Question

Show that the frequency of an electromagnetic wave undergoing reflection and transmission at an interface of two dielectrics remains invariant.

Answer 1

Plane electromagnetic wave reads as:

$$\vec{E} = \vec{E_0}\, e^{i\vec{k}\cdot\vec{x} -i\omega t}$$

Transmitted and reflected wave looks analogously, but with different wavevector $\vec{k}$ , frequency $\omega$ and amplitude $\vec{E_0}$.

Boundary conditions for EM field (the tangential component of E is continuous across the interface., the normal component of D is continuous, the normal component of B is continuous across the interface, the tangential component of H is continuous across the surface) are right at any moment of time.

Therefore, frequencies of transmitted, reflected and incident ray are equal.