= Moving-mirror Doppler shift
{title2=$\omega_r=\omega_i(1-\beta)/(1+\beta)$}
A mirror receding at <velocity> $u=\beta c$ from normally incident <light> produces two applications of the <relativistic Doppler effect>, one into the mirror rest frame and one back to the observer. In that rest frame, reflection preserves <frequency>. Consequently $\omega_r=\gamma^2(1-\beta)^2\omega_i=\omega_i(1-\beta)/(1+\beta)$. Equivalently, continuity of <wave phase> on the moving surface $z=ut$ gives $\omega_i-k_i u=\omega_r+k_r u$, with $k_i=\omega_i/c$ and $k_r=\omega_r/c$. For $|u|\ll c$, the fractional <frequency> change is $-2u/c$ to first order.
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