A model of the ring waveguide of a fixed cross-section and variable distribution of the surface impedance of waveguide's wall has been considered. For a class of circular hodographs of surface impedance the analytical solution of the corresponding boundary-value problem has been obtained. This solution has been used for simulating a 'cycle slipping' phenomenon, known from the observations of VLF signals propagating over long paths in the earth-ionosphere waveguide, with the goal of clarifying the cause for its initiation. Numerical experiments have shown that this phenomenon, in the context of the model in question, is a consequence of the interconversion of two dominant waveguide modes in circumstances where their propagation constants are close.
2. Alpert, Ya. L., E. G. Guseva, and D. S. Fligel, Propagation of Low-frequency Electromagnetic Waves in the Earth-ionosphere Waveguide, Nauka, Moscow, 1967 (in Russian).
3. Makarov, G. I., V. V. Novikov, and A. B. Orlov, "The current status of research on the VLF propagation in the earth-ionosphere waveguide," Izv. Vuzov. Radiofizika, Vol. 13, No. 3, 321-355, 1970 (in Russian).
4. Simpson, J. J. and A. Taflove, "A review of progress in FDTD Maxwell’s equations modeling of impulsive subionospheric propagation below 300KHz," IEEE Trans. Antennas Propag., Vol. 55, No. 6, 1582-1590, 2007.
5. Lynn, K. W., "Multisite observations of the VLF transequatorial propagation anomaly," Radio Science, Vol. 4, No. 3, 203-212, 1969.
6. Walker, D., "Phase steps and amplitude fading of VLF signals at dawn and dusk ," Radio Science, Vol. 69D, No. 11, 1435-1443, 1965.
7. Bahar, E. and J. R. Wait, "Propagation in a model terrestrial waveguide of nonuniform height: Theory and experiment," Radio Science, Vol. 69D, No. 11, 1445-1463, 1965.
8. Wait, J. R., "Mode conversion and refraction effects in the earth-ionosphere waveguide for VLF radio waves," Journal of Geophysical Research, Vol. 73, No. 11, 3537-3548, 1968.
9. Katsenelenbaum, B. Z., The Theory of Irregular Waveguides with Slowly Varying Parameters, Academy of Science Press, Moscow, 1961 (in Russian).
10. Bolotovskii, Ya. E. and G. I. Makarov, "Intersection of VLF-signal path and the ‘day-night’ boundary," Problems of Diffraction and Wave Propagation, No. 11, 142-158, Leningrad State University Press, Leningrad, 1972 (in Russian).
11. Pappert, R. A. and J. A. Ferguson, "VLF/LF mode conversion model calculations for air to air transmissions in the earth-ionosphere waveguide ," Radio Science, Vol. 21, No. 4, 551-558, 1986.
12. Budden, K. G., "The critical coupling of modes in a tapered earth-ionosphere wave guide ," Math. Proc. Camb. Phil. Soc., Vol. 77, 567-580, 1975.
13. Krasnushkin, P. Ye. and Ye. N. Fedorov, "About multiplicity of wave numbers of normal modes in stratified media," Radiotekhnika i Elektronika, Vol. 17, No. 6, 1129-1140, 1972 (in Russian).
14. Westerlund, S. and F. H. Reder, "VLF radio signal propagating over the Greenland ice-sheet," JATP, Vol. 35, No. 8, 1973.
15. Tyras, G., Radiation and Propagation of Electromagnetic Waves, Academic Press, New York and London, 1969.
16. Gakhov, F. D. and Yu. I. Cherskii, Convolution Equations, Nauka, Moscow, 1978 (in Russian).
17. Makarov, G. I., V. V. Novikov, and S. T. Rybachek, Radiowave Propagation in the Earth-ionosphere Waveguide and in the Ionosphere, Nauka, Moscow, 1994 (in Russian).
18. Felsen, L. B. and N. Marcuvitz, Radiation and Scattering of Waves, Prentice-Hall, Englewood Cliffs, NJ, 1973.
19. Keller, J. B., S. I. Rubinow, and M. Goldstein, "Zeros of Hankel functions and poles of scattering amplitudes," Journal of Mathematical Physics, Vol. 4, No. 6, 829-832, 1963.
20. Paknys, R., "Evaluation of Hancel functions with complex argument and complex order," IEEE Trans. Antennas Propag., Vol. 40, No. 5, 569-578, 1992.
21. Tian, Y. B. and J. Qian, "Ultraconveniently finding multiple solutions of complex transcendental equations based on genetic algorithm," Journal of Electromagnetic Waves and Applications, Vol. 20, No. 4, 475-488, 2006.
22. Hanson, G. W. and A. B. Yakovlev, "Investigation of mode interaction on planar dielectric waveguides with loss and gain," Radio Science, Vol. 34, No. 6, 1349-1359, Nov.-Dec. 1999.