Added Solutions of Boundary Value Problems for Double-Layer Guiding Structures

E xistence of the added waves in double-layered cylindrical guiding structures is considered in the article. The added waves ensure completeness of solutions of boundary value problems for layered guiding structures. It is shown that eigenvalues, corresponding added waves (or adjoint waves), can be detected at points of the Jordan’s multiplicity of the wavenumbers. At these points characteristics of two normal waves joint and complex waves originate . The added waves define frequency boundaries of a complex resonance which originates as a result of a exciting of couple of complex waves.


Introduction
The question about added waves of guidingelectrodynam ics structures is emerged in connection with boundary value problems: L(u)=0; Uν=0 (1) (L -Differentiation operator, U ν =0 -system of boundary conditions, ν=1,2,3 … n), for which we can formulate added boundary value problems, consisting of differential equations: where q=1,2 …k, and boundary conditions: Functions ϕ q in Eq. (2) and Eq. (3) are called the added (or adjoint) functions. Functions ϕ q are added to the function ϕ 0 . These functions obey Eq. (2) and boundary conditions Eq.
Solutions of the adjoined boundary value problems describe the so-called added waves or adjoint waves (AWs) [2]. In [3], it was showed that AWs origin in the points of the Jordan's multiplicity of the wavenumbers of normal waves. AWs ensure the completeness of the system of normal waves, which are violated in these points.
Completeness of system of normal waves is used for solving of diffraction problems, intended for the calculation of functional units for microwave and extremely high frequency. Completeness of diffraction basis determines correctness of solutions of the boundary value problem.
Features of AWs influence characteristics of complex resonance [4][5][6], which originates in guiding structures (the result of pair interaction of complex waves).
A steepness of front of frequency characteristic of a band-pass filter on complex resonance [7] depends on accounting of AWs.
Characteristic property of AWs is linear dependence of their amplitudes from longitudinal coordinate. First particular information about results of solution of the added boundary value problems for multilayer guiding cylindrical structures was perhaps presented in [8][9][10].

The Boundary Value Problems for Added Waves of Double-Layer Guiding Cylindrical Structures
In this paper guiding cylindrical structures with two concentric layers are discussed. The article deals with a round shielded waveguide with two concentric layers (Fig.  1a) or a round dielectric waveguide placed in infinite homogeneous medium (Fig. 1b).
These structures are used in functional units for microwave, extremely-high frequency and optical devices.
Electromagnetic fields in these guiding structures are described by longitudinal components of the Herz vectors of electric and magnetic fields, which obey the Helmholtz equation:  (9) α and β are the transverse and longitudinal wave numbers, that are related as: Eqs. (7) and (8) are added to Eqs (6) and (7), respectively. R(r) is either the Bessel function (for the radial field profile in the inner layer of the guiding structure) or a combination of cylindrical functions of the 1st and 2nd kind obeying the corresponding (Dirichlet or Neumann) boundary conditions on the shield (for the radial field profile in the outer layer) or the Hankel function for the field in the outer infinite layer of a round dielectric waveguide.
The Herz vectors (5) are written [5] in the following form: where q = 1, 2 is the layer number Terms of these functional equations, which haven't coordinate dependence, (fulfilment of condition (11)  ) ( 1 a a J a n a

Numerical Solution of Dispersion Characteristics of the Added Waves of Round Double-Layer Shielded Waveguide
A compatible solution of the systems (Eqs. 15, 16a and 16b) on the complex planes of wavenumbers was obtained using a combined approach in the search for complex roots of transcendent equations. This approach represented a combination of the Muller method and the phase variation method, thus using advantages of both rapidly finding a complex root by the first method and reliably identifying this root by the second method.
Complex wave is showed by chain curve. Fig. 4 is an oversize.

Conclusions
It is shown that multiple eigenvalues of boundary value problems can exist. The multiplicity of eigenvalues shows a possibility of a existence waves which have different features (different longitudinal dependences of field).
The AWs are characterized by a linear dependence of the field on the longitudinal coordinate. The AWs must be taken into consideration for a calculating and a constructing of components, which use the effect of the complex resonance.
We have showed that two kinds of solutions of added boundary value problems can be. The first solution is only at points of the Jordan's multiplicity of the wavenumbers of normal waves.
Second solution is at points of curve crossings of solutions of three transcendent equations which are obtained from the condition of an existence of solutions of a system of algebraic equations. Second solution of added boundary value problem is calculated for any parameters of guiding cylindrical structures.