In this second chapter, we analyze transmission problems between a dielectric and a dispersive negative material. In the first part, we consider a transmission problem between two half-spaces, filled respectively by the vacuum and a Drude material, and separated by a planar interface. In this setting, we answer the following question: Does this medium satisfy a limiting amplitude principle? This principle defines the stationary regime as the large-time asymptotic behavior of a system subject to a periodic excitation. In the second part, we consider the transmission problem of an infinite strip made of a Drude material embedded in the vacuum and analyze the existence and dispersive properties of guided waves. In both problems, our spectral analysis elucidates new and unusual physical phenomena for the considered transmission problems due to the presence of the dispersive negative material. In particular, we prove the existence of an interface resonance in the first part and the existence of slow light phenomena for guiding waves in the second part.