Suppose a police car is standing by a wall. The siren light is rotating and it will hit the wall and reflect back to the car. Does the reflected light show a Doppler effect?
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When the light reflects off the wall, it will act as if the wall is a new source for the light. If the wall itself is not moving, it will not Doppler shift the light. But if the siren is rotating, the motion of the siren might Doppler shift the light, depending on how the light source is mounted to the siren.. Reflection off the wall will retain that shift, but won't add to or subtract from it.
EDIT: Based on your comments, I may have misunderstood the situation you are describing. If the light is pointed radially outward from the center of the siren, and the siren is rotating, then the instantaneous velocity of the light source is perpendicular to the direction of propagation of the light. In that case, there will be no Doppler shift. The reflected beam would also have no Doppler shift.
Regardless of how exactly the light source is moving or rotating, if the "wall" is an atom or an electron, and it moves or scatters when the light reflects off of it, then that will cause an additional Doppler shift.
answered Apr 27, 2014 at 16:55
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$\begingroup$ If an observer is standing by the wall at some point and sees the light from the rotating siren then at the time the light was emitted, the siren was going perpendicular to the line of sight between the siren and the observer. Wouldn't that cancel the Doppler effect? And so the reflected beam then also won't show any Doppler effect? $\endgroup$– MaesumiCommented Apr 28, 2014 at 15:46
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$\begingroup$ If at some time $t$ the siren is at angle $\theta$ does the light emitted at that moment travel in direction of angle $\theta$ or would the angular velocity of the siren change the direction? $\endgroup$– MaesumiCommented Apr 28, 2014 at 15:53
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$\begingroup$ @Maesumi, I added some information to my answer in response to your comments. Does that help clarify? $\endgroup$ Commented May 11, 2014 at 19:27