Doppler Effect

Problem

A police siren emits a frequency of 640 Hz. When the police car approaches you at a speed of 25 m/s, what frequency do you perceive when:

1) you are not moving

2) you are moving towards the police car at 5 m/s

3) you are moving away from the police car at 5 m/s.

Assume the speed of sound is 343 m/s.

Solution

Use the doppler shift formula

$f = f_0 \left(\frac{v \pm v_r}{v \pm v_s}\right)$

where $f$ is the shifted frequency, $f_0$ is the frequency of the source, $v$ is the speed of sound, $v_r$ is the speed of the receiver, and $v_s$ is the speed of the source.

Part 1

$f=f_{0}\left({\frac {v}{v-v_{s}}}\right)$

$f=640\left({\frac {343}{343-25}}\right)$

$f=690\,Hz$

Part 2

$f=f_{0}\left({\frac {v+v_{r}}{v-v_{s}}}\right)$

$f=640\left({\frac {343+5}{343-25}}\right)$

$f=700.38\,Hz$

Part 3

$f=f_{0}\left({\frac {v-v_{r}}{v-v_{s}}}\right)$

$f=640\left({\frac {343-5}{343-25}}\right)$

$f=680.25\,Hz$