_{2}^{-}(\Omega).$	(38)

It is possible to define transmittances that indicate scattering through the junction as $\mathcal{T}^{+}(\Omega) = 1+\mathcal{R}^{-}(\Omega)$ and $\mathcal{T}^{-}(\Omega) = 1+\mathcal{R}^{+}(\Omega).$

These expressions are equally valid when either acoustic section is cylindrical, rather than conical. Replacing cone $1$

by a cylindrical section, $x_{1} = \infty$ and $A_{1}$ is given by the cylinder's cross-sectional area at the discontinuity.

Whereas the junction scattering coefficients for cylindrical bore diameter discontinuities were real and constant, these expressions are frequency-dependent and must be transformed to discrete-time filters for time-domain implementation.

Figure 11(a) illustrates the general scattering junction implementation for diameter and taper discontinuities in conical bores.

**Figure 11:** (a) The scattering junction for diameter and taper discontinuities in conical bores; (b) The one-multiply scattering junction for a taper discontinuity only [after (Välimäki, 1995)].
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Because $\mathcal{R}^{-}(\Omega)$ and $\mathcal{R}^{+}(\Omega)$ are different, the one-multiply form of the scattering junction implementation is not possible.

However, if the wavefront surface areas are approximated by cross-sectional areas at the discontinuity, which is reasonable only for small changes in taper rate and cross-section, the reflectances $\mathcal{R}^{-}$ and $\mathcal{R}^{+}$ become identical for a discontinuity of taper only. In this case, a one-multiply scattering junction implementation is possible, as shown in Fig. 11(b) (Välimäki, 1995).

Strictly speaking, however, the propagating wave fronts in cones are spherical and thus $\mathcal{R}^{-}$ and $\mathcal{R}^{+}$ will never be identical, even for a simple taper discontinuity.

Equation (33) can be transformed to the time domain, resulting in the reflection function

$\displaystyle r^{-}(t) = \frac{B-1}{B+1}\delta(t) - \frac{2 B}{(B+1)} \gamma e^{-\gamma t} \epsilon(t),$ (39)

where $\delta(t)$ is the Dirac impulse and $\epsilon(t)$ is the Heaviside unit step function (Martínez and Agulló, 1988).

An appropriate discrete-time filter is found by making the bilinear transform frequency variable substitution in Eq. (33) with the result

$\displaystyle \mathcal{R}^{-}(z) = \frac{B-1}{B+1} - \frac{2 B \gamma / (\alpha + \gamma)}{(B+1)} \left(\frac{1 + z^{-1}}{1 - a_{1}z^{-1}} \right),$ (40)

where $\gamma$ is as given in Eq. (34), $\alpha$ is the bilinear transform constant, and

$\displaystyle a_{1} = \frac{\alpha - \gamma}{\alpha + \gamma}$ (41)

is the first-order filter pole location in the $z-$

plane.

Unfortunately, $\mathcal{R}^{-}(z)$ is unstable for negative $\gamma,$ which occurs any time cone $2$

has a lower rate of taper than cone $1.$

Equivalently, $\mathcal{R}^{-}(\Omega)$ has no causal inverse Fourier transform for negative $\gamma.$

This corresponds to a growing exponential in the reflection function $r^{-}(t).$

The reflectance is a junction characteristic that is “blind” to the boundary conditions upstream or downstream from it. Physically realistic boundary conditions in such cases, however, will always limit the time duration over which the growing exponential can exist.

Experimental measurements of reflection functions due to discontinuities have verified this general behavior (Agulló et al., 1995).

The problem here in terms of digital waveguide modeling, is that the growing exponentials become unstable digital filters in the discrete-time domain.