Beat Frequency for a Laser

[Hells_Kitchen]

Homework Statement

A laser emits a monochromatic beam of wavelength $$\lambda$$ , which is reflected normally from a plane mirror, receding at speed $$v$$. What is the beat frequency between the incident and reflected light?


I know that the $$f_{beat} = |f_2 - f_1|$$.

When the wave hits the plane mirror it has frequency
$$f_1 = \frac{v_0}{\lambda}$$.
Then, when it bouces off due to the doppler effect the wavelength becomes:

$$\frac{\lambda_1}{\lambda} = 1 - \frac{v}{c}$$.
Furthermore then,

$$f_2 = \frac{v}{\lambda_1} = \frac{v}{\lambda - \frac{\lambda v}{c}} = \frac{v*c}{\lambda*c - \lambda*v}$$.


So if i then find $$f_{beat} = |f_2 - f_1| = |\frac{v c}{\lambda c - \lambda v} -\frac{v_0}{\lambda}|$$ This result does not match the book result which is:

$$f_{beat} = 2*(\frac{v}{c})*v_0$$.

I was wondering if anyone could help me with this problem on what I have done wrong.

Thanks!

 

[tiny-tim]

Hi Hells_Kitchen!

(what's v₀? Do you mean ν₀ ?)

I don't understand your formula for f₂.

Hint: the image of the laser (on the other side of the mirror) is receding at a speed of … ?

 

[tomcue]

Your calculations seem to be correct, but there may be a slight error in the book's answer. The beat frequency between the incident and reflected light should be equal to the difference between the frequencies of the two waves, which is given by |f_2 - f_1|. Therefore, your final result is correct and matches the equation for beat frequency, while the book's answer seems to have an extra factor of 2. It is possible that this is a typographical error in the book. I would suggest double-checking with your instructor or consulting other reliable sources for confirmation. Keep up the good work in your calculations and problem-solving skills!