Evolution equations for analytical study of digital signals in waveguides

The excitation and propagation problem of the digital signals in a hollow waveguide is considered by an analytical time-domain method. The waveguide is geometrically regular along Oz axis, and its cross section is a closed singly connected domain. The waveguide surface is a perfect electric conductor. A complete set of TE and TM waveguide modes is obtained in Time Domain (TD) directly. Every modal field is deduced as the sum of its longitudinal and transverse vector components, where each component is the product of two factors. One factor is an element of the waveguide modal basis, which is a vector function of the transverse waveguide coordinates. The other one is a modal amplitude of appropriate field component, which is a scalar function of time t and the axial coordinate z. All the elements of the modal basis are specified via two scalar potentials. They are eigensolutions (normalized in a proper way) of Dirichlet and Neumann boundary eigenvalue problems for the Laplacian. Every element of the modal basis satisfies appropriate boundary conditions over the waveguide surface. The modal amplitudes are solutions of a system of evolution partial differential equations. The problem of Walsh function signals in the waveguide is solved explicitly in compliance with the causality principle and the special theory of relativity.