Abstract-An 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 T E and T M 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 definition of complete (in Hilbert space, L 2 (S)) 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.

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Abstract-Excitation of the electromagnetic fields by a wide-band current surge, which has a beginning in time, is studied in a cavity bounded by a closed perfectly conducting surface. The cavity is filled with Debye or Lorentz dispersive medium. The fields are presented as the modal expansion in terms of the solenoidal and irrotational cavity modes with the time-dependent modal amplitudes, which should be found. Completeness of this form of solution has been proved earlier.The systems of ordinary differential equations with time derivative for the modal amplitudes are derived and solved explicitly under the initial conditions and in compliance with the causality principle. The solutions are obtained in the form of simple convolution (with respect to time variable) integrals. Numerical examples are exhibited as well.

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