## 2 Theoretical Background
### Analysis of a Circular-Rectangular Waveguide  
In the previous two sections, we demonstrated the capability of the MAR-based solution to provide an independent estimate of the correctness of results obtained by other methods. In other words, we may assert that the MAR-based solution should be recognized and treated as a benchmark solution. We wish to illustrate this assertion with one more example concerning higher-order modal characteristics of circular-rectangular coaxial waveguides [32]. The authors of this paper carried out a rigorous analysis combining the orthogonal expansion method and the Galerkin method for finding the higher-order eigenmodes in a circular-rectangular (C-R) waveguide. Despite a faultless theoretical part, the calculation of the cut-off frequencies seems to have significant errors, which are especially apparent for the higher modes. In contrast to the validated results of the previous sections, the claimed accuracy of the results presented in [32] cannot be validated in full. With the aim of validating results in [32], we modelled the outer conductor by a rectangle with rounded-off corners and the inner conductor by a circle, as shown in Figure 7. We consider the waveguide with a shifted inner conductor (\(d\
eq 0\)); the case examined in [32] is a particular case (namely, \(d=0\)) of this more general problem. The cut-off wavenumbers were calculated in [32] for a waveguide with an outer conductor of exactly rectangular cross section. The MAR requires that the contours of the conductors are smooth, so our calculations employed rectangular structures with corners rounded off by circular arcs of a suitably chosen radius \(\varrho\). To eliminate the influence of the rounded-off corners, we performed a sequence of calculations as follows. We used the same size structures as [32]: the inner circular conductor having radius \(r_{0}=0.636\) cm, the outer rectangular conductor having sides \(a=2.54\) cm, \(b=5.08\) cm; the rounding radius was chosen to be \(\varrho=0.1,\ 0.02\) and \(0.005\) cm. It was observed that the seventh decimal digit stabilizes when \(\varrho\) does not exceed \(0.02\) cm; further reduction in this parameter does not change the cut-off wavenumbers when calculated with an accuracy of \(7\) significant decimal digits. The results obtained by the MAR and the method of [32] are presented in Table 5. Notice that when \(\varrho\) takes the value \(0.02\) cm and \(0.005\) cm, all the computed cut-off wavenumbers \(k_{c}\) satisfy \(k_{c}\varrho\leq 0.08\) and \(k_{c}\varrho\leq 0.02\), i.e., the corner radius is electrically small and even very small in the latter case. The average element size of \(0.1016\) cm used by authors of [32] in the finite element method for cross validation is five times the size of the round-off radius (\(0.02\) cm). Related studies of the impact of rounding are given in [33], where it is shown that the perturbations at wavenumber \(k\) to the surface distribution and scattering pattern of the rectangular corner are \(O\big{(}(k_{Q})^{2/3}\big{)}\) and \(O\big{(}(k_{Q})^{4/3}\big{)}\), respectively, as \(k_{Q}\to 0\)  
Figure 7: Geometry of a circular–rectangular waveguide with shifted inner conductor. A comparison of the final two columns of Table 5 shows that the claim in [32] of 7 significant decimal digits of accuracy cannot be sustained. It seems likely that the discrepancy is due to an insufficient number of terms taken in the Bessel-Fourier series given in Equations (29) and (30) of [32]. In a parallel way, a discrepancy would arise in using the _MAR_ if an improper choice of the truncation number \(N\) were made. The correctness of a given number of decimal digits is verified by stabilization of the last decimal digit under successive increments of the truncation number. The computation of higher modes with larger cut-off wavenumbers requires larger truncation numbers to keep the same number of correct decimal digits as lower modes. The data collected in Table 6 illustrate this assertion.