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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.

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.

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

f’ = f (V ± V 0 ) / (V ± V s )

• f’ is the observed frequency
• f is the actual frequency
• V is the velocity of the waves
• V 0 is the velocity of the observer
• V s is the velocity of the source

Source Approaching an Observer at Rest

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

f’ = f [V / (V – V s )]

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

Observer Approaching a Stationary Source

In this situation, V s equals 0:

f’ = f (V +V 0 ) / V

Observer Moving Away From a Stationary Source

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

f’ = f (V -V 0 ) / 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 +V 0 ) / V or f (1 +V 0 /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.

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.

λ R = λ S [(1-β) / (1+β)] 1/2

• λ R  is the wavelength seen by the receiver
• λ S  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.

• 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.

Related Posts

• Sound Waves

Doppler Effect

Doppler effect or Doppler shift is a phenomenon that is observed whenever the source of waves is moving with respect to an observer. For example, an ambulance crossing you with its siren blaring is a common physical demonstration of the Doppler Effect. In this article, let us understand the intricacies of the Doppler effect in detail.

Doppler Effect Explained

Doppler effect is an important phenomenon in various scientific disciplines, including planetary science. The Doppler effect or the Doppler shift describes the changes in the frequency of any sound or light wave produced by a moving source with respect to an observer.

Doppler effect in physics is defined as the increase (or decrease) in the frequency of sound, light, or other waves as the source and observer move towards (or away from) each other.

Waves emitted by a source travelling towards an observer get compressed. In contrast, waves emitted by a source travelling away from an observer get stretched out. Christian Johann Doppler first proposed the Doppler Effect (Doppler Shift) in 1842.

Doppler Effect Examples

Let us imagine the following scenario:

Case 1: Two people A and B, are standing on the road, as shown below in the picture.

Which person hears the sound of the revving engine with a greater magnitude?

Person A hears the sound of the revving engine with a greater magnitude than person B. Person B, standing behind the car, receives fewer waves per second (because they’re spread out), resulting in a low-pitched sound. But, person A who is in front of the car, receives more of those soundwave ripples per second. As a result, the frequency of the waves is higher, which means the sound has a higher pitch.

Case 2: Now let us consider the following situations:

Situation 1: How is the pattern of waves formed when you suddenly jump into a pond? Situation 2: How is the pattern of waves formed when you are walking in a pond?

The image given below highlights the difference between wave patterns in both situations.

The difference in the wave pattern is due to the source’s movement in the second case. This is what the Doppler effect is. In the Doppler effect, the frequency received by the observer is higher during the approach, identical when the relative positions are the same, and keeps lowering on the recession of the source.

In this video let’s see how relative motion between the source and the observer changes the frequency of sound waves giving rise to the Doppler Effect.

Doppler Effect Formula

Doppler effect is the apparent change in the frequency of waves due to the relative motion between the source of the sound and the observer. We can deduce the apparent frequency in the Doppler effect using the following equation:

While there is only one Doppler effect equation, the above equation changes in different situations depending on the velocities of the observer or the source of the sound. Let us see below how we can use the equation of the Doppler effect in different situations.

(a) Source Moving Towards the Observer at Rest

(b) Source Moving Away from the Observer at Rest

(c) Observer Moving Towards a Stationary Source

(d)Observer Moving Away from a Stationary Source

Doppler Effect Solved Problems

Two trains A and B are moving toward each other at a speed of 432 km/h. If the frequency of the whistle emitted by A is 800 Hz, then what is the apparent frequency of the whistle heard by the passenger sitting in train B. (The velocity of sound in air is 360 m/s).

The source and the observer are moving toward each other, hence.

Converting 432 km/h into m/s we get 120 m/s.

Substituting the values in the equation, we get

\(\begin{array}{l}f=800(\frac{360+120}{360-120})=1600\,Hz\end{array} \)

2. A bike rider approaching a vertical wall observes that the frequency of his bike horn changes from 440 Hz to 480 Hz when it gets reflected from the wall. Find the speed of the bike if the speed of sound is 330 m/s.

Let the bike approach the wall with speed u.

Then the apparent frequency received by the wall can be calculated as

For the reflected wave,

Substituting (1) in (2), we get

Simplifying, we get

Let us see more solved examples in the video given below

Uses of Doppler Effect

Many people mistake the Doppler effect to be applicable only for sound waves. It works with all types of waves including light. Below, we have listed a few applications of the doppler effect:

• Medical Imaging
• Blood Flow Measurement
• Satellite Communication
• Vibration Measurement
• Developmental Biology
• Velocity Profile Measurement

Doppler Effect Limitations

• Doppler Effect is applicable only when the velocities of the source of the sound and the observer are much less than the velocity of sound.
• The motion of both source and the observer should be along the same straight line.

Doppler Effect In Light

Doppler effect of light can be described as the apparent change in the frequency of the light observed by the observer due to relative motion between the source of light and the observer. For sound waves, however, the equations for the Doppler shift differ markedly depending on whether it is the source, the observer, or the air, which is moving. Light requires no medium, and the Doppler shift for light travelling in a vacuum depends only on the relative speed of the observer and source.

Red Shift and Blue Shift

• When the light source moves away from the observer, the frequency received by the observer will be less than the frequency transmitted by the source. This causes a shift towards the red end of the visible light spectrum. Astronomers call it the redshift .
• When the light source moves towards the observer, the frequency received by the observer will be greater than the frequency transmitted by the source. This causes a shift towards the high-frequency end of the visible light spectrum. Astronomers call it the blue shift .

The below video provides an in-depth analysis of Doppler Effect for JEE Advanced 2023

Frequently Asked Questions – FAQs

What is the doppler effect in physics, who discovered the doppler effect, can doppler effect be observed in both longitudinal and transverse waves, how can the doppler effect be applied to everyday life.

A few daily life examples of the Doppler effect are: a) When you stand beside a police radar. b) The Doppler effect is used by meteorologists to track storms. c) Doctors use the Doppler Effect in hospitals to diagnose heart problems. d) Traffic police make use of the doppler effect a radar gun to check the speed of the oncoming vehicles.

Why is the Doppler Effect used in hospitals?

How does the doppler effect prove that the universe is expanding.

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17.7 The Doppler Effect

Learning objectives.

By the end of this section, you will be able to:

• Explain the change in observed frequency as a moving source of sound approaches or departs from a stationary observer
• Explain the change in observed frequency as an observer moves toward or away from a stationary source of sound

The characteristic sound of a motorcycle buzzing by is an example of the Doppler effect . Specifically, if you are standing on a street corner and observe an ambulance with a siren sounding passing at a constant speed, you notice two characteristic changes in the sound of the siren. First, the sound increases in loudness as the ambulance approaches and decreases in loudness as it moves away, which is expected. But in addition, the high-pitched siren shifts dramatically to a lower-pitched sound. As the ambulance passes, the frequency of the sound heard by a stationary observer changes from a constant high frequency to a constant lower frequency, even though the siren is producing a constant source frequency. The closer the ambulance brushes by, the more abrupt the shift. Also, the faster the ambulance moves, the greater the shift. We also hear this characteristic shift in frequency for passing cars, airplanes, and trains.

The Doppler effect is an alteration in the observed frequency of a sound due to motion of either the source or the observer. Although less familiar, this effect is easily noticed for a stationary source and moving observer. For example, if you ride a train past a stationary warning horn, you will hear the horn’s frequency shift from high to low as you pass by. The actual change in frequency due to relative motion of source and observer is called a Doppler shift . The Doppler effect and Doppler shift are named for the Austrian physicist and mathematician Christian Johann Doppler (1803–1853), who did experiments with both moving sources and moving observers. Doppler, for example, had musicians play on a moving open train car and also play standing next to the train tracks as a train passed by. Their music was observed both on and off the train, and changes in frequency were measured.

What causes the Doppler shift? Figure 17.30 illustrates sound waves emitted by stationary and moving sources in a stationary air mass. Each disturbance spreads out spherically from the point at which the sound is emitted. If the source is stationary, then all of the spheres representing the air compressions in the sound wave are centered on the same point, and the stationary observers on either side hear the same wavelength and frequency as emitted by the source (case a). If the source is moving, the situation is different. Each compression of the air moves out in a sphere from the point at which it was emitted, but the point of emission moves. This moving emission point causes the air compressions to be closer together on one side and farther apart on the other. Thus, the wavelength is shorter in the direction the source is moving (on the right in case b), and longer in the opposite direction (on the left in case b). Finally, if the observers move, as in case (c), the frequency at which they receive the compressions changes. The observer moving toward the source receives them at a higher frequency, and the person moving away from the source receives them at a lower frequency.

We know that wavelength and frequency are related by v = f λ , v = f λ , where v is the fixed speed of sound. The sound moves in a medium and has the same speed v in that medium whether the source is moving or not. Thus, f multiplied by λ λ is a constant. Because the observer on the right in case (b) receives a shorter wavelength, the frequency she receives must be higher. Similarly, the observer on the left receives a longer wavelength, and hence he hears a lower frequency. The same thing happens in case (c). A higher frequency is received by the observer moving toward the source, and a lower frequency is received by an observer moving away from the source. In general, then, relative motion of source and observer toward one another increases the received frequency. Relative motion apart decreases frequency. The greater the relative speed, the greater the effect.

The Doppler effect occurs not only for sound, but for any wave when there is relative motion between the observer and the source. Doppler shifts occur in the frequency of sound, light, and water waves, for example. Doppler shifts can be used to determine velocity, such as when ultrasound is reflected from blood in a medical diagnostic. The relative velocities of stars and galaxies is determined by the shift in the frequencies of light received from them and has implied much about the origins of the universe. Modern physics has been profoundly affected by observations of Doppler shifts.

Derivation of the Observed Frequency due to the Doppler Shift

Consider two stationary observers X and Y in Figure 17.31 , located on either side of a stationary source. Each observer hears the same frequency, and that frequency is the frequency produced by the stationary source.

Now consider a stationary observer X with a source moving away from the observer with a constant speed v s < v v s < v ( Figure 17.32 ). At time t = 0 t = 0 , the source sends out a sound wave, indicated in black. This wave moves out at the speed of sound v . The position of the sound wave at each time interval of period T s T s is shown as dotted lines. After one period, the source has moved Δ x = v s T s Δ x = v s T s and emits a second sound wave, which moves out at the speed of sound. The source continues to move and produce sound waves, as indicated by the circles numbered 3 and 4. Notice that as the waves move out, they remained centered at their respective point of origin.

Using the fact that the wavelength is equal to the speed times the period, and the period is the inverse of the frequency, we can derive the observed frequency:

As the source moves away from the observer, the observed frequency is lower than the source frequency.

Now consider a source moving at a constant velocity v s , v s , moving toward a stationary observer Y , also shown in Figure 17.32 . The wavelength is observed by Y as λ o = λ s − Δ x = λ s − v s T s . λ o = λ s − Δ x = λ s − v s T s . Once again, using the fact that the wavelength is equal to the speed times the period, and the period is the inverse of the frequency, we can derive the observed frequency:

When a source is moving and the observer is stationary, the observed frequency is

where f o f o is the frequency observed by the stationary observer, f s f s is the frequency produced by the moving source, v is the speed of sound, v s v s is the constant speed of the source, and the top sign is for the source approaching the observer and the bottom sign is for the source departing from the observer.

What happens if the observer is moving and the source is stationary? If the observer moves toward the stationary source, the observed frequency is higher than the source frequency. If the observer is moving away from the stationary source, the observed frequency is lower than the source frequency. Consider observer X in Figure 17.33 as the observer moves toward a stationary source with a speed v o v o . The source emits a tone with a constant frequency f s f s and constant period T s . T s . The observer hears the first wave emitted by the source. If the observer were stationary, the time for one wavelength of sound to pass should be equal to the period of the source T s . T s . Since the observer is moving toward the source, the time for one wavelength to pass is less than T s T s and is equal to the observed period T o = T s − Δ t . T o = T s − Δ t . At time t = 0 , t = 0 , the observer starts at the beginning of a wavelength and moves toward the second wavelength as the wavelength moves out from the source. The wavelength is equal to the distance the observer traveled plus the distance the sound wave traveled until it is met by the observer:

If the observer is moving away from the source ( Figure 17.34 ), the observed frequency can be found:

The equations for an observer moving toward or away from a stationary source can be combined into one equation:

where f o f o is the observed frequency, f s f s is the source frequency, v v is the speed of sound, v o v o is the speed of the observer, the top sign is for the observer approaching the source and the bottom sign is for the observer departing from the source.

Equation 17.18 and Equation 17.19 can be summarized in one equation (the top sign is for approaching) and is further illustrated in Table 17.4 :

where f o f o is the observed frequency, f s f s is the source frequency, v v is the speed of sound, v o v o is the speed of the observer, v s v s is the speed of the source, the top sign is for approaching and the bottom sign is for departing.

Interactive

The Doppler effect involves motion and a video will help visualize the effects of a moving observer or source. This video shows a moving source and a stationary observer, and a moving observer and a stationary source. It also discusses the Doppler effect and its application to light.

Example 17.8

Calculating a doppler shift.

(a) What frequencies are observed by a stationary person at the side of the tracks as the train approaches and after it passes?

(b) What frequency is observed by the train’s engineer traveling on the train?

• Enter known values into f o = f s ( v v − v s ) : f o = f s ( v v − v s ) : f o = f s ( v v − v s ) = ( 150 Hz ) ( 340 m/s 340 m/s − 35.0 m/s ) . f o = f s ( v v − v s ) = ( 150 Hz ) ( 340 m/s 340 m/s − 35.0 m/s ) . Calculate the frequency observed by a stationary person as the train approaches: f o = ( 150 Hz ) ( 1.11 ) = 167 Hz . f o = ( 150 Hz ) ( 1.11 ) = 167 Hz . Use the same equation with the plus sign to find the frequency heard by a stationary person as the train recedes: f o = f s ( v v + v s ) = ( 150 Hz ) ( 340 m/s 340 m/s + 35.0 m/s ) . f o = f s ( v v + v s ) = ( 150 Hz ) ( 340 m/s 340 m/s + 35.0 m/s ) . Calculate the second frequency: f o = ( 150 Hz ) ( 0.907 ) = 136 Hz . f o = ( 150 Hz ) ( 0.907 ) = 136 Hz .
• It seems reasonable that the engineer would receive the same frequency as emitted by the horn, because the relative velocity between them is zero.
• Relative to the medium (air), the speeds are v s = v o = 35.0 m/s . v s = v o = 35.0 m/s .
• The first Doppler shift is for the moving observer; the second is for the moving source.

Significance

For the engineer riding in the train, we may expect that there is no change in frequency because the source and observer move together. This matches your experience. For example, there is no Doppler shift in the frequency of conversations between driver and passenger on a motorcycle. People talking when a wind moves the air between them also observe no Doppler shift in their conversation. The crucial point is that source and observer are not moving relative to each other.

Check Your Understanding 17.9

Describe a situation in your life when you might rely on the Doppler shift to help you either while driving a car or walking near traffic.

The Doppler effect and the Doppler shift have many important applications in science and engineering. For example, the Doppler shift in ultrasound can be used to measure blood velocity, and police use the Doppler shift in radar (a microwave) to measure car velocities. In meteorology, the Doppler shift is used to track the motion of storm clouds; such “Doppler Radar” can give the velocity and direction of rain or snow in weather fronts. In astronomy, we can examine the light emitted from distant galaxies and determine their speed relative to ours. As galaxies move away from us, their light is shifted to a lower frequency, and so to a longer wavelength—the so-called red shift . Such information from galaxies far, far away has allowed us to estimate the age of the universe (from the Big Bang) as about 14 billion years.

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This is a simulation of the Doppler effect. You can set both the initial position and the velocity of the source (the small blue dot). and the initial position and the velocity of the observer (green rectangle), and then see the pattern of waves emitted by the source as the waves wash over the observer. The source emits a frequency of 100 Hz when the source is at rest. fo represents the observed frequency (the one heard by the observer).

Simulation first posted on 1-27-2017. Written by Andrew Duffy

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What is the Doppler Effect?

The Doppler effect can be observed for any type of wave - water wave, sound wave, light wave, etc. We are most familiar with the Doppler effect because of our experiences with sound waves. Perhaps you recall an instance in which a police car or emergency vehicle was traveling towards you on the highway. As the car approached with its siren blasting, the pitch of the siren sound (a measure of the siren's frequency) was high; and then suddenly after the car passed by, the pitch of the siren sound was low. That was the Doppler effect - an apparent shift in frequency for a sound wave produced by a moving source.

The Doppler Effect in Astronomy

The Doppler effect is of intense interest to astronomers who use the information about the shift in frequency of electromagnetic waves produced by moving stars in our galaxy and beyond in order to derive information about those stars and galaxies. The belief that the universe is expanding is based in part upon observations of electromagnetic waves emitted by stars in distant galaxies. Furthermore, specific information about stars within galaxies can be determined by application of the Doppler effect. Galaxies are clusters of stars that typically rotate about some center of mass point. Electromagnetic radiation emitted by such stars in a distant galaxy would appear to be shifted downward in frequency (a red shift ) if the star is rotating in its cluster in a direction that is away from the Earth. On the other hand, there is an upward shift in frequency (a blue shift ) of such observed radiation if the star is rotating in a direction that is towards the Earth.  

• Traveling Waves vs. Standing Waves

The Doppler Effect

This topic is part of the HSC Physics course under the section Sound Waves.

HSC Physics Syllabus

• analyse qualitatively and quantitatively the relationships of the wave nature of sound to explain the Doppler's effect

Doppler Effect for Sound Waves Explained

Understanding the Doppler Effect

Imagine you're standing on a street corner and an ambulance speeds by with its siren blaring. You've likely noticed that as the ambulance approaches, the siren sounds higher-pitched, but as it passes and moves away, the pitch lowers. This phenomenon is the Doppler Effect at play.

What is the Doppler Effect?

The Doppler Effect refers to the change in frequency (or wavelength) of a wave concerning an observer who is moving relative to the source of the wave. It's a phenomenon observed with all types of waves, including sound, light, and radio waves.

When the source of a wave is moving towards the observer, the waves get "compressed", leading to a higher frequency (shorter wavelength). Conversely, if the source is moving away from the observer, the waves get "stretched", resulting in a lower frequency (longer wavelength).

For sound waves, a higher frequency is perceived as a higher pitch, while a lower frequency sounds like a lower pitch.

The formula to calculate the observed frequency `f'`  due to the Doppler Effect for any wave including sound is

$$f' = \frac{v_{\text{wave}} + v_{\text{observer}}}{v_{\text{wave}} - v_{\text{source}}} f$$

• `f` = original frequency of the source
• `v_{\text{wave}}` = velocity of the wave
• `v_{\text{observer}}` = velocity of the observer relative to the medium (positive if the observer is moving towards the source).
• `v_{\text{source}}`​ = velocity of the source relative to the medium (positive if the source is moving towards from the observer).

It is important to be aware that there are various forms of the formula for Doppler's effect which all give the same calculated values when used correctly. HSC Physics students are advised to learn the formula provided above as that is given by the NESA data sheet.

Important values from NESA data sheet:

• The speed of sound in air = `340` m/s
• The speed of light and electromagnetic waves in a vacuum = `3 xx 10^8` m/s

Calculation Example

An ambulance with a siren frequency of 700 Hz.

Calculate the observed frequency of the siren  if

(a) the ambulance is moving towards a stationary observer at 30 m/s

(b) the ambulance is moving at 30 m/s towards an observer who is moving away from the ambulance at 5 m/s

(c) the ambulance is moving away from a stationary observer at 30 m/s.

Solution to part (a):

$$f' = \frac{v_{\text{wave}} + v_{\text{observer}}}{v_{\text{wave}} - v_{\text{source}}} f$$ 

Since the ambulance (source) is moving towards the observe, its velocity is positive in the formula:

$$f' = \frac{340 + 0}{340 - 30} \times 700 = 768 \, \text{Hz}$$

Solution to part   (b):

Since the observer is moving away from the ambulance, his velocity is negative in the formula:

$$f' = \frac{340 + (-5)}{340 - 30} \times 700 = 756 \, \text{Hz}$$

Solution part ( c):

Since the ambulance is moving away from the observer, its velocity is negative in the formula:

$$f' = \frac{340 + 0}{340 - (-30)} \times 700 = 643 \, \text{Hz}$$

RETURN TO MODULE 3: Waves and Thermodynamics

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Five Doppler Effect Experiments with a Smartphone

Updated: Feb 15

Discovered in 1842, the Doppler effect has established itself as an essential investigation tool in modern science. This article details five activities adapted to different learning levels, to be carried out in class, at home or outdoors, simply using a smartphone or tablet. We will also provide specific advice to optimize their implementation. These practical experiments offer a unique opportunity to grasp the concrete applications of the Doppler effect in our daily lives, as well as its role in more advanced fields such as the detection of exo-planets.

A little history - Studying the Doppler effect with a smartphone - Demonstrating the Doppler effect - Doppler measurements from a recording - Measuring the speed of a cyclist - Identification of exo-planets - Measurement height - Doppler effect and acoustic beats - Conclusion

A little history ...

In 1842, Christian Doppler, an Austrian physicist, proposed a new theory: the frequency of a wave (whether sound or light) is affected by the movement of the source relative to the observer. This frequency shift is directly proportional to the speed:

Δf = f.V mobile /V wave where V mobile is the speed of the mobile and V wave is the speed of the wave.

Initially, Christian Doppler's theory was met with skepticism. However, irrefutable proof was soon provided by the meteorologist Buys-Ballot: in 1845, he organized a spectacular experiment by placing musicians on the platform of a train traveling at a speed of 70 km/h and making them play a constant note. Each person on the train journey was able to notice the change in frequency of the sounds emitted by the orchestra as the train passed in front of them, thus convincing themselves that the Doppler effect was not an illusion.

In developing his theory, Christian Doppler hoped to explain the color variations of stars by the change in their light frequency due to their speed relative to the Earth. We now know that the temperature of stars is the main factor in their coloring. However, there is indeed a relativistic Doppler effect for light, which is an extension of the classic Doppler effect, taking into account the effects of Einstein's special relativity: in special relativity, time expansion and length contraction must be considered. The relativistic Doppler effect can then be described by the following formulas for an object moving away from the source (red Doppler):

Δf  = f* [(1+ β)/(1-β)] ½ - f, with β = v/c, v relative speed and c speed of light

Today, the Doppler effect is used in many technologies, such as weather radar, medical imaging, and for control and security. It has proven to be a valuable tool for astronomers, allowing them to understand celestial movements and discover new objects such as exoplanets. From the beginnings in Doppler's laboratory to modern observatories peering into the depths of space, the Doppler effect has shaped our understanding of the universe, providing us with windows into the motion and composition of celestial bodies.

Why study the Doppler effect with a smartphone

If the study of the Doppler effect for light proves difficult, if not impossible, outside of a laboratory, anyone can very easily set up experiments showing its effect on sound waves. All we will need for these experiments is a sound source and a frequency meter, two instruments that are easily available with a smartphone or tablet.

For the sound source, you can use your phone's speaker, or better, a connected speaker, more robust and compact. It is easier for analysis to work with pure, easily identifiable sounds. The FizziQ application includes a synthesizer which can be found in the Tools menu and which can be connected to an external speaker. To ensure precise measures, we will prefer to use a high frequency, greater than 1000 hertz, but not too high so as not to tire the eardrums. Of course, the sound volume must be adjusted so that it is comfortable for the experimenter.

For frequency measurements, we will use the frequency meter in the FizziQ application. This uses the microphone of the smartphone or tablet. Smartphone microphones are extremely sensitive and analyze sound waves precisely, capturing nearly 44,000 pieces of information per second. These characteristics, combined with the significant computing power of digital tools, make it possible to obtain precise data on the shapes and periods of sound waves. Note that with the FizziQ application you can emit pure sound and at the same time analyze the sounds with the microphone. In the majority of cases we therefore only need two telephones.

FizziQ offers several measures for the frequency of a sound wave: fundamental frequency, which is calculated in FizziQ with a Yin algorithm; the dominant frequency, which is the frequency of greatest intensity in the spectrum and which is calculated by a Fourier series transformation; and the frequency spectrum, which allows complex sounds to be analyzed.

One of the frequent problems encountered when testing the Doppler effect is ambient noise which disrupts the measurements. This is especially true when working outdoors. It is therefore necessary to favor a quiet place such as a dead end or a parking lot, and to work with pure frequencies to facilitate frequency measurements.

On the ground, the temptation to take direct measurements is strong. However, sound measurements are delicate, particularly outdoors and in groups. We therefore recommend making sound recordings during the experiments and analyzing these recordings later, in the laboratory or in the classroom. This approach not only saves time, and helps adjust measurement methods in a quiet environment, but also facilitates the sharing of data between different groups, thus ensuring effective and enriching collaboration.

Finally, for those who live in the city, do not have access to quiet places to carry out the experiments, or do not have the time to go our in the field, they can use sound files present in the sound library of the FizziQ application or available on the internet. This use also has the advantage of being able to make reproducible measurements.

Thanks to modern technology, students, science enthusiasts and teachers have at their disposal powerful tools to address the question of the Doppler effect and its direct and powerful applications. Let us now move on to the experiments that can be carried out to understand this phenomenon and its applications.

Exhibit the Doppler effect

Our first experiment consists simply of highlighting the concept of the Doppler effect. It could not be easier ! Download the FizziQ application on a smartphone (or any other application giving access to a sound synthesizer). In the Tool tab, we select the Synthesizer and generate a pure sound with a frequency of 1000 hertz. We then wave the smartphone in front of us using large movements from left to right then from right to left. We clearly hear a shift in the sound: from the lowest to the highest when the smartphone gets closer, then more serious when it moves away.

It can be verified that the Doppler effect is also present if the detector moves, rather than the sound source. By shaking the receiving smartphone, we notice frequency shifts in the same way. Finally, we will ensure that if the two smartphones are shaken together but without one moving relative to the other, the Doppler effect is then zero. It is therefore the relative movement of the source in relation to the receiver which creates the Doppler effect.

If you wish, you can carry out a more spectacular experiment: you place a smartphone in a plastic bag and perform rotations at arm's length with the bag. If we place ourselves perpendicular to the axis of rotation, we will clearly hear the difference in frequency when the bag approaches and when it moves away. On the other hand, if we position ourselves exactly a few meters in the axis of rotation, we will not hear a change in frequency because the speed of the smartphone along this axis is zero if the rotation is uniform.

Finally, let's highlight by measuring the change in frequency that we perceive by ear. To do this, we use a second smartphone on which we have also installed the FizziQ application. On this second smartphone, we select the Dominant Frequency in the Microphone instrument, and we can see that the frequency increases when the source moves towards the sensor and decreases when it moves away. We have clearly demonstrated the Doppler effect.

Doppler measurements from a recording

Studying the Doppler effect has never been easier since the advent of digital tools. Simply download an application that measures frequencies and play a sound file containing a Doppler effect recording on another smartphone or connected speaker. In a few minutes, students can make a first measurement and apply the theoretical formulas learned in class. There are many files available on the internet. The easiest to analyze are those that use a mobile emitting pure sound. If the sound is complex we will use fundamental frequency measurement or a history of the frequency spectrum.

The FizziQ app contains everything you need to study the Doppler effect:

The Sounds library in the Tool menu offers the choice of two different Doppler effect sounds: a moving mobile emitting a pure sound of 1000 hertz, and the sound of a sound pendulum.

To make frequency measurements we will use the dominant frequency measurement or the fundamental frequency measurement in the Measurements tab. These measurements will be recorded over the necessary period of time.

In FizziQ it is possible to do both sound generation and measurement at the same time. we therefore need only one smartphone to do the analysis.

The experiment notebook allows you to analyze graphs and data, write text, add photos and share the notebook in PDF. It is also interesting to export the data to Excel.

Thanks to the power of modern digital tools, it is very easy for teachers to put the theory of the Doppler effect into practice in a few minutes after the theoretical course. However, it is even more educational for students to make their own sound file, and it is ultimately simpler to do than you might think...

Measuring the speed of a cyclist using the Doppler effect

How to carry out a life-size Doppler effect experiment? What precautions should we take? What are the best activities? We will see that even if life-size Doppler effect experiments can sometimes be difficult to carry out, with a little perseverance we can carry out very interesting measurements and the challenge of making these measurements is of great educational interest.

An easy-to-perform experiment uses a bike, a connected speaker and a smartphone. We attach the connected speaker to the front of the bike, and we emit on this speaker a pure sound, for example with a frequency of 1000 hertz, generated by the sound synthesizer of the fizziQ application. The cyclist then rides at a constant speed and passes near an operator who measures the frequency. On FizziQ, we will record the frequency as the bike passes by by pressing the REC button. By measuring the frequency before and after the bicycle passes, we deduce the average frequency and the frequency shift, then the speed of the mobile.

To check the measurements taken, you can also record the GPS speed, either with another smartphone, or using the dual measurement option, Duo mode, an option found in the Tools menu. Be careful to select the frequency as the first instrument because it is this which dictates the acquisition frequency.

How can you carry out this experiment with the maximum chance that your field visit will not be a failure?

Favor environments without external noise and use pure sound for the broadcast. A park, a cul-de-sac or a school parking lot could do the trick.

Rather than trying to take measurements on site, make an audio recording of the cyclist's passage, an audio file that will be shared and analyzed in class. So everyone can do their own analysis.

Make sure the speaker broadcasts in all directions, not directly in front, and pay attention to the volume level that poses a health hazard.

Some students will question whether these measurements are the same as those made by the gendarmerie to measure car speeds. Doppler radar works by emitting radio waves (very low wavelength waves) towards vehicles moving on the road. When these radio waves come into contact with a moving vehicle, they are reflected and return to the radar. By measuring the change in frequency of these reflected waves compared to those emitted, the Doppler effect allows the radar to determine the speed of the targeted vehicle.

Identification of exo-planets

The first exoplanet was discovered by astronomers Michel Mayor and Didier Queloz in 1995. This breakthrough paved the way for the search for other worlds beyond our own solar system and more than 5,000 new planets have been identified to date . Given their distance, it is impossible to detect them visually but their presence can nevertheless be detected by measurement. There are several methods for detecting exo-planets: the transit method which consists of measuring the decrease in the luminosity of a star when the planet passes in front of it, astrometry which measures the small oscillations of a star but requires very high precision in measurements, and variations in the speed of stars by Doppler effect measurement.

When a planet orbits a star, gravity causes the two bodies to exert mutual attraction. Even though the star is much more massive and seems little influenced by the planet, it actually moves back and forth around a common point, called the center of mass of the system. This tiny stellar swing manifests itself as a regular oscillation, synchronized with the orbit of the planet. This effect, although subtle, causes periodic variations in its speed through space. These variations slightly modify the color (or wavelength) of the light emitted by the star because of the Doppler effect. By observing the star's spectral lines, which are very precise lines in its light spectrum characteristic of certain chemical elements, astronomers can detect these tiny color changes and accurately calculate the star's radial velocity. The magnitude of the shifts also gives indications of the mass of the planet, because a more massive planet will induce a more pronounced movement of the star. Furthermore, by observing the periodicity of this movement, we can deduce the orbital period of the planet, and, by applying the laws of celestial mechanics, such as Kepler's third law and Newton's principles of universal gravitation, scientists can determine key characteristics of the exoplanet, such as its mass and the shape of its orbit.

To understand this phenomenon we can do an experiment on sound rather than light. In this experiment we study the frequency variations of a rotating sound pendulum. We place a smartphone set to measure the fundamental (or dominant) frequency and at a distance of one meter we rotate a pendulum composed of a sound source emitting a pure sound of 1000 hertz. Analyzing the frequency allows us to obtain two pieces of information which will tell us about the diameter of the circle described by the weighing pendulum.

This experiment shows that at a distance we can know valuable information about distant objects, provided that they follow very specific physical laws. Here, we know that the mobile describes a circle and therefore the tangential speed and the period make it possible to deduce the radius of the circle traveled. In the case of exo-planets, it is the knowledge of Newton's laws which will make it possible to deduce the mass and the distance from the star.

To find out more, we can consult TP on the star Pegasus 51: https://faculty.uca.edu/njaustin/PHYS1401/Laboratory/exoPlanet.pdf

Height measurement by Doppler effect

Can we know the height of a building using the Doppler effect? This question will undoubtedly bring to mind the anecdote about Niels Bohr, then a student, who was asked how to measure the height of a building using a barometer. Faced with this question, the young Bohr imagined a catalog of solutions, some of which were humorous by deliberately omitting the solution his teacher expected and which used the dependence of atmospheric pressure on altitude.

One solution consists of dropping a device generating a sound source from the top of the building and measuring the frequency of the sound at ground level. By Doppler effect, by knowing the frequency of the source we will determine the landing speed, and as we also know the law of gravitation, we can deduce the height of the building.

Indeed h = 1/2.g.T ²   , V mobile = g.T and on the other hand Δf = f.V mobile /V wave

from where h = ( Δf.V wave /f) ² /(2.g)

with h, height of the building, T duration of the fall, Vmobile speed of the object in free fall, g the acceleration of gravity i.e. 9.81 m/s2 and V wave the speed of sound i.e. 340 m/s.

Of course there is no question of dropping a smartphone from the top of a building but you can experience a height of 2m by placing a cushion to absorb the shock of the falling sound source. This sound source can be a small connected speaker which emits a sound of 1000 hertz for example.

Doppler effect and acoustic beats

We have seen that we can measure the speed of an object emitting sound by measuring its frequency, but can we also measure this speed if we do not have a frequency meter?

An interesting tool that musicians have used for many centuries to measure small frequency shifts is the acoustic beat phenomenon; concept that we discussed in another article: the acoustic beat . An acoustic beat is a regular variation in sound intensity, easily detectable by ear, that occurs when two pure tones are emitted at the same time with a small shift in frequency. If this offset is less than 20 hertz, we can hear the regular and periodic variations due to interference between the two sound waves. For higher offsets, the phenomenon is highlighted by a sound intensity level which shows the characteristic periodic variations in intensity.

If we now consider a moving mobile emitting a pure sound of a certain frequency f. For a stationary observer the wave is shifted by a frequency Δf due to the Doppler effect. For speeds less than 10 m/s, this variation will be of the order of a few tens of hertz. If at the same time this transmitter emits a sound of the same frequency f, the two waves will interfere and create a beat of frequency Δf which can be measured using the sound level measurement. We therefore have a way to measure the frequency of the Doppler shift, without measuring the frequency of the signal, but by measuring its intensity, the result of the interference of two sound sources of the same frequency, one in motion and the other stationary.

Let's make this montage with a sound pendulum. We attack at the end of a pendulum a sound source of a certain frequency f. We then place a sound source of the same frequency f as that emitted by the sound pendulum next to the lowest point of the pendulum, at rest we will only hear one frequency. But if the pendulum oscillates, due to the Doppler effect the sound emitted by the pendulum will be shifted depending on the speed of the pendulum relative to the sound source, and a beating phenomenon will appear. The frequency of the beat will be maximum when the pendulum passes through its lowest point, and minimum (and zero), at its highest point when the speed is zero. We deduce the maximum speed by Doppler effect V max = c/(T*f) with c the speed of sound, T the period of the beat and f the frequency used.

The experiment was carried out with a small speaker connected as a mobile, a frequency of 300 hertz, and the use of a smartphone with FizziQ both as a fixed sound source and as a tool for measuring sound intensity. We found a speed of 2.83 m/s. As it is a pendulum we have a simple way to check this result. Indeed, for a pendulum the maximum speed depends on the height h at which the pendulum is released. By conservation of mechanical energy while neglecting friction, the speed at the lowest point is then V max = (2*g*h) ½, with h the height for which the mobile is released. In our example the theoretical speed is V theo = 2.8 m/s, therefore a value very close to that which we calculated using the acoustic beat method.

The combination of the Doppler effect and acoustic beats was popularized by Ulysse Delabre who used it to estimate the speed of sound. the details can be found in this video: https://www.canal-u.tv/chaines/univ-bordeaux/les-smartphones/18-les-smartphones-determination-de-la-vitesse-du-son-par

Exploring the Doppler effect through the use of smartphones offers an educational perspective rich in possibilities. This educational approach allows complex scientific concepts to be approached in a practical and interactive way, while taking advantage of modern technology. Students can develop their understanding of fundamental physics principles while acquiring essential skills in observation, measurement and data analysis. This educational approach, by integrating ubiquitous mobile technology into students' daily lives, also provides a unique opportunity to spark their interest in science and encourage them to consider careers in fields related to science, technology, engineering and mathematics (STEM), but also to open their eyes to the technologies that are used in everyday life.

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Shop Experiment Doppler Effect: Sound Waves Experiments​

Doppler effect: sound waves.

Experiment #23 from Physics with Video Analysis

Introduction

Have you ever noticed that if a car with a horn blowing is moving past you rapidly that the sound waves emitted by the horn seems to change frequency? In 1842 an Austrian physicist, Hans Christian Doppler, observed and analyzed the same phenomenon for sound emitted by a passing train. Hence the phenomenon is known as the Doppler Effect .

A similar effect is found for the propagation of light and other electromagnetic waves from moving sources. In fact highway patrol officers use radar to measure Doppler shifts in radio waves so they can determine how fast vehicles are moving.

In this activity, you will work with movies of a honking car moving fairly rapidly on a straight level road past a stationary microphone.

In this activity, you will

• Verify that the Doppler Equations can be used to predict the ratio ( f F / f B ) of the apparent car horn frequencies before and after the car passes the microphone.

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This experiment is #23 of Physics with Video Analysis . The experiment in the book includes student instructions as well as instructor information for set up, helpful hints, and sample graphs and data.

Learn about the Doppler Effect

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Astronomers study the light from distant objects in order to understand them. Light moves through space at 299,000 kilometers per second, and its path can be deflected by gravity as well as absorbed and scattered by clouds of material in the universe. Astronomers use many properties of light to study everything from planets and their moons to the most distant objects in the cosmos. 

Delving into the Doppler Effect

One tool they use is the Doppler effect. This is a shift in the frequency or wavelength of radiation emitted from an object as it moves through space. It's named after Austrian physicist Christian Doppler who first proposed it in 1842. 

How does the Doppler Effect work? If the source of radiation, say a star , is moving toward an astronomer on Earth (for example), then the wavelength of its radiation will appear shorter (higher frequency, and therefore higher energy). On the other hand, if the object is moving away from the observer then the wavelength will appear longer (lower frequency, and lower energy). You have probably experienced a version of the effect when you heard a train whistle or a police siren as it moved past you, changing pitch as it passes by you and moves away.

The Doppler effect is behind such technologies as police radar, where the "radar gun" emits light of a known wavelength. Then, that radar "light" bounces off a moving car and travels back to the instrument. The resulting shift in wavelength is used to calculate the speed of the vehicle. ( Note: it is actually a double shift as the moving car first acts as the observer and experiences a shift, then as a moving source sending the light back to the office, thereby shifting the wavelength a second time. )

When an object is receding (i.e. moving away) from an observer, the peaks of the radiation that are emitted will be spaced farther apart than they would be if the source object were stationary. The result is that the resulting wavelength of light appears longer. Astronomers say that it is "shifted to the red" end of the spectrum.

The same effect applies to all bands of the electromagnetic spectrum, such as radio , x-ray or gamma-rays . However, optical measurements are the most common and are the source of the term "redshift". The more quickly the source moves away from the observer, the greater the redshift . From an energy standpoint, longer wavelengths correspond to lower energy radiation.

Conversely, when a source of radiation is approaching an observer the wavelengths of light appear closer together, effectively shortening the wavelength of light. (Again, shorter wavelength means higher frequency and therefore higher energy.) Spectroscopically, the emission lines would appear shifted toward the blue side of the optical spectrum, hence the name blueshift .

As with redshift, the effect is applicable to other bands of the electromagnetic spectrum, but the effect is most often times discussed when dealing with optical light, though in some fields of astronomy this is certainly not the case.

Expansion of the Universe and the Doppler Shift

Use of the Doppler Shift has resulted in some important discoveries in astronomy. In the early 1900s, it was believed that the universe was static. In fact, this led Albert Einstein to add the cosmological constant to his famous field equation in order to "cancel out" the expansion (or contraction) that was predicted by his calculation. Specifically, it was once believed that the "edge" of the Milky Way represented the boundary of the static universe.

Then, Edwin Hubble found that the so-called "spiral nebulae" that had plagued astronomy for decades were not nebulae at all. They were actually other galaxies. It was an amazing discovery and told astronomers that the universe  is much larger than they knew.

Hubble then proceeded to measure the Doppler shift, specifically finding the redshift of these galaxies. He found that that the farther away a galaxy is, the more quickly it recedes. This led to the now-famous Hubble's Law , which says that an object's distance is proportional to its speed of recession.

This revelation led Einstein to write that his addition of the cosmological constant to the field equation was the greatest blunder of his career. Interestingly, however, some researchers are now placing the constant back into general relativity .

As it turns out Hubble's Law is only true up to a point since research over the last couple of decades has found that distant galaxies are receding more quickly than predicted. This implies that the expansion of the universe is accelerating. The reason for that is a mystery, and scientists have dubbed the driving force of this acceleration dark energy . They account for it in the Einstein field equation as a cosmological constant (though it is of a different form than Einstein's formulation).

Other Uses in Astronomy

Besides measuring the expansion of the universe, the Doppler effect can be used to model the motion of things much closer to home; namely the dynamics of the Milky Way Galaxy .

By measuring the distance to stars and their redshift or blueshift, astronomers are able to map the motion of our galaxy and get a picture of what our galaxy may look like to an observer from across the universe.

The Doppler Effect also allows scientists to measure the pulsations of variable stars, as well as motions of particles traveling at incredible velocities inside relativistic jet streams emanating from supermassive black holes .

Edited and updated by Carolyn Collins Petersen.

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Doppler’s vision

Petzval’s attack, vogel’s spectrum, voigt’s transformation, doppler spectroscopy, the fall and rise of the doppler effect.

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David D. Nolte; The fall and rise of the Doppler effect. Physics Today 1 March 2020; 73 (3): 30–35. https://doi.org/10.1063/PT.3.4429

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Of all the eponymous discoveries that emerged from 19th-century physics—Young’s fringes, the Biot–Savart law, the Fresnel lens, the Carnot cycle, the Faraday effect, Maxwell’s equations, Michelson’s interferometer, and many more—only one is heard daily on the evening news: the Doppler effect. 1 The effect, which describes the change in a wave’s frequency heard by an observer moving relative to the wave source, is shown in figure 1 . You experience the effect as you wait by the roadside for a train to pass by or a jet to fly overhead. Albert Einstein may have the most famous name in physics, but Christian Doppler’s is probably the most commonly used. That’s ironic because Doppler was hounded by a pompous nemesis, ridiculed for his effect, stripped of his university position, and forced to abandon Vienna in public disgrace and declining health. He finally retreated to Venice and died a few months later.

The Doppler effect in light backscattering is a relativistic effect that involves two different frames. The mirror in the first frame (a) sees a redshifted photon emitted from a receding source. The moving mirror in the second frame (b) re-emits the photon, which is redshifted again relative to the receiver, and produces twice the effect. The source frequency is f 0 , the Doppler frequency shift on backscattering is Δ f Doppler , and the ratio of velocity v to the speed of light is β . (Image by David Nolte.)

Despite Doppler’s ignominious end, today his effect tells scientists of Earth’s motion across the universe, allows physicists to cool atoms in laser traps to a fraction of a degree Kelvin, and is used to detect alien planets orbiting distant stars. With Doppler light scattering, scientists can see the flow of blood in arteries, and they are beginning to personalize chemotherapy by measuring tiny Doppler shifts from the motion of components inside living cells. 2 So why did his peers reject his idea, even years after his death, and how has it been rehabilitated so thoroughly that we now stake our lives on it? The answer begins with a troubled career that almost failed to launch.

Doppler was born in 1803 in Salzburg, Austria, to a long-standing family of stonemasons. By the age of 30, he was at the end of a temporary mathematics assistantship at the Imperial and Royal Polytechnic Institute (now TU Wien) in Vienna and could not find work except as a bookkeeper at a cotton factory. The Austrian empire in the early 19th century was a sprawling bureaucratic state with layers of regulations and armies of able applicants for any position. Doppler was lost in that environment despite his advanced education. His applications for permanent technical posts were denied, and he despaired of ever finding a suitable life, so he decided to emigrate to the US. He sold most of his possessions to pay for his journey and visited the US consulate in Munich to obtain the necessary paperwork. But on his return to Austria, on the eve of leaving Europe for an uncertain future, he received an offer for a teaching position in Prague, which he took in 1835.

He began to publish scholarly papers and in 1837 was appointed supplementary professor of higher mathematics and geometry at the Prague Polytechnical Institute (now Czech Technical University); in 1841 he was promoted to full professor of applied geometry. There he met Bernard Bolzano—a political agitator and a mathematician who developed rigorous concepts of mathematical limits. He is famous today for his part in the Bolzano–Weierstrass theorem in functional analysis. Bolzano presided as chairman over a meeting of the Royal Bohemian Society of Sciences on 25 May 1842, the day Doppler read a landmark paper on the color of stars to a meager assembly of only five regular members of the society.

Doppler had become fascinated by astronomy and by the phenomenon of stellar aberration. It was discovered by James Bradley in 1727 and could be explained by Earth’s motion around the Sun combined with the finite speed of light, which causes the apparent position of a distant star to change slightly through a year. As Doppler studied Bradley’s work, he wondered how Earth’s relative motion would affect the color of the star. By making the simple analogy of a ship traveling with or against a series of ocean waves, he concluded that the frequency of the wave peaks hitting the ship’s bow was no different from the peaks of light waves impinging on the eye. He concluded that the color of light would be shifted slightly to the blue if the eye was approaching towards, and to the red if it was receding from, the light source.

His interest in astronomy had made Doppler familiar with binary stars in which the relative motion of the light source might be large enough to cause color shifts. In fact, the star catalogs included examples of binaries that had complementary red and blue colors. Therefore, his paper, published in the Proceedings of the Royal Bohemian Society of Sciences a few months after he read it to the society, was titled “Über das farbige Licht der Doppelsterne und einiger anderer Gestirne des Himmels” (“On the colored light of the double stars and certain other stars of the heavens”). 3 Although Doppler was mistaken in his assumption that stellar motion would cause a change in the broad-spectrum color of a star, his derivation of frequency shifts was correct. Figure 2 shows Doppler’s own drawings of his effect at high speeds.

These drawings , from Doppler’s 1847 paper, 17 illustrate his effect at high speeds and anticipate the Mach cone, which was predicted and photographed by Ernst Mach 40 years later. They also show shock-wave focusing for sources with angular velocity. (Harvard University Biodiversity Heritage Library, public domain.)

Many who heard of Doppler’s theory did not believe it. Subsequently, on a cold February morning in 1845, Dutch scientist Christoph Buys Ballot, who had recently received his doctorate from the University of Utrecht, loaded an open train car with seasoned musicians and sent them blowing their horns down the railroad line between Utrecht and Maarssen. Buys Ballot did not think that stars would change color by moving, but having no means to test the effect on light, he decided to test it on sound. Unfortunately, the musicians were pelted with hail and snow, which prevented them from blowing their horns properly, so the experiment was reconvened in the milder month of June. That time, with Buys Ballot riding the footplate of the locomotive and the car of trumpeters holding a steady note, musicians standing beside the tracks could hear the approaching note a half-tone higher and the receding note a half-tone lower. The experiment validated Doppler’s theory for sound. 1 Buys Ballot published a paper describing the experiment, 4 but he still refused to acknowledge that light could change color despite the close analogy between sound and light.

Doppler’s prolific scientific output, combined with influential supporters who valued the importance of his work, brought him to the attention of the emperor of Austria, Franz Joseph, newly crowned after his uncle was forced to abdicate during the revolutions of 1848. Reforming education was a top priority for some of the emperor’s advisers, and they persuaded him to found Austria’s first institute of physics and to name Doppler as its first director. Excited by the prospects and full of ideas, Doppler threw himself into his new position. As a member of the Austrian Academy of Sciences, he proposed a prize for the development of photography to advance scientific inquiry. Unfortunately, photographic lenses were the specialty of another member, Joseph Petzval. His supporters in the academy quashed Doppler’s prize proposal, possibly at Petzval’s instigation. 1 More trouble from Petzval awaited Doppler and his effect.

At a meeting of the academy on 22 January 1852, Petzval read a paper criticizing Doppler’s theory. At a later meeting on 21 May 1852, about 60 members and guests assembled to hear both sides of the argument. The large audience for the mock trial of the Doppler effect stands in ironic contrast to the mere five members of the Bohemian Society who first heard Doppler’s ideas 10 years earlier. Petzval’s speech, which was published later, attacked Doppler’s theory for both sound and light. Petzval thought that no great science could come from a few simple lines of algebra: In his view, all natural phenomena were the manifestations of underlying differential equations. From that premise, he proposed a principle for the conservation of oscillation time in undulatory phenomena. Although he was a mathematician of some talent, he was adrift as a natural philosopher. Petzval conflated a source and receiver in relative motion with a stationary source and receiver embedded in a moving medium. He argued that the pure notes of a well-tuned orchestra would be just as harmonious to an audience on a blustery day as on a calm one; the notes would be unaffected by the wind’s motion. Ernst Mach later said that Doppler would agree but quipped that if the orchestra were falling from a great height, the audience would hear the piece in F major rather than E major. 5  

Doppler was nonplussed by Petzval’s attack. His principle already had been verified for acoustic waves by Buys Ballot and by John Scott Russell, a UK railroad and naval engineer who later discovered solitons propagating in a canal. Furthermore, Hippolyte Fizeau in France had proposed the same theory for light in 1848. Unaware of Doppler’s work, he had made the insightful prediction that the effect would be observable in shifts of narrow emission lines from stars in motion rather than from their overall change in color. Fizeau presented his results in a lecture to the Philomatic Society of Paris on 29 December 1848. Hence, the effect is sometimes called the Doppler–Fizeau effect.

Doppler defended himself against Petzval’s onslaught simply by asking his opponent whether an observed phenomenon must be deemed nonexistent if it cannot be derived from differential equations. As reasonable as that argument is, a majority of the academy sided with Petzval, and only a few others, including Andreas von Ettingshausen, put up a defense for Doppler.

The academy’s final decision was scheduled to take place during a meeting on 21 October 1852; once again Petzval was allowed to make his case. Doppler was unable to attend: The stress and disappointment of the Petzval affair had taken its toll on his health, which collapsed after years of battling tuberculosis. When academy members learned that he was making arrangements for a trip to Venice to improve his health, some mistakenly viewed it as a retreat from the fray and a concession of defeat. Members found in favor of Petzval and pronounced that Doppler’s theory must be “abandoned, since it is false, as has been demonstrated.” 1 Ten days later Doppler was officially stripped of his directorship of the Physics Institute of Vienna and replaced by Ettingshausen, but Doppler was already en route to Venice where he would die of his disease only four months later.

That might have been the end of the affair at the Physics Institute of Vienna, but Ettingshausen wasn’t ready to abandon the Doppler effect just because a committee said it didn’t exist. Several years later, he suggested to his student Ernst Mach that he construct a laboratory apparatus to directly demonstrate the acoustic Doppler effect. Mach built and tested a rotating-reed system with tubing that delivered air to the reed, causing it to vibrate at its natural frequency while directing its sound to a stationary observer. As the device spun, the reed approached and receded from the observer, who could hear the rapidly rising and falling tones. 6 Petzval continued denying the effect and accused Mach of youthful foolishness and of throwing away his chances at a career by pursuing an abandoned theory. 1 In response, Mach devised an even more ingenious apparatus, which allows one to listen in one direction to the rising and falling tones and in an orthogonal direction, in which the reed and observer are relatively stationary, to a constant pitch. That arrangement demonstrates even Petzval’s preferred principle of frequency conservation. Despite such demonstrations, Petzval was never satisfied, and over the succeeding years Mach had to contend with persistent confusion and disbelief by many others until he finally refused to discuss the effect further.

Although experimental support for the acoustic Doppler effect accumulated steadily, corresponding demonstrations of the optical Doppler effect were slow to emerge. The breakthrough came in 1868 from William Huggins. He was an early pioneer in astronomical spectroscopy and was famous for discovering that some bright nebulae—planetary nebulae in our own galaxy—consist of atomic gases whereas others consist of unresolved emitting stars. Huggins corresponded with James Clerk Maxwell to confirm the soundness of Doppler’s arguments, which Maxwell corroborated using his new electromagnetic theory. In May 1868 Huggins read a paper to the Royal Society of London reporting on observed shifts in the star’s spectral lines. 7  

The importance of Huggins’s report on the Doppler effect from Sirius was more psychologically important than scientifically accurate because it convinced the scientific community that the optical Doppler effect existed. Only one year later, Joseph Norman Lockyer, codiscoverer of helium, observed a shift in the spectral lines of solar prominences—the high-speed motion of luminous gases ejected from the Sun. 8 Lockyer didn't mention the associated Doppler effect, and because there was no method to confirm the speed of the prominences, his observations were not a definitive demonstration of the optical Doppler effect.

A German astronomer, Hermann Vogel, began working with a new spectrograph that optically projected the spectrum from one side of the Sun next to a reversed spectrum from a point on the opposite side. That doubled the visible effect of the Doppler shift on sharp spectral lines, and Vogel was able to calculate an equatorial rotation speed of the Sun that closely matched the value obtained from the motion of sunspots. Vogel’s results were published in 1872 as the first conclusive demonstration of the optical Doppler effect. 9  

Vogel was also working to improve the measurements of the radial velocity of stars—the speed along the line of sight—and was acutely aware that the many values quoted by Huggins and others for stellar velocities were nearly the same as the uncertainties in the measurement process. Using the human eye to observe the spectral lines was the chief problem. Astronomers had begun using photographic plates on telescopes, and Vogel adapted that new technology to the radial velocities problem. He installed photographic capabilities in the telescope and spectrograph at the Potsdam Observatory in 1887 and made observations of Doppler line shifts in stars through 1890. Vogel published an initial progress report in 1891, and his definitive paper in 1892 provided the first accurate stellar radial velocities. 10 Fifty years after Doppler read his paper to the Royal Bohemian Society of Sciences, the Doppler effect had become an established workhorse of quantitative astrophysics. Aristarkh Belopolsky, a Russian astronomer, finally achieved a laboratory demonstration of the phenomenon in 1901 by constructing a device with a narrow-linewidth light source and rapidly rotating mirrors. 11  

At the January 1887 meeting of the Royal Society of Science in Göttingen, Germany, Woldemar Voigt delivered a paper in which he derived the longitudinal optical Doppler effect in an incompressible medium. He was responding to results published in 1886 by Albert Michelson and Edward Morley on their measurements of the Fresnel drag coefficient using an improved version of the 1851 Fizeau experiment that propagated light through moving water. Voigt pointed out that the wave equation for light is invariant under his transformations, shown in figure 3 . From a modern vantage point, physicists immediately recognize, to within a scale factor, the Lorentz transformation of relativity theory. Of particular note is the last equation, 12 which introduces a position-dependent time as an observer moves with speed κ relative to the speed of light ω . Therefore, Voigt derived the longitudinal Doppler effect by considering relativistic effects a few months prior to the epic Michelson and Morley experiment of 1887 on ether drift and two years before George Fitzgerald proposed length contraction.

The transformations that keep the wave equation for light invariant were stated in 1887 by Woldemar Voigt. 12 His factor q is the inverse of the Lorentz factor γ = 1 /q . Voigt’s equations are identical to the Lorentz transformation if each equation is divided by q . (Image from ref. 12 .)

Voigt’s derivation takes a classic approach that is still used in today’s textbooks to derive the Doppler effect. Twenty years later, Einstein completed the relativistic description of the Doppler effect by predicting the transverse Doppler effect for a source moving along a line perpendicular to an observer’s line of sight. 13 That effect had not been predicted by either Doppler or Voigt.

Almost two centuries have elapsed since Doppler published his simple idea using the analogy of a ship plowing through a series of ocean waves, and the idea now underlies our most sensitive forms of optical metrology of dynamical systems. Far beyond Doppler weather radar, the effect’s applications extend from the ultrasmall, using Doppler cooling of atoms in the laboratory, to the ultralarge, using Doppler measurements of stellar wobble in the search for exoplanets. Until the launch of the Kepler satellite in 2008, most exoplanets had been discovered by detecting the Doppler shifts caused by small radial velocity variations as a star and an exoplanet orbit the system’s center of mass. Using the Doppler wobble technique they reported in 1995, shown in figure 4 , Michel Mayor and Didier Queloz discovered the first exoplanet, orbiting the star 51 Pegasi. 14 They received the 2019 Nobel Prize in Physics for their work (see Physics Today , December 2019, page 17 ). Radial velocities as small as 3 m/s are resolved if measured over many years.

The orbital motion of the star 51 Pegasi was detected by the Doppler wobbles of the star and its Jupiter-mass planet relative to the system’s center of mass (CM) (a–c) . The velocity modulation is 60 m/s (d) . (Panels a–c by David Nolte; panel d is adapted from ref. 14 .)

On a larger scale, the velocity curves of stars within galaxies, which provide some of the most compelling evidence for the existence of dark matter, are observed by Doppler spectroscopy. The relative velocities of the galaxies themselves, such as the streaming of the Virgo cluster of galaxies toward the Great Attractor, are also determined through the Doppler effect. At the largest scale, the Hubble effect is a cosmological redshift caused by the expansion of space rather than an actual Doppler effect. But the motion of Earth, 370 km/s relative to the local cosmic microwave background (CMB), is observed as the large-scale Doppler dipole anisotropy, as shown in figure 5 . Doppler fluctuations caused by local motions in the early universe contributed to the small-scale CMB anisotropy that helps to determine the early uniformity of mass distributions and the fraction of dark matter in the universe.

Anisotropy in the cosmic microwave background . The Doppler dipole anisotropy (left) is caused by the motion of Earth. The small-angle anisotropy (right), after subtracting the dipole, is caused partly by Doppler scattering of photons in the early universe. (Images courtesy of NASA.)

In the life sciences, the acoustic Doppler effect is used in ultrasound imaging, first demonstrated in the 1960s for blood flow measurement, 15 and is now used routinely for Doppler imaging of internal motions, including the Doppler fetal monitor that detects a newborn’s heartbeat in prenatal care. The optical Doppler effect is a major feature of dynamic light scattering to detect the directed motion of blood in optical tomography. The intracellular motions in living tissues produce Doppler signatures down to 10 mHz for speeds of several nanometers per second. 16 Subtle changes in intracellular speeds may eventually help doctors select the best treatments for cancer patients. Thus Doppler’s eponymous effect has achieved a form of immortality he could never have imagined as he retreated from Vienna on his final journey to Italy, watching St. Stephen’s steeple receding into the distance at a redshift of several MHz, though he could not perceive it.

I thank Olivier Darrigol for his helpful comments during the preparation of this article.

David Nolte is a distinguished professor of physics and astronomy at Purdue University in West Lafayette, Indiana, and is the author of Galileo Unbound: A Path Across Life, the Universe and Everything (2018) on the history of dynamics.

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Title: michelson-morley experiment, doppler effect, aberration of light and the aether concept.

Abstract: After an overview of various citations relevant in the context of photon propagation, the relativistic Doppler effect and the addition theorem of velocities are first derived taking into account momentum and energy conservation. Clocks and the aberration of light are treated next, before the lengths of rods and the Lorentz transformations are discussed. The Michelson-Morley experiment is described at rest and in motion with respect to a preferred aether system, first under the assumption of an operation in vacuum. It is concluded that the aether concept is fully consistent with the formal application of the Special Theory of Relativity (STR). Whether a determination of the speed of the laboratory system relative to the aether is possible, is considered next either for an operation of the experiment in vacuum or in a medium with an index of refraction not equal to one. In both cases, the answer appears to be negative.

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