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In (1+1)D Minkowski spacetime, with coordinates (t,x),

let's say there is an incoming plane wave of frequency [tex]\omega[/tex],

[tex]\phi_{in}(t,x)=e^{-i\omega (t+x)}[/tex].

There is a mirror, [tex]x=z(t)[/tex]

It reflects the incoming plane wave and emits an outgoing plane wave.

Question:

why is the outgoing wave

[tex]\phi_{out}=e^{-i\omega (2\tau_u-u)}[/tex],

where

[tex]u=t-x[/tex],

[tex]\tau_u-z(\tau_u)=u[/tex],

i.e. it is the retarded time.

??

For mirror at constant velocity v, this reduces to

[tex]\phi_{out}=e^{-i\omega\frac{1+v}{1-v}\cdot u}[/tex],

the two Doppler shifts are obvious.

But how can I prove the general expression?

Thanks

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# Wave reflected by a mirror

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