Need help finding origin of an equation

[jinksys]

I am doing a Michelson interferometer lab which instructs me to use the equation Fb=(Fl*v)/c, where Fb is the beat frequency, Fl is the frequency of the laser, and v is the velocity of a oscillating mirror. The interferometer has one stationary mirror and a mirror that is mounted on a speaker that oscillates from a signal provided by a function generator. I'm trying to find the origin of the Fb=... equation.

 

[Dickfore]

It's derived using the relativistic Doppler effect formula.

When the light is incident on the mirror, the frequency in the rest frame of the mirror is:
<br /> f&#039; = f_L \, \sqrt{\frac{1 + \beta}{1 - \beta} }, \ \beta = \frac{v}{c}<br />

Then, as the mirror re-emits the light, the new frequency is:
<br /> \tilde{f} = f&#039; \, \sqrt{\frac{1 + \beta}{1 - \beta} } = f_L \, \frac{1 + \beta}{1 - \beta}<br />
Then, use the fact that \beta \ll 1 and perform an expansion in powers of \beta. To first order we have:
<br /> \frac{1 + \beta}{1 - \beta} = (1 + \beta) ( 1 - \beta)^{-1} = (1 + \beta) \left[ 1 + (-1) (-\beta) + O(\beta^2) \right] = 1 + 2 \beta + O(\beta^2)<br />
<br /> \tilde{f} \approx f_L + 2 \, f_L \, \beta<br />

Then, consider the interference of the incoming light ray and the reflected ray with a slightly bigger frequency.

<br /> u = A \, \cos \left[ 2 \pi f_L \left(t - \frac{x}{c} \right) \right] - A \, \cos \left[ 2 \pi \tilde{f} \left(t + \frac{x}{c} \right) \right] <br />

You should use simple trigonometry to see that this represents a beating wave, with a beat frequency:
<br /> f_B = \frac{\tilde{f} - f_L}{2}<br />
which reproduces your formula.

 

[jinksys]

Thank you, I really appreciate it!