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Projecting the impedance matrix

The equations for projection of the impedance matrix were derived in section 2.6. Remember that the labels (0), (1) and (2) refer to planes 0, 1 and 2 in figure 4.1.
Figure 4.1: Detail of a waveguide consisting of straight sections of length $d$ joined discontinuously
\begin{figure}\begin{center}
\epsfig{figure=chapter4/disc.eps, width=5cm} \end{center}\end{figure}

By way of summary, the equation for projection across a discontinuity is

\begin{displaymath}
Z^{(1)} = F Z^{(2)} F^{\mathrm{T}}, \mbox{\hspace{1cm}}
S_1 < S_2
\end{displaymath} (4.3)

where $S_1 = \pi R_1^2$, $S_2 = \pi R_2^2$ are the cross-sectional areas and
\begin{displaymath}
Z^{(1)} = V^{-1} Z^{(2)} (V^{\mathrm{T}})^{-1}, \mbox{\hspace{1cm}}
S_1 > S_2.
\end{displaymath} (4.4)

The projection matrices are given by
\begin{displaymath}
F_{nm}(\beta) =
\frac{2 \beta \gamma_m J_1(\beta \gamma_m)}
{(\beta^2 \gamma_m^2 - \gamma_n^2)J_0(\gamma_m)}
\end{displaymath} (4.5)

where $\beta = R_1/R_2$ with $F(0,0)=1$ and $V_{nm}(\beta) = F_{nm}(1/\beta)$.

The equation for projection through a distance $d$ is

\begin{displaymath}
Z^{(0)} = (Z^{(1)} + i D_3 Z_c)(i D_3 Z_c^{-1} Z^{(1)} + I)^{-1}
\end{displaymath} (4.6)

where
\begin{displaymath}
D_3(n,m) = \left\{ \begin{array}
{r@{\quad:\quad}l}
\tan(k_n d) & n = m, \\
0 & n \neq m.
\end{array} \right.
\end{displaymath} (4.7)

Here
\begin{displaymath}
k_n = \left\{ \begin{array}
{r@{\quad:\quad}l}
-\sqrt{k^2 ...
...{R}\right)^2}
& k > \frac{\gamma_n}{R}.
\end{array} \right.
\end{displaymath} (4.8)

is the wavenumber of the $n$th mode along the tube neglecting the effect of losses; the corresponding expression for lossy propagation is given in section 2.4.1. $\gamma_n$ is the $n$th zero of the Bessel function $J_1$ as tabulated in appendix A.


This thesis has moved to Jonathan Kemp Thesis at http://www.kempacoustics.com/thesis
Please change your bookmark/reference to reflect this change as this site may be discontinued
next up previous contents
Next: Numerical implementation Up: Review of input impedance Previous: The radiation impedance matrix   Contents
Jonathan Kemp 2003-03-24