Need help finding origin of an equation

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In summary, the conversation discussed the use of the equation Fb=(Fl*v)/c in a Michelson interferometer lab, which is derived using the relativistic Doppler effect formula. The equation is used to find the beat frequency of the interferometer, which involves a stationary mirror and an oscillating mirror. The formula is derived by considering the frequency of incident and re-emitted light rays and using simple trigonometry to find the beat frequency.
  • #1
jinksys
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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.
 
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  • #2
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:
[tex]
f' = f_L \, \sqrt{\frac{1 + \beta}{1 - \beta} }, \ \beta = \frac{v}{c}
[/tex]

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

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

[tex]
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]
[/tex]

You should use simple trigonometry to see that this represents a beating wave, with a beat frequency:
[tex]
f_B = \frac{\tilde{f} - f_L}{2}
[/tex]
which reproduces your formula.
 
  • #3
Thank you, I really appreciate it!
 

1. What is the purpose of finding the origin of an equation?

Understanding the origin of an equation can provide insight into how and why it was developed, as well as its significance and potential applications. It can also help verify the accuracy and validity of the equation.

2. How can I determine the origin of an equation?

The origin of an equation can often be traced back to the original publication or paper where it was first introduced. Other methods include consulting textbooks, scientific databases, and experts in the field.

3. What factors should I consider when trying to find the origin of an equation?

Some important factors to consider include the time period in which the equation was developed, the field of study it relates to, and the individuals who contributed to its development.

4. Can an equation have multiple origins?

Yes, it is possible for an equation to have multiple origins. This can occur if the equation was independently discovered or developed by different individuals or if it was adapted or modified from a previous equation.

5. Why is it important to properly credit the origin of an equation?

Crediting the origin of an equation is important for acknowledging and honoring the contributions of the individuals who developed it. It also helps to ensure the accuracy and credibility of scientific research and advancements.

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