Electromagnetic wave in two media

[The Flobjka]

So if an EM waves travels from one medium to the other, it refracts depending on Snell's law. But what happens when it doesn't travel from one to the other, but goes right along the boundary so that one end of the amplitude is in each medium? As the wave "waved", would it have different speeds at different points in the wave? Is it even possible?

 

[UltrafastPED]

The wave will separate into two strands ... just like a column of soldiers with one file on the rode, and the other in the mud.

There are other, transient effects inside of the media - some related to interference, some due to evanescent "dragging". You will find a very complete analysis when you study fiber optic behavior in a graduate level optics course.