Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

Spectral method in the theory of wave propagation in cellular periodic waveguides

  • 32 Accesses


The spectral method is substantiated for the particular example of normal wave determination in a corrugated waveguide with rectangular-profile ribs. We establish the rib conditions, prove an analog of the Paley—Wiener theorem for Fourier series, and use the theorem to prove equivalence of the solutions of the original boundary-value problem and the dispersion equation. Some topics connected with the existence of the characteristic value of the dispersion equation and with the convergence of the approximate method are explored.

This is a preview of subscription content, log in to check access.

Literature Cited

  1. 1.

    O. A. Val'dner, N. P. Sobenin, B. V. Zverev, and I. S. Shchedrin, A Handbook of Iris Waveguides [in Russian], Atomizdat, Moscow (1977).

  2. 2.

    I. E. Efimov and G. A. Shermina, Waveguide Transmission Lines [in Russian], Svyaz’, Moscow (1979).

  3. 3.

    W. Walkinshaw, "Notes on ‘Waveguides for slow waves’," J. Appl. Phys.,20, No. 6, 634–636 (1949).

  4. 4.

    P. E. Krasnushkin and S. P. Lomnev, "Methods for exact computation of homogeneous cellular waveguides," Radiotekh. Élektron.,11, No. 6, 1051–1065 (1966).

  5. 5.

    M. I. Gans, "A general proof of Floquet's theorem," IEEE Trans.,MTT-13, No. 3, 384–385 (1965).

  6. 6.

    I. A. Shishmarev, An Introduction to the Theory of Elliptic Equations [in Russian], Moscow State Univ. (1979).

  7. 7.

    F. Stummel, "Perturbation of domains in elliptic boundary value problems," Lect. Notes Math.,503, 110–136 (1976).

  8. 8.

    R. Mittra and S. Lee, Analytical Methods of Waveguide Theory [Russian translation], Mir, Moscow (1974).

  9. 9.

    L. A. Vainshtein, Electromagnetic Waves [in Russian], Sovet-skoe Radio, Mocsow (1957).

  10. 10.

    G. P. Tolstov, Fourier Series [in Russian], Nauka, Moscow (1980).

  11. 11.

    A. F. Nikiforov and V. B. Uvarov, Special Functions of Mathematical Physics [in Russian], Nauka, Moscow (1978).

  12. 12.

    Ya. I. Khurgin and V. P. Yakovlev, Finite Functions in Physics and Engineering [in Russian], Nauka, Moscow (1971).

  13. 13.

    E. V. Chernokozhin and Yu. V. Shestopalov, "On the Fredholm property of an integral operator with a kernel having a weak singularity," Vestn. Mosk. Gos. Univ., Ser. Vychisl. Mat. Kibern., 23–28 (1982).

  14. 14.

    A. S. Markus, "On holomorphic operator functions," Dokl. Akad. Nauk SSSR,119, No. 6, 1099–1102 (1958).

  15. 15.

    A. S. Markus and E. I. Sigal, "On multiplicity of the characteristic value of an analytical operator function," Mat. Issled.,5, No. 3, 129–147 (1970).

  16. 16.

    V. P. Trofimov, "On root subspaces of operators analytically dependent on a parameter," Mat. Issled.,5, No. 3, 117–125 (1968).

  17. 17.

    G. M. Vainikko and O. O. Karma, "On rate of convergence of approximate methods in the eigenvalue problem with nonlinear occurrence of the parameter," Zh. Vychisl. Mat. Mat. Fiz.,14, No. 6, 1393–1408 (1974).

Download references

Additional information

Translated from Vychislitel'naya Matematika i Matematicheskoe Obespechenie EVM, pp. 186–198, 1985.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Il'inskii, A.S., Mutallimov, M.M. Spectral method in the theory of wave propagation in cellular periodic waveguides. Comput Math Model 1, 95–103 (1990). https://doi.org/10.1007/BF01128320

Download citation


  • Mathematical Modeling
  • Wave Propagation
  • Computational Mathematic
  • Fourier Series
  • Industrial Mathematic