A initial-boundary value problem for the system of Maxwell's equations with time derivative is formulated and solved rigorously for transient modes in a hollow waveguide. It is supposed that the latter has perfectly conducting surface. Cross section, S, is bounded by a closed singly-connected contour of arbitrary but smooth enough shape. Hence, the TE and TM modes are under study. Every modal field is a product of a vector function of transverse coordinates and a scalar amplitude dependent on time, t, and axial coordinate, z. It has been established that the study comes down to, eventually, solving two autonomous problems. i) A modal basis problem. Final result of this step is de nition of complete (in Hilbert space, L2) set of functions dependent on transverse coordinates which originates a basis. ii) A modal amplitude problem. The amplitudes are generated by the solutions to Klein-Gordon equation (KGE), derived from Maxwell's equations directly, with t and z as independent variables. The solutions to KGE are invariant under relativistic Lorentz transforms and subjected to the causality principle. Special attention is paid to various ways that lead to analytical solutions to KGE. As an example, one case (among eleven others) is considered in detail. The modal amplitudes are found out explicitly and expressed via products of Airy functions with arguments dependent on t and z.
2. Tretyakov, O. A., "Evolutionary waveguide equations," Sov. J. Comm. Tech. Electron. (English Translation of Elektrosvyaz and Radiotekhnika), Vol. 35, No. 2, 7-17, 1990.
3. Tretyakov, O. A., "Essentials of nonstationary and nonlinear electromagnetic field theory," Analytical and Numerical Methods in Electromagnetic Wave Theory, M. Hashimoto, M. Idemen, O. A. Tretyakov (eds.), Chap. 3, Science House Co. Ltd., Tokyo, 1993.
4. Kristensson, G., "Transient electromagnetic wave propagation in waveguides," Journal of Electromagnetic Waves and Applications, Vol. 9, No. 5-6, 645-671, 1995.
6. Aksoy, S. and O. A. Tretyakov, "Evolution equations for analytical study of digital signals in waveguides," Journal of Electromagnetic Waves and Applications, Vol. 17, No. 12, 1665-1682, 2003.
7. Aksoy, S. and O. A. Tretyakov, "The evolution equations in study of the cavity oscillations excited by a digital signal," IEEE Trans. Antenn. Propag., Vol. 52, No. 1, 263-270, Jan. 2004.
8. Slater, J. C., "Microwave electronics," Rev. Mod. Phys., Vol. 18, No. 4, 441-512, 1946.
9. Kisun'ko, G. V., Electrodynamics of Hollow Systems, VKAS-Press, Leningrad, 1949 (in Russian).
10. Kurokawa, K., "The expansion of electromagnetic fields in cavities," IRE Trans. Microwave Theory Tech., Vol. 6, 178-187, 1958.
11. Felsen, L. B. and N. Marcuvitz, Radiation and Scattering of Waves, Prentice Hall, Englewood Cliffs, NJ, 1973.
12. Borisov, V. V., Transient Electromagnetic Waves, Leningrad Univ. Press, Leningrad, 1987 (in Russian).
13. Tretyakoy, O. A., "Evolutionary equations for the theory of waveguides," IEEE AP-S Int. Symp. Dig., Vol. 3, 2465-2471, Seattle, Jun. 1994.
14. Shvartsburg, A. B., "Single-cycle waveforms and non-periodic waves in dispersive media (exactly solvable models)," Phys. Usp., Vol. 41, No. 1, 77-94, Jan. 1998.
15. Slivinski, A. and E. Heyman, "Time-domain near-field analysis of short-pulse antennas --- Part I: Spherical wave (multipole) expansion," IEEE Trans. Antenn. Propag., Vol. 47, 271-279, Feb. 1999.
16. Nerukh, A. G., I. V. Scherbatko, and M. Marciniak, "Electromagnetics of Modulated Media with Application to Photonics," National Institute of Telecommunications Publishing House,Warsaw, Poland, 2001.
17. Geyi, W., "A time-domain theory of waveguides," Progress In Electromagnetics Research, Vol. 59, 267-297, 2006.
18. Erden, F. and O. A. Tretyakov, "Excitation by a transient signal of the real-valued electromagnetic fields in a cavity," Phys. Rev. E, Vol. 77, No. 5, 056605, May 2008.
19. Polyanin, A. D. and A. V. Manzhirov, Handbook of Mathematics for Engineers and Scientists, Chapman & Hall/CRC Press, Boca Raton, FL, 2006.
20. Torre, A., "A note on the Airy beams in the light of the symmetry algebra based approach," J. Opt. A: Pure Appl. Opt., Vol. 11, 125701, Sep. 2009.
21. Polyanin, A. D., Handbook of Linear Partial Differential Equations for Engineers and Scientists, Chapman & Hall/CRC Press, Boca Raton, FL, 2002.
22. Miller, Jr., W., Symmetry and Separation of Variables, Addison-Wesley Publication Co., Boston, MA, 1977.
23. Vallee, O. and M. Soares, "Airy Functions and Applications to Physics," Imperial College Press, London, England, 2004.
24. Kalnins, E., "On the separation of variables for the Laplace equation in two- and three-dimensional Minkowski space," SIAM J. Math. Anal., Vol. 6, 340-374, 1975.