# Need help finding origin of an equation

1. Feb 8, 2012

### 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.

2. Feb 8, 2012

### 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:
$$f' = f_L \, \sqrt{\frac{1 + \beta}{1 - \beta} }, \ \beta = \frac{v}{c}$$

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

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

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

You should use simple trigonometry to see that this represents a beating wave, with a beat frequency:
$$f_B = \frac{\tilde{f} - f_L}{2}$$

3. Feb 8, 2012

### jinksys

Thank you, I really appreciate it!