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Loss-less propagation

The boundary condition for the eigenfunction of the $n$th higher mode defined in equation (2.31) is:

\begin{displaymath}
\frac{\partial\psi_n}{\partial x} = 0, \mbox{\hspace{1cm}} x = -a,a
\end{displaymath} (2.63)


\begin{displaymath}
\frac{\partial\psi_n}{\partial y} = 0, \mbox{\hspace{1cm}} y = -b,b
\end{displaymath} (2.64)

The solution is most conveniently expressed by separating it into the $x$ dependent and $y$ dependent parts:
\begin{displaymath}
\psi_n = \phi_{n_x}\sigma_{n_y}
\end{displaymath} (2.65)

where
\begin{displaymath}
\phi_{n_x} = \left\{ \begin{array}
{r@{\quad:\quad}l}
1 & ...
...\sqrt{2} \cos(n_x \pi x / a) & n_x>0. \\
\end{array} \right.
\end{displaymath} (2.66)


\begin{displaymath}
\sigma_{n_y} = \left\{ \begin{array}
{r@{\quad:\quad}l}
1 ...
...\sqrt{2} \cos(n_y \pi y / b) & n_y>0. \\
\end{array} \right.
\end{displaymath} (2.67)

Performing the differentiation from equation (2.31) gives the corresponding eigenvalues as
\begin{displaymath}
\alpha_n=\sqrt{\left(\frac{n_x \pi}{a}\right)^2
+ \left(\frac{n_y \pi}{b}\right)^2}.
\end{displaymath} (2.68)

As with circular cross-section it is possible to use a lossy boundary condition to give lossy versions of $\psi$ and $\alpha$ but the effect of losses will be noticeable in the $z$ direction only and will therefore be represented entirely by the choice of $z$ direction wavenumber, $k_n$.

\begin{displaymath}
k_n = \left\{ \begin{array}
{r@{\quad:\quad}l}
-\sqrt{k^2 ...
...right)^2
+ \left(n_y \pi / b \right)^2.
\end{array} \right.
\end{displaymath} (2.69)


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: Lossy propagation Up: Solutions for a uniform Previous: Solutions for a uniform   Contents
Jonathan Kemp 2003-03-24