Question

A plane electromagnetic wave of frequency ω₀ falls normally on the surface of a mirror approaching with a relativistic velocity V. Making use of the Doppler formula, find the frequency of the reflected wave. Simplify the obtained expression for the case V << c.

Answer 1

We define the frame S (the lab frame) by the condition of the problem. In this frame the mirror is moving with velocity v (along say x - axis) towards left and light of frequency ω₀ is approaching it from the left. We introduce the frame S ' whose axes are parallel to those of S but which is moving with velocity v along x axis towards left (so that the mirror is at re st in S'). In S' the frequency of the incident light is

In S.' the reflected light still has frequency ω₀ but it is now moving towards left. When we transform back to S this reflected light has the frequency