# Doppler Effect Definition, Formula, and Examples

In physics, the Doppler effect or Doppler shift is the change in the frequency of a wave due to the relative motion between the wave source and an observer. For example, an approaching siren has a higher pitch and a receding siren has a lower pitch than the original source. Light approaching a viewer is shifted toward the blue end of the spectrum, while receding light shifts toward red. While most often discussed relating to sound or light, the Doppler effect applies to all waves. The phenomenon gets its name for Austrian physicist Christian Doppler, who first described it in 1842.

### History

Christian Doppler published his findings in a paper titled “Über das farbige Licht der Doppelsterne und einiger anderer Gestirne des Himmels” (“On the colored light of binary stars and some other stars of the heavens”) in 1842. Doppler’s work focused on the analysis of light from binary stars. He observed that the colors of the stars changed depending on their relative motion.

### What Is the Doppler Effect?

In simple terms, the Doppler effect is the change in the pitch or frequency of a sound or light wave as the source or observer moves. When a source of waves (such as a car engine or a star) is moving closer to an observer, the frequency of the waves increases. The frequency of the wave increases, so the pitch of sound becomes higher or the wavelength of light becomes more blue. Conversely, when the source moves away from the observer, the frequency decreases. Sound pitch becomes lower or light becomes redder.

### How the Doppler Effect Works

Waves approaching an observer are compressed, which increases their frequency. On the other hand, waves from a source moving away from an observer get stretched. When the distance between waves increases, frequency decreases.

### The Doppler Effect and Sound Waves

Examples of the Doppler effect in sound waves occur in everyday scenarios such as a passing siren or a train whistle. When a police car with a siren drives past an observer, the pitch of the siren appears to rise as the car approaches and then drop as it moves away.

#### Formulas

The frequency the observers depends on the actual frequency, the velocity of the observer, and the velocity of the source:

f’ = f (V ± V0) / (V ± Vs)

Here:

• f’ is the observed frequency
• f is the actual frequency
• V is the velocity of the waves
• V0 is the velocity of the observer
• Vs is the velocity of the source

#### Source Approaching an Observer at Rest

When the observer has a velocity of zero, then V0 = 0.

f’ = f [V / (V – Vs)]

#### Source Moving Away from an Observer at Rest

When the observer has a velocity of 0, V0 = 0. Because the source moves away, the velocity has a negative sign.

f’ = f [V / (V – (-Vs))] or f’ = f [V / (V +Vs)]

#### Observer Approaching a Stationary Source

In this situation, Vs equals 0:

f’ = f (V +V0) / V

#### Observer Moving Away From a Stationary Source

The observer is moving away from the source, so the velocity is negative:

f’ = f (V -V0) / V

#### Doppler Example Problem

For example, a boy runs toward a music box. The box produces sound with a frequency of 500 Hz. The boy runs toward the box at a speed of 2 m/s. What frequency does the boy hear? The velocity of sound in air is 343 m/s.

Since the boy approaches a stationary object, the correct formula is:

f’ = f (V +V0) / V or f (1 +V0/V)

Putting in the numbers:

f’ = 500 sec-1 [1 + (2 m/s / 343 m/s)] = 502.915 sec-1 = 502.915 Hz

### Doppler Effect in Light

In light waves, the Doppler effect is known as red shift or blue shift, depending on whether the source is moving away from or toward the observer. When a star or galaxy moves away from the observer, its light shifts to longer wavelengths (red shift). Conversely, when the source moves toward the observer, its light shifts to shorter wavelengths (blue shift). Red shift and blue shift are important in astronomy, as they provide information about the movement and distance of celestial objects.

#### Formula

The formula for the Doppler effect in light differs from the formula for sound because light (unlike sounds) needs no medium for propagation. Also, the equation is relativistic because light in a vacuum travels at (you guessed it) the speed of light. The frequency (or wavelength) shift depends only on the relative speeds of the observer and source.

λR = λS [(1-β) / (1+β)]1/2

• λR is the wavelength seen by the receiver
• λS is the wavelength of the source
• β = v/c = velocity / speed of light

How Fast to Make a Red Light Look Green

Explore the Doppler effect in light and calculate how fast you have to go so that a red traffic light appears green. (No, it won’t get you out of a ticket.)

### Practical Applications of the Doppler Effect

The Doppler effect has numerous practical applications. In astronomy, it measures the speed and direction of celestial objects such as stars and galaxies. Meteorology uses the Doppler effect for finding wind speeds by analyzing the Doppler shift of radar waves. In medical imaging, Doppler ultrasound visualizes blood flow in the body. Other uses include sirens, radar, vibration measurement, and satellite communication.

### References

• Ballot, Buijs (1845). “Akustische Versuche auf der Niederländischen Eisenbahn, nebst gelegentlichen Bemerkungen zur Theorie des Hrn. Prof. Doppler (in German)”. Annalen der Physik und Chemie. 142 (11): 321–351. doi:10.1002/andp.18451421102
• Becker, Barbara J. (2011). Unravelling Starlight: William and Margaret Huggins and the Rise of the New Astronomy. Cambridge University Press. ISBN 9781107002296.
• Percival, Will; et al. (2011). “Review article: Redshift-space distortions”. Philosophical Transactions of the Royal Society. 369 (1957): 5058–67. doi:10.1098/rsta.2011.0370
• Qingchong, Liu (1999). “Doppler measurement and compensation in mobile satellite communications systems.” Military Communications Conference Proceedings / MILCOM. 1: 316–320. ISBN 978-0-7803-5538-5. doi:10.1109/milcom.1999.822695
• Rosen, Joe; Gothard, Lisa Quinn (2009). Encyclopedia of Physical Science. Infobase Publishing. ISBN 978-0-8160-7011-4.