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Method for calculation of input impedance

We have presented the equations for the pressure and axial velocity in a duct in terms of the unknowns $P_n(z)$ and $U_n(z)$. Now we define the left hand side (negative $z$ side) of a waveguide of arbitrary cross-section as the input end and the right hand side (positive $z$ side) as the output end. The waveguide is then approximated by a series of uniform sections as shown in figure 2.8 (cylinders in circular geometry and rectangular sections in rectangular geometry).
Figure 2.8: Horn approximated by a series of cylinders
\begin{figure}\begin{center}
\epsfig{figure=chapter2/horncyl.eps, width=5cm} \end{center}\end{figure}

In plane wave acoustics the impedance is defined as the ratio of the acoustic pressure and volume velocity. For the multimodal case we define the acoustic impedance matrix, $Z$, as follows:

\begin{displaymath}
{\mathbf P} = Z {\mathbf U}
\end{displaymath} (2.95)

so that
\begin{displaymath}
P_n = \sum^\infty_{m=0} Z_{nm} U_m
\end{displaymath} (2.96)

and therefore $Z_{nm}$ is the factor of contribution to the pressure amplitude of the $n$th mode due to the volume velocity amplitude of the $m$th mode. The vectors and matrices are infinite but must be truncated before numerical implementation.



Subsections
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: Projection across a discontinuity Up: Multimodal propagation in acoustic Previous: Solutions for rectangular cross-section   Contents
Jonathan Kemp 2003-03-24