## 2 Theoretical Background
### Cut-Off Wavenumbers of an Elliptical Waveguide  
When the eccentricity \(e\) of the waveguide cross section increases from 0 (a circular waveguide) to 0.1 (a slightly deformed circular waveguide), the \(TM_{c11}\) and \(TM_{s11}\) modes, which are degenerate in the circular waveguide, split into two independent modes in the elliptic waveguide. The resonance peaks reveal the approximate values of the cut-off wavenumbers: \(kb\approx 3.832\)\((e=0)\) and \(kb\approx 3.837\), \(3.846\)\((e=0.1)\). These and the other approximate values of the cut-off wavenumbers found serve as initial values for finding more precise results in solving the dispersion equation \(detA_{N}=0\) as the truncation number \(N\) is increased. A typical sequence of approximate solutions \(\gamma_{c01}^{(n)}\) of the dispersion equation \(detA_{N}=0\) for the wavenumber \(kb=\gamma_{c01}\) of the lowest even mode \(TM_{c01}\) is shown in Table 1 (\(\gamma_{c01}\) is the eigenvalue (cut-off wave number) of the mode \(TM_{c01}\)). It demonstrates convergence as \(N\) increases. Application of [PERSON]'s delta-squared process (see, for example, [31]) shows that \(\gamma_{c01}^{(N)}\approx\gamma_{c01}+cN^{-p}\) for some constant \(c\) independent of \(N\) and an exponent \(p\) very close to 3; the extrapolated value of \(2.410889876\), achieved with the values \(\gamma_{c01}^{(N)}\) computed for \(N=64,\ 128\) and \(256\), agrees, to 9 decimal places, with that achieved in the computation \begin{table}
\begin{tabular}{c c} \hline \hline
**Truncation Number \(N\)** & **Computed Value \(\tau_{c01}^{(N)}\) of \(\gamma_{c01}\)** \\ \hline \hline
32 & 2.41063303918760 \(-\) i. 0.000000066967616 \\ \hline
64 & 2.410857871457190 \(-\) i. 0.000000001040154 \\ \hline
128 & 2.410885878808779 \(-\) i. 0.0000000000016227 \\ \hline
256 & 2.410889376581807 \(-\) i. 0.0000000000000253 \\ \hline
512 & 2.410889813714743 \(-\) i. 0.000000000000004 \\ \hline
1024 & 2.410889868251958 \(-\) i. 0.000000000000000 \\ \hline
2048 & 2.410889875181512 \(-\) i. 0.0000000000000000 \\ \hline
4096 & 2.410889876035204 \(-\) i. 0.0000000000000000 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Convergence of approximations \(\gamma_{c01}^{(N)}\) to wavenumber \(\gamma_{c01}\) of the lowest even mode \(TM_{c01}\). of \(\gamma_{c01}^{(4096)}\) with the much larger truncation number of 4096. This suggests that Aitken extrapolation is an effective tool for achieving high accuracy for modest truncation numbers. Let us compare the results of the calculation of the first three cut-off wavenumbers \(\gamma_{c01},\ \gamma_{c11}\) and \(\gamma_{s11}\) of the elliptical waveguides having eccentricities \(e=0.1,\ 0.5\) and \(0.9\) with those obtained in [28]. In this paper, the eigenmode sequence for an elliptical waveguide with arbitrary eccentricity was validated against the values obtained by direct calculation of the parametric zeros of the modified Mathieu functions of the first kind. The results are presented in terms of cut-off wavelengths \(\lambda_{c}/b\), which we have expressed in terms of the cut-off relative wavenumbers \(k_{c}b=2\pi/(\lambda_{c}/b)\); the comparison is displayed in Table 2. The results show near-perfect coincidence. There is a maximum difference of one unit in the last (eighth) decimal digit for some values of the cut-off wavenumbers; most probably, the difference is due to a rounding error resulting from the rearrangement of the results in [28]. The computations in this and subsequent sections were performed on a standard modern PC with sufficient memory for the inversion of a matrix of order 4096 by the usual process of Gaussian elimination. The programming used the MATLAB (R2023b) environment (including its procedures for Gaussian elimination and the Fast Fourier Transform) with an unoptimized code. For a fixed truncation number \(N\), the computational time did not depend on the eccentricity \(e\) of the ellipse nor on the relative wave number \(ka\).