<p>We have studied the transmission of a wave packet through finite discontinuous media with dissipation using analytical methods. The focus is on the behavior of an electromagnetic pulse moving from air to a finite conducting slab. The wave equation governs the pulse’s evolution in the free medium, while the telegrapher’s equation is applied in the slab. The presence of the Joule effect in the slab affects the wave packet’s propagation. By solving three discontinuous partial differential equations analytically using Fourier representation and the attenuated spatial wave packet formalism, we derive reflection and transmission coefficients as functions of the telegrapher’s equation parameters. Experimental results for the transmission coefficient varying the absorption energy are presented, utilizing a demineralized water tank with different salt concentrations as the slab. The results are consistent with our theoretical model.</p>