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Solutions for circular cross-section

The matrices $F$ and $V$ defined in (2.80) can be found analytically for circular cross-section using the standard integral in equation (A.1) of appendix A. A full derivation is given in appendix B. The result is that each element is a function of $\beta = R_1/R_2$ and the element $(n,m)$ is 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} (2.85)

where $\beta = R_1/R_2$ and $F(0,0)=1$ and
\begin{displaymath}
V_{nm}(\beta) = F_{nm}(1/\beta)
\end{displaymath} (2.86)

When the change in cross-section tends to zero (ie. $\beta$ tends to 1) we obtain
\begin{displaymath}
F \approx I - \epsilon Q
\end{displaymath} (2.87)

with $I$ being the identity matrix (a diagonal matrix with all the entries having a value of 1). $\epsilon$ is the fractional change in cross-section, $\epsilon = (S_2 - S_1)/S_1$ and $Q$ is a matrix whose elements are given by
\begin{displaymath}
Q_{nm} = \left\{ \begin{array}
{r@{\quad:\quad}l}
0 & n = ...
...m^2 - \gamma_n^2} & \mbox{otherwise}. \\
\end{array} \right.
\end{displaymath} (2.88)


Back to Kemp Acoustics Home next up previous contents
Next: Solutions for rectangular cross-section Up: Multimodal equations at a Previous: Multimodal equations at a   Contents
Jonathan Kemp 2003-03-24