What Causes Beat Frequency in Reflected Laser Light?

[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 … ?