## 2 Theoretical Background
### Waveguide with an Infinitesimally Thin Strip Placed Inside at Varying Locations
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline  & \(d/_{w}\) & \(\mathbf{TM_{1,1}}\) & \(\mathbf{TM_{1,2}}\) & \(\mathbf{TM_{1,3}}\) & \(\mathbf{TM_{2,1}}\) & \(\mathbf{TM_{2,2}}\) & \(\mathbf{TM_{1,4}}\) & \(\mathbf{TM_{2,3}}\) \\ \hline Thin strip & 0.000 & 4.699720 & 4.804192 & 6.475592 & 6.880896 & 7.024814 & 7.710194 & 7.853981 \\ \hline  & 0.005 & 4.702703 & 4.805858 & 6.482328 & 6.888568 & 7.025587 & 7.713700 & 7.857580 \\ \cline{2-9}  & \(\Delta_{m,n}\) & 0.000635 & 0.000347 & 0.001040 & 0.001115 & 0.000110 & 0.000455 & 0.000458 \\ \cline{2-9}  & 0.01 & 4.705672 & 4.807522 & 6.488118 & 6.896279 & 7.026372 & 7.717212 & 7.861226 \\ \cline{2-9}  & \(\Delta_{m,n}\) & 0.001266 & 0.000693 & 0.001934 & 0.002236 & 0.000222 & 0.000910 & 0.000922 \\ \cline{2-9}  & 0.02 & 4.711575 & 4.810844 & 6.499754 & 6.911824 & 7.027974 & 7.724254 & 7.868661 \\ \cline{2-9}  & \(\Delta_{m,n}\) & 0.002522 & 0.001385 & 0.003731 & 0.004495 & 0.000450 & 0.001824 & 0.001869 \\ \cline{2-9}  & 0.05 & 4.728984 & 4.820755 & 6.535065 & 6.959426 & 7.033045 & 7.745534 & 7.892164 \\ \cline{2-9}  & \(\Delta_{m,n}\) & 0.006227 & 0.003448 & 0.009184 & 0.011413 & 0.001172 & 0.004584 & 0.004862 \\ \hline \hline \end{tabular}
\end{table} 
Table 8: Cut-off wavenumbers for \(TM_{m,n}\)-modes in the rounded rectangular waveguide of sides \(a=1\) and \(b=2\); rounding radius \(\varrho=0.01\) with various inner conductors with the centre located at \((x_{0},y_{0})=(0,0)\) and orientation angle \(\alpha=0^{{}^{\prime}}\). Each column shows the cut-off wavenumber for the infinitesimally thin strip of semi-width \(w\), the corresponding cut-off wavenumber and the relative difference \(\Delta_{m,n}\) when the strip is replaced by an elliptic strip of semi-width \(w\) equal to its semi-major axis and thickness \(2d\) equal to its minor axis. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline  & \(d/_{w}\) & \(\mathbf{TM_{1,1}}\) & \(\mathbf{TM_{1,2}}\) & \(\mathbf{TM_{1,3}}\) & \(\mathbf{TM_{2,1}}\) & \(\mathbf{TM_{2,2}}\) & \(\mathbf{TM_{1,4}}\) & \(\mathbf{TM_{2,3}}\) \\ \hline Thin strip & 0.000 & 4.699720 & 4.804192 & 6.475592 & 6.880896 & 7.024814 & 7.710194 & 7.853981 \\ \hline  & 0.01 & 4.730638 & 4.826366 & 6.506220 & 6.928066 & 7.030347 & 7.749254 & 7.870873 \\ \cline{2-9}  & \(\Delta_{m,n}\) & 0.006579 & 0.004616 & 0.004730 & 0.006855 & 0.000788 & 0.005066 & 0.002151 \\ \cline{2-9}  & 0.02 & 4.754750 & 4.843135 & 6.536790 & 6.972584 & 7.036501 & 7.780812 & 7.888865 \\ \cline{2-9}  & \(\Delta_{m,n}\) & 0.011709 & 0.008106 & 0.009451 & 0.013325 & 0.001664 & 0.009159 & 0.004442 \\ \cline{2-9}  & 0.05 & 4.816639 & 4.886328 & 6.631948 & 7.058027 & 7.107963 & 7.867458 & 7.950136 \\ \cline{2-9}  & \(\Delta_{m,n}\) & 0.024878 & 0.017097 & 0.024145 & 0.025742 & 0.011836 & 0.020397 & 0.012243 \\ \hline \hline \end{tabular}
\end{table} 
Table 9: Cut-off wavenumbers for \(TM_{m,n}\)-modes in the rounded rectangular waveguide of sides \(a=1\) and \(b=2\); rounding radius \(\varrho=0.01\) with various inner conductors with the centre located at \((x_{0},y_{0})=(0,0)\) and orientation angle \(\alpha=0^{{}^{\prime}}\). Each column shows the cut-off wavenumber for the infinitesimally thin strip of semi-width \(w\), the corresponding cut-off wavenumber and the relative difference \(\Delta_{m,n}\) when the strip is replaced by a rounded rectangular strip of semi-width \(w\) and thickness \(2d\). to the parameter \(d/w\); for example, as \(d/w\) increases from 0.005 to 0.05, the cross-sectional area increases tenfold. The same behaviour is observed in the relative shifts \(\Delta_{m,n}\); for example, as \(d/w\) increases from 0.005 to 0.05, the relative shift \(\Delta_{1,1}\) increases very nearly by tenfold, from 0.000635 to 0.006227. Let us denote by \(\Delta_{m,n}(d/w)\) the dependence of the relative shift of the TM\({}_{m,n}\)-mode on the parameter \(d/w\).