# Doppler Shift in Michelson-Interferometer

I've used a michelson interferometer with a moveable mirror in order to verify the doppler effect for light waves. But whilst writing the theory for my lab report I encounterd an issue I'm not able to resolve: I get a frequency shift 2x as all my peers. I think I account the Doppler shift twice but can't pinpoint the mistake in my argument, neither can my friends. I'll try to keep it as brief as possible:

Let mirror A be the mirror moving with velocity v and B be the stationary mirror. The lightray reflected in B is trivial. Let's think about the ray reflected in A. In order to get the right frequency $$f'$$ of the reflected wave I make a lorentz transform into the frame of reference of the mirror, the wave gets reflected and after that I transform back into the lab frame, in order to recover $$f'(f,v)$$. From the invariance of $$k_\alpha$$, which I proved successfully, results a transfromation rule for the frequencies of Frame 1 into Frame 2 $$$$f_1=f_2\frac{\sqrt{1-\frac{v^2}{c^2}}}{1-\frac{v}{c}\cos\phi}$$$$ which is a well known result. For the special case this means $$$$f'=\left(f\frac{\sqrt{1-\frac{v^2}{c^2}}}{1-\frac{v}{c}}\right)\frac{\sqrt{1-\frac{v^2}{c^2}}}{1-\frac{v}{c}}=f\frac{1-\frac{v^2}{c^2}}{1-\frac{v}{c}}=f+2f\frac{v}{c}+\mathcal{O}\left(\frac{v}{c}\right)^2$$$$ Now all of my peers just take the superposition of this wave and the wave reflected by the other mirror to get the interference pattern on the mirror and are happy. But I tried accounting for the phase shifts of both waves. More precisely the relative phase shift. Starting from the beam splitter this phase shift is $$$$\Psi(t)=kd_A(t)+k'd_A(t)-2kd_B$$$$ where $$d_x$$ is the distance from the beam splitter to the mirror x. All constant terms are not relevant since they just change the initial values of the interference pattern. But the term $$kd_A(t)+k'd_A(t)$$ hast the not constant part $$$$(k+k')vt$$$$ which turns out to be a frequency shift aquivalent up to the linear term in the formula for $$f'$$ above! Which would imply, that I get a frequency shift twice as high as everyone else.

But this can't be true, because this experiment has been performed many times before at this university. I'd be very helpful for clarification. Does the lorentz transform argument account for the second effect already?