electromagnetism - Electromagnetic waves at interface between dielectrics

Jackson's Classical Electrodynamics textbook claims that the simple existence of boundary conditions (doesn't matter which they are) between incident, reflected and refracted waves that must be satisfied at every points of the interface and at every time leads to

$$\omega=\omega '=\omega '' $$

$$(\mathbf{k}\cdot \mathbf{x})_{z=0}=(\mathbf{k}'\cdot \mathbf{x})_{z=0}=(\mathbf{k}''\cdot \mathbf{x})_{z=0} $$

where $\omega$ and $\mathbf{k}$ are respectively the angular frequency and the wave vector of the incident wave, $\omega'$ and $\mathbf{k}'$ of the refracted wave and $\omega''$ and $\mathbf{k}''$ of the reflected wave. $z=0$ identifies the interface between two different dielectrics materials.

I really can't see how these relations are found. Can someone please show me the math behind?

EDIT:

Let's say I ignore what Jackson says and I use the continuity condition for the electric fields

$$ \mathbf{E}e^{i(\mathbf{k}\cdot \mathbf{x}-\omega t)}+ \mathbf{E}''e^{i(\mathbf{k}''\cdot \mathbf{x}-\omega'' t)}=\mathbf{E}'e^{i(\mathbf{k}'\cdot \mathbf{x}-\omega' t)} $$

This one holds only if there's not surface charge on the interface. Does that means that if there's some surface charge the above relations are false and then also the Snell's law is false?

Answer

This is required for the electric field to be a consistent and continuous function across the boundary. If you add together all of the waves on either side, you get an equation that looks essentially of the form $$ A_\mathrm{i}e^{i(k_x^{(\mathrm{i})}x-\omega^{(\mathrm{i})}t)}+ A_\mathrm{r}e^{i(k_x^{(\mathrm r)}x-\omega^{(\mathrm r)}t)} = A_\mathrm{t}e^{i(k_x^{(\mathrm t)}x-\omega^{(\mathrm t)}t)} .$$ (Ignoring the effect of the $y$ coordinate, of course.) Here you can, and it's important to do, fiddle with the amplitudes to make sure that the left-hand side is equal to the right-hand side, but unless the exponents on all of the exponentials match exactly, there is absolutely no way that you'll be able to maintain the equality across all times $t$ and all positions $x$ along the boundary. Thus, you require all the temporal frequencies, and all the along-the-surface wavenumbers, to match.