General penetrable-edge conditions are derived from a new approach to diffraction by a dielectric wedge. The increased computational efficiency, to be expected from satisfying these edge conditions explicitly, is illustrated with numerical evaluations of the surface fields on wedge cylinders, and by comparison of measured and computed resonant frequencies of cylindrical cavities loaded with dielectric sectors. Additional evidence is provided for the apparent validity of our recently suggested technique for calculating the E-polarised field scattered from the vicinity of the apex of a penetrable wedge.