For simplicity we will consider an incoming plane wave. At the interface we will have three different waves, the incoming (k), the reflected (k') and the refracted (k").
These three wave must have the same value at the interface for
all times, because at the interface they are one and the same thing.
Looking at the expression of a plane wave the condition, this implies that the scalar product of the 'k' and 'r' must have the same value.This implies that the three wave vectors are all co-planar (the three light beam are in one plane). To understand we consider a coordinate system in which the interface forms the xz-plane and the in coming wave vector 'k' lies in the xy plane. The three angles between the vectors and the interface normal are (abc). We can then write the boundary condition as (Sine equation)
Now, as k and k' are in the same medium the must
have the same absolute value. The angle of incidence therefore equals
the angle of reflection, an observation most should have made by
the time they read this a webside.
If the propagation speed (Phase velocity) in the two media is different, otherwise there would be no refraction, the absolute value of k" differes from the absolute of k. Consequently the angle of refraction differs from the incident angle. The ratio of their sinuses is given by Snell's law of refraction.