Lorentz transformation of frequency

[Maximtopsecret]

Homework Statement

Light (plane wave) reflects from the mirror moving along X-axis with speed V. The wave is orthogonal to the mirror (φ=0°).
Write the law for frequency change.

Homework Equations

I know Lorenz transformation for frequency.

The Attempt at a Solution

All I do not know is how to apply the mentioned equation. I suppose if we denote ω'₁ is the frequency of a falling wave in the mirror's system, ω'₂ is the frequency of a reflected wave in that system, ω'₁ = ω'₂. Also I suppose ω'₂ will be equal to the frequency of reflected wave in the stationary system. Am I wrong?

 

[MarcusAgrippa]

Convert the frequency $\omega_1$ of the wave in the lab frame to $\omega_1'$ in the frame of the mirror. The, as you correctly state, the reflected wave has $\omega_2' = \omega_1'$. Convert the frequency $\omega_2'$ of the wave in the mirror frame back to the lab frame. The conversion is done by noting that the wave phase is an absolute invariant - all observers agree on what is a crest and a trough.

 

[Maximtopsecret]

Convert the frequency $\omega_1$ of the wave in the lab frame to $\omega_1'$ in the frame of the mirror. The, as you correctly state, the reflected wave has $\omega_2' = \omega_1'$. Convert the frequency $\omega_2'$ of the wave in the mirror frame back to the lab frame. The conversion is done by noting that the wave phase is an absolute invariant - all observers agree on what is a crest and a trough.

Should it be like that?

 

[MarcusAgrippa]

View attachment 84379

Should it be like that?

Looks correct.