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# Doppler Effect

## Waves emitted from a moving source are perceived are perceived at a higher or lower frequency by a stationary observer.

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Doppler Shift

At normal speeds, when an object generates sound, the sound travels away from the object and people hear the same sound that was generated by the object. In some unusual cases, like this airplane, the object can travel as fast or faster than the sounds it makes. The sound wave moves along with this airplane; as the airplane continues to generate more sound, this sound is added to the old sound. When this extra large sound wave front arrives, the sound is much louder than the sound that was originally generated. When this happens with airplanes, the sound is called a “sonic boom.”

### Doppler Shift

If an emitted sound is moving at a significant fraction of the speed of sound relative to the receiver, it is possible for the sound emitted by a source and the sound received by a receiver to be different. This is called a Doppler shift.

Suppose the trumpet player in the sketch below is playing a constant note with a wavelength equal to 1.00 m/s. If you are sitting down the road a ways and the truck is stationary, then the wavelength of the sound that reaches you will be 1.00 m long and you will hear a note corresponding to that wavelength. Suppose now, that the speed of sound is 333 m/s and this truck is moving toward you at 166. m/s. When the trumpet player begins playing the tone, the first crest will move toward you at 333 m/s. In the first 0.003 seconds, that first crest will move exactly 1.00 m (which happens to be one wavelength for this sound). After the trumpet player generates the first crest, however, the truck also moves and in the first 0.003 seconds, the truck moves 0.50 m. After the first 0.003 seconds, the trumpet player creates a second crest and sends it on its way. When this second crest is generated, the first crest is only 0.50 m away. If the truck did not move, the crests would reach your ear with a wavelength of 1.00 m, but because the truck moves toward you at very high speed, the crests reach your ear at a distance of only 0.50 m apart. Therefore, the sound you hear will have shorter wavelengths than the sound that was emitted by the trumpet player. The frequency of the sound you hear would be double the frequency that was being emitted.

If the truck was moving away from you, the opposite change of frequency would occur.  That is, the received wavelength would be longer than the one emitted and the received frequency would be lower than the frequency emitted. You may have noticed frequency changes in automobile sounds when you stand near a highway. Or that an emergency vehicle's siren is different when approaching you than when leaving.

You can also hear the Doppler shift in the classroom simply by striking a tuning fork and then moving the tuning sharply toward you or away from you.  Even though the tuning fork generates a constant tone, you will hear wavering tones as you move the tuning fork around.

In terms of the listener, the train sketched on the left above is not moving and therefore the wavelengths and frequency heard in all directions will be the same as the frequency being emitted. For the train on the right, however, since the train is moving toward the right, the wavelengths measured to the right will be shorter than those emitted and the wavelengths measured to the left will be longer than those emitted.

Christian Doppler (1803 – 1853) did experiments in 1842 with trumpeters playing a single note as they sat on a railroad flatcar and were pulled back and forth past a stationary observer. The Doppler effect also occurs when the source of the sound is stationary and the observer is moving (with a significant fraction of the speed of sound). The following formulas have been developed for calculating the observed frequency:

When the source is moving toward you, $f_o=\frac{f}{1-\frac{v_R}{v}}$ .

when the source is moving away from you, $f_o=\frac{f}{1+\frac{v_R}{v}}$ .

In the equations $f_o$ is the perceived frequency,  $f$ is the emitted frequency,  $v_R$ is the relative velocity and  $v$ is the speed of sound.

Example Problem: The speed of sound is 340. m/s and a train whistle with a frequency of 512 Hz is emitted from a train coming toward you at 40.0 m/s. What frequency will you hear?

Solution: $f_o=\frac{f}{1-\frac{v_R}{v}}=\frac{512 \ s^{-1}}{1-\frac{40.0 \ m/s}{340. \ m/s}}=580 \ Hz$

#### Summary

• The Doppler shift, or Doppler effect, occurs when a sound's emission and the sound's receiver travel relative to each other at a significant speed compared to the speed of sound.
• If the sender and receiver are getting closer together, the perceived frequency will be higher than the emitted frequency, given by the equation   $f_o=\frac{f}{1-\frac{v_R}{v}}$ .
• If the sender and receiver are getting farther apart, the perceived frequency will be lower than the emitted frequency, given by the equation  $f_o=\frac{f}{1+\frac{v_R}{v}}$ .

#### Practice

The following video demonstrates the doppler effect with car horns. Use this resource to answer the questions that follow.

1. What is necessary for a wave source / receiver in order to observe the Doppler effect?
2. How are the waves made with his finger in water similar to the sound waves? How are they different?

#### Review

1. What is the frequency heard by a person driving at 15 m/s toward a blowing factory whistle if the emitted frequency is 800. Hz and the speed of sound is 340. m/s?
2. While standing near a railroad crossing, a person hears a distant train horn.  According to the train’s engineer, the frequency emitted by the horn is 440 Hz and the train is traveling toward the railroad crossing at 20.0 m/s.  If the speed of sound is 343 m/s, what frequency will the observer hear?
3. After the train passed the person at the crossing, what frequency would he hear?