## 2 Theoretical Background
### Analysis of a Circular-Rectangular Waveguide
As indicated by the bold digits, the cut-off wavenumbers for three modes are computed with increasing accuracy as the truncation number is increased. The process stops when computed wavenumbers stabilize at 6 decimal digits. The radius \(\varrho\) of curvature of the rounded corner used is 0.02 cm. Air-Filled Rectangular Waveguides with Arbitrary Located Inner Conductors: Metallic Strips and Strip-like Elliptic Cylinders  
In this section, we investigate the cut-off wavenumbers in which inner conductors of various shapes and locations occupy the waveguide. The geometry is shown in Figure 8. Inner conductors of three different cross sections are considered: (a) an infinitely thin strip; (b) a thick strip, which is rectangular with rounded corners; and (c) an inner conductor of elliptic cross section. Inner conductors may take arbitrary locations inside the outer conductor; their cross section may also be rotated around their symmetry axis relative to the \(x-y\) axis, subject to the restriction that the inner conductor does not make electrical contact with the walls of the rectangular waveguide. The sides of the rectangular waveguide are \(a\) and \(b,~{}(b>a)\); its corners are rounded with a radius of curvature \(\varrho_{0}\). The centre of \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Truncation Number \(N\)** & \(\mathbf{TM_{11}}\) & \(\mathbf{TM_{22}}\) & \(\mathbf{TM_{23}}\) \\ \hline
128 & 1.99478678 & 2.87210785 & 3.98418755 \\ \hline
512 & 1.99480566 & 2.87217868 & 3.98437637 \\ \hline
1024 & 1.99480584 & 2.87217966 & 3.98437902 \\ \hline
2048 & 1.99480588 & 2.87217979 & 3.98437935 \\ \hline
4096 & & 2.87217980 & 3.98437939 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Computed values of the cut-off wavenumbers \(k_{c}\) (cm\({}^{-1}\)) for three modes as a function of truncation number. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**TM-Mode** & \begin{tabular}{c} **MAR,** \\ \(\varrho\) = 0.1 cm** \\ \end{tabular}  & \begin{tabular}{c} **MAR,** \\ \(\varrho\) = 0.02 cm** \\ \end{tabular}  & \ \begin{tabular}{c} **MAR,** \\ \(\varrho\) = 0.005 cm** \\ \end{tabular}  & \begin{tabular}{c} **[PERSON]. [PERSON] et al. [30]** \\ \end{tabular}  \\ \hline \(\textit{TM}_{11}\) & 1.9948124 & 1.9948058 & 1.9948058 & 1.9948099 \\ \hline \(\textit{TM}_{21}\) & 1.9953724 & 1.9953659 & 1.9953659 & 1.9953650 \\ \hline \(\textit{TM}_{12}\) & 2  
symmetry of the inner conductor is located at \((x_{0},y_{0})\), and its angle \(\alpha\) (\(0\leq\alpha\leq 2\pi\)) of orientation is measured by the anti-clockwise angle of its major axis of symmetry relative to the \(x-y\) axis. For each type of inner conductor, a common parameter will be used to describe its width \(l\), semiwidth \(w=l/2\) and thickness \(2d\); these are defined as follows. The infinitesimally thin strip is characterized by its width \(l\); its thickness \(2d\) is zero. For the conductor of elliptic cross section with minor and major semi-axes \(a^{\prime}\) and \(b^{\prime}\), respectively, \(l\) is defined to be \(2b^{\prime}\), and thickness \(2d\) is defined to be \(2a^{\prime}\). The thick strip is characterized as a rectangle of width \(l\) and thickness \(2d\); its corners are rounded with a radius of curvature \(e_{l}\). In all calculations, we set \(a=1\) unit so that other parameters may be interpreted as normalized values; for example, the non-dimensional ratio \(b/a=2\) simply becomes \(b=2\); the non-dimensional cut-off relative wavenumbers \(k_{c}a\) become the values \(k_{c}\). The idealized problem (a) with the infinitesimally thin strip is not directly soluble by the method described in this paper; it is solved using the so-called \"cavity solution\" of [28] when the thin strip is regarded as a part of a larger smooth closed contour. The results are discussed in the next section.