Most electrodynamics textbooks derive the behavior of light on the boundary of two mediums with the starting hypothesis that there are only three waves: incident, transmitted and reflected. Why only these three?
Griffiths certainly does not talk about it; I can't find the explanation in Zangwill and Jackson either. Quote from Zangwill, "Everyday experience tells us that an incident plane wave which approaches medium 2 from medium 1 'splits' into a reflected wave confined to medium 1 and a refracted wave confined to medium 2." (pg.588)
I wonder how to derive this result theoretically. If we employ a left-right symmetry argument, naively from the ray picture, there could be another transmitted ray going off to the left, or even maybe a ray that goes straight up and down. From the wave picture, we get continuous translation symmetry on one axis and discrete translation symmetry on the other two, but these symmetries do not seem to help.