  The problem of the emergence of wave dispersion due to the heterogeneity of a
transmission line (TL) is considered. An exactly solvable model helps to better
understand the physical process of a signal passing through a non-uniform
section of the line and to compare the exact solution and solutions obtained
using various approximate methods. Based on the transition to new variables,
the developed approach made it possible to construct exact analytical solutions
of telegraph equations with a continuous distribution of parameters, which
depend on the coordinate. The flexibility of the discussed model is due to the
presence of a number of free parameters, including two geometric factors
characterizing the lengths of inhomogeneities in values of the inductance L and
of the capacitance C. In the new variables, the spatiotemporal structure of the
solutions is described using sine waves and elementary functions, and the
dispersion is determined by the formulas of the waveguide type. The dispersive
waveguide-like structure characterized by refractive index N and cut-off
frequency $\Omega$. The exact expressions for the complex reflection and
transmission coefficients are derived. These expressions describe phase shifts
for reflected and transmitted waves. The following interesting cases are
analyzed: the passage of waves without phase change, the reflectionless passage
of waves, and the passage of signals through a sequence of non-uniform
sections. The developed mathematical formalism can be useful for the analysis
of a wider range of problems.
