In the present paper, we have presented a Maxwellian boundary-type solution for total internal reflection with unbounded incident waves at an interface between two nonabsorbing media, in which the instantaneous, time varying, and time averaged radiant fluxes have been determined at all points in the two media. Solutions for the s and p polarizations were found for which the instantaneous tangential E and H components and normal components of the radiant flux were continuous in crossing the interface. From these radiant fluxes, it was possible to derive equations for the flow lines, to determine the instantaneous radiant fluxes along these flow lines, and to see how the methods of propagation differed in the two media and for the two polarizations. At the interface, the flow lines and their radiant fluxes experience unusual reflection and refraction processes, follow curved flow lines in the second medium, and return into the first medium with boundary conditions, which are mirror images of those at the points of incidence. These unfamiliar processes in the second medium are due to inhomogeneous waves, whose properties have not been understood. When these instantaneous solutions are extended to time varying and time averaged radiant fluxes, it is interesting to see how incident planes of constant radiant flux and phase experience such complex processes in the second medium and are still able to generate other reflected planes of constant radiant flux and phase in the first medium. These ideas prescribe specific detailed functions for the E and H fields and radiant fluxes in the second medium, which help to answer many long standing questions about the physical processes involved in total internal reflection.
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