A rich variety of beams exist whose induced waveguides are axially uniform. They span vastly different physical situations. We give examples of several discoveries that were motivated by the linear perspective. The first is for a beam which is the second mode (one node) of its induced waveguide. The second is for a beam composed of two fundamental modes (no nodes) of the induced waveguide. The third is for a beam composed of the fundamental and the second mode. The fourth concerns light guiding light, where one beam acts as a waveguide for the other. Readers concerned only with the framework of the linear perspective should pass to Section 4.