A bifurcated circular waveguide problem

Wave propagation in a bifurcated impedance-lined cylindrical waveguide

“…We have also extended the usual Sturm-Liouville method in the appendices to give useful information on the disposition of the poles and zeros of the complicated special-function eigenvalue equations that arise from third-type boundary conditions with complex coefficients. We note that by letting ξ → ∞ we obtain the solution to the problem of the radiation from a rigid semi-infinite duct into an infinite lined duct that was given by [14]. Finally, we intend to deal with the important extensions of this work for the electromagnetic communication in subsurface tunnels in a future publication.…”

“…We have been able to numerically evaluate the reflection coefficient for the dominant surface-wave mode, which is of practical importance in applications, and which involves complicated split functions (that arise from the Wiener-Hopf technique). This is achieved by numerically evaluating the split functions defined in terms of suitable Cauchy integrals, rather than the usual method of infinite products; see the references cited in [14]. The numerical evaluation of the infinite product would normally be a non-trivial matter because of the infinite number of complex factors.…”

“…1 To the above conditions we add the incident field and those conditions at infinity which are relevant to the nature of the propagating modes that the various duct regions can sustain. It is not difficult to show, by using the method of separation of variables [14] that the appropriate modal expansions in the various duct regions at infinity are stated below. For z → −∞, 0 ≤ r ≤ a:…”

“…The satisfaction of the above conditions (2) to (14) will result in a unique solution to the boundary-value problem formulated.…”

“…The cut Riemann sheet on which we shall work is defined by 0 ≤ arg κ ≤ π . The contour of integration is indented below the point −k and above the point k. 2 The quantities A(ν), B(ν),and C(ν) are as yet unknown; however, the edge condition (14) requires that as |ν| → ∞…”