A plane electromagnetic wave of frequency `omega_(0)` falls normally on the surface of a mirror approching 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 `Vlt lt c`.

Explanation

We define the frame `S` (the lab frame) by the condition of the problem. In this frame thee mirror is moving with velocity `v`(along say `x`-axis) towards left and light of frequecny `omega_(0)` 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 rest in `S'`). In `S'` the frequency of the incident light is
`omeha_(1) = omega_(0) ((1+v//c)/(1-v//c))^(1//2)`
In `S'` the reflected light still has frequency `omega_(1)` but it is now moving twoards left. When we transform back to `S` this reflected light has the frequency
`omega = omega_(1) ((1+v//c)/(1-v//c))^(1//2) = omega_(0) ((1+v//c)/(1-v//c))`
In the nonrelativistic limit
`omega ~~ omega_(0) (1+(2v)/(c))`