## 2 Theoretical Background
### Waveguides of Arbitrary Cross Section Filled by M Metallic Conductors  
In this section, we consider the more complex case, when a positive number \(M\) of metallic conductors is placed inside the waveguide. As before, the PEC waveguide is the outer boundary that is the closed smooth contour \(L_{0}\) of arbitrary shape. It encloses \(M\) inner conductors, which also have PEC boundaries that are arbitrarily variably shaped smooth contours \(L_{1},L_{2},\ldots,\;L_{M}\) (see Figure 2); the contours are mutually non-intersecting. Using the superposition principle and classical potential theory, we seek the solution for the \(z\)-component of the electric field \(E_{z}(q)\) in the domain \(D\) as a sum of individual contributions \(E_{z}^{(m)}(q),\;m=0,1,2,\ldots,M\) from each conductor (including the outer conductor \(m=0\)) in the form of single-layer potentials  
\[U(q)=\sum_{m=0}^{M}E_{z}^{(m)}(q)=\sum_{m=0}^{M}\int_{L_{m}}G_{2}(k|p_{m}-q|)Z _{m}(p_{m})dl_{p_{m}} \tag{25}\]  
where \(Z_{m}(p_{m}),\;m=0,1,2,\ldots,M,\) are unknown density functions, and the argument of the Green's function \(G_{2}\) depends on the distance \(|p_{m}-q|\;\) between observation point \(q\) and points \(p_{m}\) on the contour \(L_{m}\):  
\[G_{2}(k|p_{m}-q|)=-\frac{i}{4}H_{0}^{(1)}(k|p_{m}-q|). \tag{26}\]  
Figure 2: Waveguide of arbitrary cross section with \(M\) metallic inserts. We now enforce the Dirichlet boundary condition on each of the \(M+1\) contours:  
\[U\left(q\right)\rfloor_{q\in L_{s}}=0,\;s=0,1,2,\ldots,M. \tag{27}\]  
This produces \(M+1\) coupled boundary integral equations  
\[\sum_{m=0}^{M}\int_{L_{m}}G_{2}(k|p_{m}-q_{s}|)Z_{m}(p_{m})dl_{p_{m}}=0,\;q_{s} \in L_{s} \tag{28}\]  
where the integral equations are indexed by \(s=0,1,2,\ldots,M\). Observe that exactly one term in each of the set (28) of integral Equation has a (logarithmically) singular kernel and that the kernels appearing in the remaining terms are non-singular and smooth. Thus, for further consideration, it is convenient to rearrange each of these integral equations in the following form:  
\[\int_{L_{s}}G_{2}(k|p_{s}-q_{s}|)Z_{s}(p_{s})dl_{s}+\sum_{m=0,m\
eq s}^{M}\int_{ L_{m}}G_{2}(k|p_{m}-q_{s}|)Z_{m}(p_{m})dl_{p_{m}}=0,\;q_{s}\in L_{s} \tag{29}\]  
If we set \(s=0\) in (29), the sum  
\[\sum_{m=1}^{M}\int_{L_{m}}G_{2}(k|p_{m}-q_{0}|)Z_{m}(p_{m})dl_{p_{m}}\]  
(in which \(q_{0}\in L_{0}\), thus containing terms with non-singular kernels) may be interpreted as a smooth perturbation of Equation (5) describing the hollow waveguide. Noting that all the contours are mutually non-intersecting, we may treat all the integral equations in (29) in the same manner without particularly distinguishing that corresponding to the outer waveguide. We employ parameterization in a similar manner to that of Section 2.1. On each contour \(L_{m}\) the points \(q(\theta_{m})=(x_{m}(\theta_{m}),\;y_{m}(\theta_{m}))\) are represented by smooth periodic functions \(x_{m},\;y_{m}\) so that \(x_{m}(-\pi)=x_{m}(\pi)\;\) and \(y_{m}(-\pi)=y_{m}(\pi)\), with a corresponding differential of arc length  
\[l_{m}(\tau_{m})=\sqrt{\left(x_{m}^{{}^{\prime}}(\tau_{m})\right)^{2}+\left(y_ {m}^{{}^{\prime}}(\tau_{m})\right)^{2}}.\]  
The coupled system of integral Equation (29) then takes the form  
\[\int_{-\pi}^{\pi}H_{0}^{(1)}(kR_{ss}(\theta_{s},\tau_{s}))z_{s}(\tau_{s})d \tau_{s}+\sum_{m=0,m\
eq s}^{M}\int_{-\pi}^{\pi}H_{0}^{(1)}(kR_{ms}(\theta_{s },\tau_{m}))z_{m}(\tau_{m})d\tau_{m}=0, \tag{30}\]  
where \(s=0,\,1,\,2,\,\ldots,M\); here  
\[R_{ss}(\theta_{s},\tau_{s})=\sqrt{\left(x_{s}(\theta_{s})-x_{s}(\tau_{s}) \right)^{2}+\left(y_{s}(\theta_{s})-y_{s}(\tau_{s})\right)^{2}}\]  
is the distance between two points on the contour \(L_{s}\) parametrized by \(\theta_{s}\) and \(\tau_{s}\); when \(m\
eq s\), and  
\[G_{2}(k|p_{m}-q|)=-\frac{i}{4}H_{0}^{(1)}(k|p_{m}-q|). \tag{26}\]0  
is the distance between two points lying on the non-intersecting contours \(L_{m}\) and \(L_{s}\)--this is always strictly positive, being at least the positive minimum distance between these conductors.