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Explanation of the Doppler Effect Equations for Sound by Ron Kurtus - Succeed in Understanding Physics. **Key words:** frequency, wavelength, velocity, speed, angle, cosine, physical science, School for Champions. Copyright © Restrictions

# Doppler Effect Equations for Sound

by Ron Kurtus (revised 21 November 2012)

The Doppler Effect for sound is the change in frequency or pitch that you hear from a moving source. It will be either higher or lower than the emitted frequency, depending on the direction the source is moving.

The *Doppler Effect equations* allow you to calculate the frequency of the sound as a source moves toward you, away from you, or even at an angle with respect to the line-of-sight. You need to know the initial frequency, the velocity of the source and the speed of sound, as well as the angle between the source and the line-of-sight.

Although you can hear the difference in frequencies, you really need to use a microphone and a frequency meter, such as an oscilloscope, to determine the initial and final frequencies.

Note: This is a good idea for a science project.

Questions you may have include:

- What are the equations when the source is moving toward you?
- What happens when the source is moving away from you?
- What are the equations when the source is moving at an angle to you?

This lesson will answer those questions. Useful tool: Units Conversion

## Source moving toward you

When the source of sound is moving toward you, the pitch you hear is higher than what was emitted from the source and the wavelength is shorter than emitted.

Notethat the speed of the source must bethe speed of sound. An aircraft flying at the speed of sound or greater creates a sonic boom, which is a different effect.less than(

See Traveling Faster than Sound for more information.)

### Observed frequency

The equation for the observed frequency of sound when the source is traveling **toward you** is:

f_{O}= f_{S}c/(c − v_{S})

or

f_{O}= f_{S}/(1 − v_{S}/c)

where

**f**is the observed frequency_{O}**f**is the emitted frequency_{S}**c**is the velocity of sound**v**is the velocity of the source toward you_{S}**c > v**(_{S}**v**is less than_{S}**c**)

Note: Althoughcis usually denoted as the speed of light, it is also often used as the speed or velocity of sound. Since we usevas the velocity of the source, using_{S}cas the velocity of sound avoids confusion.

Notethat this equation does not work if the speed of the source is equal to the speed of sound. In such as case, you would be dividing by 0, which is impossible.

The change in frequency is:

Δf = f_{S}/(1 − c/v_{S}_{})

where **Δf = f _{S} − f_{O}**.

### Observed wavelength

The Doppler Effect equation for the observed wavelength when the source is traveling **toward you** is:

λ_{O}= λ_{S}(c − v_{S})/c

or

λ_{O}= λ_{S}(1 − v_{S}/c)

where

**λ**is the observed wavelength (Greek symbol lambda)_{O }**λ**is the emitted wavelength_{S}

The change in wavelength is:

Δλ = λ_{S}v_{S}/c

where **Δλ = λ _{S} − λ_{O}**.

### Source velocity

If you know the resulting frequency, you can find the speed of the source moving **toward you**:

v_{S}= c(f_{S}/f_{O}+ 1)

### Example

If a vehicle is coming toward you at 96 km/hr (60 miles per hour) and sounds its horn that blares at 8000 Hz, what is the frequency of the sound you hear when the speed of sound is 340 m/s (1115 ft/s)?

Notethat the units of speed must all be the same in the equation. You can also round off numbers to get an answer that is close enough for all practical purposes.

Convert kilometers per hour to meters per second:

96 km/hr = 96000 m/hr

Since 1 hour = 3600 seconds,

96000 m/hr = 96000/3600 = 26.7 m/s

Calculate the frequency:

f_{o}= fc/(c − v_{t}) = 8000*340/(340 − 26.7)

f_{o}= 8682 Hz

In other words, the frequency you hear is about 682 Hz higher than the actual sound of the horn.

## Source moving away from you

When the source of sound is moving away from you, the pitch you hear is lower, the frequency is slower and the wavelength is longer than what was emitted from the source.

Notethat the equations are the same as when the source is moving toward you, except that the"−"sign is replaced by a"+"sign to indicate the change in direction of the source. In the case of velocity, the"+"sign is replaced by a"−"sign.

### Observed frequency

The equation for the observed frequency of a waveform when the source is traveling **away from you** is:

f_{O}= f_{S}/(1 + v_{S}/c)

Δf = f_{S}/(1 + c/v_{S}_{})

where **v _{S} **is the velocity of the source away from you.

### Observed wavelength

The equation for the observed wavelength when the source is traveling **away from you** is:

λ_{O}= λ(1 + v_{S}/c)

Δλ = −λ_{S}v_{S}/c

### Source velocity

The equation for the velocity of the source, when it is traveling **away from you** is:

v_{S}= c(f_{S}/f_{O}− 1)

## Source moving at an angle

When the source is moving directly toward you or directly away from you, its relative velocity is constant with respect to you. Thus, the observed frequency would be constant. But in the usual case of hearing a sound from a moving vehicle, you are at an angle to the line of motion of the source. Otherwise the vehicle would hit you.

The velocity of the source with respect to you is:

v_{Sr}= v_{S}cosθ

where

**v**is the velocity of the source with respect to you_{Sr}**v**is the actual velocity of the source_{S}**cosθ**is the cosine of the angle between the actual direction of the source and your line of sight to the source**θ**is the Greek symbol theta, representing the angle

Velocity of source at an angle

When you aren't standing in the direct path of the moving source, you must substitute **v _{Sr}** for

**v**in the Doppler Effect equations. As the vehicle or source moves, the angle and the relative velocity change. This is the reason that you hear the sound of a siren change pitch as the vehicle comes toward you and passes by.

_{S}### Example

If you were standing 10 meters from the road, and the car in the above example blares its horn when it is 50 meters down the road, what would be the frequency of the sound that you heard?

The sine of **θ** would be **10/50 = 0.2**. Thus **θ = 11.5** degrees and **cosθ = 0.98**.

v_{sr}= v_{s}cosθ = 26.7 * 0.98 = 26.2

f_{o}= fv/(v − v_{t}) = 8000*340/(340 − 26.2)

f_{o}= 8668 Hz

That is **14 Hz** less—or a slightly lower pitch—than what would be heard if you were standing in the road.

## Summary

The pitch of the sound you hear from a moving source will be either higher or lower than the emitted frequency, depending on the direction the source is moving. This is called the Doppler Effect. Knowing the initial frequency, the velocity of the source and the speed of sound, equations are available that allow you to calculate the new frequency. The angle between the source and the line-of-sight adds another factor to the equations.

Be observant

## Resources and references

### Websites

### Books

**Top-rated books on Doppler Effect**

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## Doppler Effect Equations for Sound