## 2 Theoretical Background
### Waveguide with an Infinitesimally Thin Strip Placed Inside at Varying Locations
Beyond the critical value \(w_{0}\) one observes the emergence of a resonance duplet in \(\mathit{cond}\ A_{256}(ka)\) corresponding to modal splitting: when \(w=0.3\), the resonance duplet maxima in the graph of the condition number arise at values \(\left(ka\right)_{1}=6.98625\) and \(\left(ka\right)_{2}=6.99220\); when \(w=0.4\), the corresponding values are \(\left(ka\right)_{1}=7.02256\) and \(\left(ka\right)_{2}=7.02266\). Figure 9: Cut-off wavenumbers \(\gamma_{m,n}\) of the infinitesimally thin metallic strip, with centre \((x_{0},y_{0})=(0,0)\) and orientation angle \(\alpha=0^{\circ}\), as a function of semi-width \(w\). The emergence of a new duplet of cut-off wavenumbers is undoubtedly because of the gradual transformation of the original waveguide, of sides \(a\) and \(b\) with \(b/a=2\), into a two-sectioned waveguide, comprising two coupled waveguides, which are square of sides \(a\) and \(b^{\prime}=b/2\), with \(b^{\prime}/a=1\). The basic resonance peak of the condition number graph (_cond_\(A_{256}(ka)\)) remains in a stable position (at \(ka=7.02480\)) whatever the semi-width \(w\) of the strip. The emergence of a new duplet of the resonance peaks in the graph reflects the splitting of the \(TM_{1,2}\)- and \(TM_{2,1}\)-modes in each of the square waveguides. Upon further extension of the strip, so that its ends approach the waveguide walls, the \(TM_{2,2}\)- and \(TM_{1,4}\)-modes of the rectangular waveguide (\(b/a=2\)), become the degenerate \(TM_{1,2}\)- and \(TM_{2,1}\)-modes for the square waveguide (\(b^{\prime}/a=1\) with \(b^{\prime}=b/2\)), as seen in Figure 11d. The calculated cut-off wavenumbers for the cases represented in Figure 11 are collected in Table 7. The calculation was performed with an accuracy of 6 significant decimal digits. Figure 11: Evolution of the resonance duplet in the condition number _cond_\(A_{256}(ka)\) as \(w\) increases: (**a**) \(w=0.25\), (**b**) \(w=0.3\), (**c**) \(w=0.4\), and (**d**) \(w=0.48\). Figure 10: Cut-off wavenumbers \(\gamma_{m,\alpha}\) of the infinitesimally thin metallic strip, with centre \((x_{0},y_{0})=(0,0)\) and orientation angle \(\alpha\), as a function of semi-width \(\gamma_{1,4}(w)\): (**a**) \(\alpha=30^{\degree}\), (**b**) \(\alpha=90^{\degree}\). What Is the Difference in Cut-Off wavenumbers between a Strip of Small but Non-Zero Thickness and an Infinitesimally Thin Strip? When its thickness is small compared with the wavelength, the electromagnetic response of an infinitesimally thin strip is a reasonable approximation to that of a strip of small but non-zero thickness; however, in practice, it is highly desirable to have some indication of the deviation of the cut-off wavenumbers calculated for an infinitesimally thin strip from those obtained for a strip of non-zero thickness. In this section, we will examine the following two thin strip-like conductors of non-zero thickness: elliptic strips (elliptic cylinders with small aspect ratios of the minor \(a^{\prime}\) and major \(b^{\prime}\) semi-axes, i.e., \(a^{\prime}/b^{\prime}\ll 1\)); and cylinders of rectangular cross section of width \(2w\) and thickness \(2d\) with rounded off semi-circular ends, as shown in Figure 12.  
Table 8 compares the calculated cut-off wavenumbers for the first seven modes of the rounded rectangular waveguide sides with \(a=1\) and \(b=2\) when the inner conductor is an infinitesimally thin strip, or an elliptic strip of aspect ratio \(a^{\prime}/b^{\prime}=d/w\) taking the values 0.005, 0.01,0.02 and 0.05. The strip is located symmetrically inside the waveguide and is oriented at an angle \(\alpha=0^{\circ}\). The semi-width parameter \(w\) for each type of strip is fixed at 0.25.