Optical Doppler effect in a medium

The acoustic and optical Doppler effects are unified into one single formula, via which the Doppler shift for EM waves in a medium can be obtained without having to address the issue of complicated interactions with the dense matter. Since the medium provides an absolute rest frame, this formula is expectedly different from that of the vacuum. But it also does not trivially adopt the form for sound waves, because the latter fails to include the optical shift in vacuum as a limiting case. A universal two-way Doppler shift is derived by solving the wave equation in an infinitely hard well with a moving boundary. Since the mathematical solution does not care about the detailed nature of the wave, its features are shared by sound and light in vacuum or a medium.