Equations for reflection at a moving planar interface
Assume a wave from a source in vacuum of the form:
where 
where q is the angle of the light wave propagation with
respect to the x axis and
c is the speed of the wave fronts and w is their radian frequency.
Now consider that the interface is moving at a rate v=bc away from the light source along the x direction. Then, using the simple scalar transformations of special relativity, with the primed quantity being that in the moving interface frame of reference we have the following equations.
Source frame: Moving frame:
(1)
(2)
(3)
Using left sides of equations 1, 2, and 3 but with nx’ negative, we transform w, nx and ny back to the frame of the light source and renaming these with double primed notation:
Therefore the angle of reflection, q’’, will be given by
The ratio of the wavelengths of the reflected wave to the incident wave will be the inverse of the frequency ratio:
Recall that the wave fronts
are always observed, from both the source frame and the moving interface frame,
to travel at the speed of light.[1]