I. A. Vdovichenko  Ph.D. (Phys.-Math.), Associate Professor,

Nizhny Novgorod State Technical University n.a. R.E. Alekseev

E-mail: physics@nntu.ru

G. S. Malyshev  Post-graduate Student,

Nizhny Novgorod State Technical University n.a. R.E. Alekseev

E-mail: gr1g0r@yandex.ru

N. A. Novoselova  Ph.D. (Eng.), Associate Professor,

Nizhny Novgorod State Technical University n.a. R.E. Alekseev

G. I. Shishkov  Ph.D. (Eng.), Professor,

Nizhny Novgorod State Technical University n.a. R.E. Alekseev

The notion of «proper and improper waves (oscillations)» is usually associated with the formulation of the corresponding boundary value problems. Own waves (oscillations) are described by solutions of completely homogeneous boundary value problems - a homogeneous differential equation and homogeneous boundary conditions. Invalid waves (vibrations) are associated either with inhomogeneous boundary value problems in which the inhomogeneous Helmholtz equation is solved, or with semi-homogeneous boundary value problems-a homogeneous equation and partially non-homogeneous (nonzero) boundary conditions. The improper waves of the second category by classification are characteristic, for example, for open directing structures - waves that do not satisfy the zero boundary condition at infinity. In the present paper, improper waves (oscillations) are considered, which are proposed to be classified as waves (oscillations) connected to a source.

In this paper we consider boundary self-consistent problems that describe oscillations and waves connected to a source. In the formulated problems, the inverse influence of the field on the source is taken into account, since the wave numbers in both the field functions and in the source functions are the same. Amplitudes of these waves and oscillations depend on the longitudinal coordinate.

In the article it is shown that in the range of existence of complex waves under the condition of their pair excitation in a circular twolayer shielded waveguide, a phenomenon called «complex resonance» arises that is described by a self-consistent boundary-value problem and is considered as a vibration attached to the source. The oscillation is improper, since it exists only in the presence of a source through which the power fluxes of two complex conjugate waves are closed.

1. Unger H.G. Planarnye i volokonnye opticheskie volnovody. M.: Mir. 1980.
2. Raevskij S.B., Titarenko A.A. Reshenie vneshnej kraevoj zadachi o rasprostranenii elektromagnitnyh voln v napravlyayushchej dielektricheskoj strukture proizvol'nogo poperechnogo secheniya. Zhurnal vychislitel'noj matematiki i matematicheskoj fiziki. 2009. T. 49. № 12. S. 22012213.
3. Malyshev G.S., Sedakov A.YU., Titarenko A.A. Dva metoda rascheta harakteristik poloskovogo dielektricheskogo volnovoda. Antenny. 2015. № 6. S. 6368.
4. Belanov A.S., Dianov E.M., Krivenkov V.I. Dispersiya v svetovodah so slozhnym profilem pokazatelya prelomleniya. Doklady akademii nauk. 1999. T. 364. № 1. S. 3741.
5. Raevskij S.B., Titarenko A.A. Metod elektrodinamicheskogo rascheta pryamougol'nyh zakrytyh volnovodov s proizvol'nym dielektricheskim zapolneniem. Antenny. 2007. № 2. S. 411.
6. Kleev A.I., Manenkov A.B., Rozhnev A.G. Chislennye metody rascheta dielektricheskih volnovodov (volokonnyh svetovodov): chastnye metody. Radiotekhnika i elektronika. 1993. T. 38. № 5. S. 769788.
7. Malyshev G.S., Raevskaya Yu.V., Titarenko A.A. Sravnitel'naya ocenka metodov rascheta otkrytyh dielektricheskih volnovodov. Antenny. 2017. № 7. S. 6776.
8. Malyshev G.S., Raevskaya Yu.V., Titarenko A.A. Tri metoda rascheta otkrytyh dielektricheskih volnovodov. Antenny. 2018. № 1. S. 5360.