Question

Ifwe the source then velocity changes and frequency remains const

Answer 1

I’m shocked — shocked I tell you — to find no reference to Noether’s Theorem in the otherwise excellent answers here. Several reference energy conservation, but skip over the small issue of momentum non-conservation: why is one conservation law inviolable, but breaking the other is fair game?

Noether’s theorem relates continuous symmetries of physical systems to conserved quantities in them: for example a system whose elements don’t change with time will conserve energy — and conversely one which is not time-invariant will not conserve energy. And one in which the same physical constraints apply at all space positions will conserve momentum.

A refracting boundary is a spatial discontinuity, i.e. it breaks spatial translation symmetry in the direction perpendicular to the boundary. This means that momentum in that direction need not be conserved. But the time symmetry remains intact, and hence energy is still conserved.

You can now solve the refraction problem by energy conservation, either explicitly or by shortcut: just note that E∝νE∝ν and p∝1/λp∝1/λ for frequency νν and wavelength λλ, so the frequency has to remain unchanged to conserve energy while the momentum/wavelength across the boundary can change.

What about momentum parallel to the boundary? That remains a good symmetry direction, and so momenta along it have to be conserved. In fact, applying momentum conservation parallel to the surface is another method to derive Snell’s Law.

This view also gives an immediate answer to a very interesting and insightful question from one of my students: what happens if a light ray is happily propagating through a uniform medium, when instantaneously the refractive index is changed through the whole space? In this situation — weird and idealised, but not unreasonable, e.g. the refractive index could be controlled by an external electric field — what happens? Well, there’s now no breaking of spatial translational invariance either before or after the change, but the system’s time-translation invariance has been broken. So here we would expect the momentum to be unaffected, but the energy to change since energy conservation is no longer a good symmetry. Pretty neat… and all without a single calculation.

A final shout out to Bill Otto's answer, which highlights very nicely how this pure & simplistic mathematical picture leaves out the real questions about where that locally non-conserved momentum goes (the universe as a whole being space- and time-translation invariant, and so momentum and energy are both good global symmetries), and how they lead to significant issues in real engineering systems