The Doppler Effect

The Doppler effect is the change in the observed frequency of a wave caused by relative motion between the wave source and the observer.

A familiar example is the sound of an emergency vehicle passing by. As it approaches, the siren sounds higher in pitch. After it passes and moves away, the siren sounds lower.

The siren itself does not suddenly change its emitted frequency. The difference occurs because motion changes how frequently the wavefronts reach the observer.

The basic pattern is:

approaching → higher observed frequency

moving away → lower observed frequency

The Doppler effect occurs with many types of waves, including sound, radio waves, microwaves, radar signals, ultrasound, and light.

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Wave Speed, Frequency, and Wavelength

All waves follow the relationship:

v = fλ

where:

• v = wave speed
• f = frequency
• λ = wavelength

For sound travelling through still air, the wave speed is mainly determined by the properties of the air, including temperature.

A moving sound source does not normally make the sound travel faster through the air in front of it. Instead, the motion changes the spacing between successive wavefronts.

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A Stationary Source

Consider a stationary sound source producing waves at a constant frequency.

Because the source remains in the same position between successive wave emissions, the wavefronts spread outward with equal spacing.

The wavelength is the same in every direction.

A stationary observer therefore detects the same frequency that the source emits:

f_observed = f_source

assuming there is no relative motion.

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A Moving Source

Now suppose the source moves toward an observer.

Between producing one wavefront and the next, the source moves forward.

Therefore, each new wavefront begins slightly closer to the previous wavefront in front of the source.

The waves become compressed.

Therefore:

smaller λ → higher observed f

because the wave speed in the medium remains approximately constant.

Behind the moving source, the opposite occurs.

The wavefronts become more widely spaced:

larger λ → lower observed f

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Approaching Source

When a sound source approaches a stationary observer:

• wavefronts are compressed
• wavelength decreases
• wavefronts arrive more frequently
• observed frequency increases
• observed pitch increases

Therefore:

f_observed > f_source

This explains the higher-pitched sound of an approaching siren.

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Receding Source

When the source moves away:

• wavefronts are spread apart
• wavelength increases
• wavefronts arrive less frequently
• observed frequency decreases
• observed pitch decreases

Therefore:

f_observed < f_source

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Doppler Equation for a Moving Sound Source

For a source moving toward a stationary observer:

f_observed = f_source × v/(v − v_source)

For a source moving away:

f_observed = f_source × v/(v + v_source)

where:

• f_observed = frequency detected by observer
• f_source = emitted frequency
• v = speed of sound
• v_source = speed of source

The signs reflect whether the source is approaching or receding.

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Worked Example: Approaching Source

An ambulance siren produces:

f_source = 700 Hz

The ambulance approaches at:

v_source = 25 m/s

Take the speed of sound as:

v = 340 m/s

Then:

f_observed = 700 × 340/(340 − 25)

f_observed = 700 × 340/315

f_observed ≈ 756 Hz

The observer hears a frequency higher than 700 Hz.

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Worked Example: Receding Source

After the ambulance passes, it moves away at the same speed.

f_observed = 700 × 340/(340 + 25)

f_observed = 700 × 340/365

f_observed ≈ 652 Hz

The observer therefore hears:

approaching: about 756 Hz

actual siren: 700 Hz

receding: about 652 Hz

This produces the familiar sudden change in pitch as the vehicle passes.

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Moving Observer

The Doppler effect also occurs when the observer moves while the source remains stationary.

In this case, the wavelength in the medium is not changed by the observer's motion.

Instead, the observer moves through the existing wavefronts at a different rate.

If the observer moves toward the source:

more wavefronts encountered each second → higher observed frequency

If the observer moves away:

fewer wavefronts encountered each second → lower observed frequency

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Moving Observer Equation

For a stationary source and an observer moving toward it:

f_observed = f_source × (v + v_observer)/v

For an observer moving away:

f_observed = f_source × (v − v_observer)/v

The observer's motion changes the rate at which wavefronts are encountered.

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Moving Source vs Moving Observer

The two situations produce similar frequency changes, but the physical mechanisms are different.

This distinction is especially important for sound, because sound travels through a medium.

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Both Source and Observer Moving

If both move, both effects can be included.

A useful general form for sound is:

f_observed = f_source × (v ± v_observer)/(v ∓ v_source)

The signs depend on the directions of motion.

A reliable way to choose them is to think physically:

• motion toward each other should increase observed frequency
• motion apart should decrease observed frequency

It is better to understand the physical situation than simply memorize signs.

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Worked Example: Moving Source and Observer

A horn emits:

f_source = 500 Hz

An observer moves toward the source at:

v_observer = 10 m/s

The source also moves toward the observer at:

v_source = 20 m/s

Using:

v = 340 m/s

f_observed = 500 × (340 + 10)/(340 − 20)

f_observed = 500 × 350/320

f_observed ≈ 547 Hz

Because both motions bring the source and observer together, the observed frequency is significantly higher.

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Frequency Is Not Loudness

A Doppler shift changes observed frequency.

It does not directly mean the sound has become more intense.

As an approaching vehicle gets closer, its sound may also become louder, but this is mainly because the distance to the source decreases.

Therefore:

frequency → pitch

intensity → loudness

These effects should not be confused.

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Frequency Is Not Wave Speed

Another common misconception is that the approaching waves travel faster.

For sound in the same stationary medium:

wave speed remains approximately constant

The moving source changes the wavelength, not the propagation speed.

Since:

v = fλ

a shorter wavelength corresponds to a higher frequency detected by the stationary observer.

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Doppler Shift

The difference between the observed and emitted frequency is called the Doppler shift.

It can be written:

Δf = f_observed − f_source

For an approaching source:

Δf > 0

For a receding source:

Δf < 0

Sometimes only the magnitude of the shift is required:

|Δf| = |f_observed − f_source|

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Radar

Radar uses electromagnetic waves, often in the radio or microwave region, to detect objects and measure properties such as:

• distance
• direction
• speed
• motion

A radar system transmits an electromagnetic signal toward an object.

The signal reflects from the object and returns to the detector.

If the object is moving, the reflected signal can have a different frequency.

This Doppler shift can be used to determine the object's radial velocity.

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Doppler Radar

Consider a vehicle moving toward a radar system.

The returning signal experiences a frequency shift.

The radar electronics compare:

transmitted frequency

with:

received frequency

The difference provides information about the vehicle's motion.

A larger Doppler shift generally corresponds to a greater component of velocity toward or away from the radar.

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Radial Velocity

The Doppler effect measures motion along the line between the detector and the object.

This is called radial velocity.

An object moving directly toward the detector produces a strong Doppler shift for its speed.

An object moving perpendicular to the detector's line of sight may have little or no radial Doppler shift at that instant.

Therefore, Doppler systems do not automatically measure the object's complete velocity vector.

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Doppler Radar Approximation

For reflected electromagnetic waves when the target speed is much smaller than the speed of light, the magnitude of the Doppler shift is approximately:

|Δf| ≈ 2f₀v_r/c

where:

• Δf = Doppler frequency shift
• f₀ = transmitted frequency
• v_r = radial speed
• c = speed of light

The factor of 2 occurs because the wave experiences Doppler effects on the trip to the moving target and again when the reflected wave returns.

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Applications of Doppler Radar

Doppler radar is used in:

• vehicle speed measurement
• weather monitoring
• aviation
• navigation
• motion sensors
• sports measurement
• satellite tracking

Its ability to detect motion without direct physical contact makes it extremely useful.

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Weather Radar

Weather radar can detect motion within storms.

Radar signals reflect from:

• raindrops
• snow
• ice particles
• other precipitation

Doppler shifts reveal motion toward or away from the radar.

Meteorologists can use this information to study:

• wind motion
• storm rotation
• precipitation movement
• severe weather development

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Sonar

Sonar uses sound waves rather than electromagnetic waves.

A sonar system can transmit sound through water and detect reflected echoes.

Sonar can be used to determine:

• distance
• depth
• position
• motion
• speed

If the reflecting object moves relative to the sonar system, the reflected sound experiences a Doppler shift.

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Doppler Sonar

Suppose a submarine sends a sound wave toward another moving object.

The reflected frequency can differ from the transmitted frequency.

By analyzing this shift, the system can estimate the object's radial motion.

Applications include:

• underwater navigation
• submarine detection
• ocean-current measurements
• ship-speed measurements
• underwater research

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Radar vs Sonar

Both can use the Doppler effect to obtain information about motion.

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Doppler Ultrasound

The Doppler effect has important medical applications.

Doppler ultrasound sends high-frequency sound waves into the body.

The waves can reflect from moving structures, especially blood cells.

Motion of the blood produces a Doppler shift in the reflected ultrasound.

Doctors can use this shift to estimate blood-flow velocity.

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Measuring Blood Flow

Consider ultrasound reflecting from blood moving toward the probe.

The reflected frequency is shifted upward.

Blood moving away produces a lower returned frequency.

By analyzing the frequency shift, medical equipment can determine information about:

• direction of blood flow
• speed of blood flow
• patterns of circulation

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Doppler Ultrasound Equation

A commonly used simplified relationship is:

Δf = 2f₀v cos θ / c

where:

• Δf = measured frequency shift
• f₀ = transmitted ultrasound frequency
• v = blood-flow speed
• θ = angle between the ultrasound beam and blood-flow direction
• c = speed of sound in the tissue

The angle is important.

If the beam is parallel to the flow:

θ = 0°

and:

cos 0° = 1

The Doppler shift is greatest.

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Why the Ultrasound Angle Matters

If the ultrasound beam is perpendicular to blood flow:

θ = 90°

Since:

cos 90° = 0

the ideal Doppler shift is:

Δf = 0

even though the blood is moving.

Therefore, Doppler measurements depend on the component of velocity along the ultrasound beam.

This is another example of radial motion being important.

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Medical Applications

Doppler ultrasound can help investigate:

• blood flow through arteries
• blood flow through veins
• heart function
• circulation
• fetal blood flow
• narrowing of blood vessels

It provides information without requiring direct measurement inside the blood vessel.

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Doppler Effect and Light

The Doppler effect also applies to electromagnetic waves such as light.

When a light source and observer move relative to each other, the observed frequency and wavelength can change.

For visible light:

approaching source → higher observed frequency → shorter wavelength

receding source → lower observed frequency → longer wavelength

These changes are called blueshift and redshift.

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Blueshift

A source moving toward the observer produces a shift toward shorter wavelengths.

This is called blueshift.

It does not necessarily mean the object literally appears blue.

Instead, its entire spectrum is shifted toward shorter wavelengths.

Therefore:

approaching → shorter λ → higher f → blueshift

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Redshift

A source moving away produces a shift toward longer wavelengths.

This is called redshift.

Again, the object does not necessarily appear red to the eye.

Its spectral features move toward longer wavelengths.

Therefore:

receding → longer λ → lower f → redshift

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Spectral Lines

Stars and galaxies produce spectra containing identifiable spectral lines.

Laboratory measurements tell scientists the normal wavelengths of these lines.

Astronomers compare:

rest wavelength

with:

observed wavelength

If the observed lines shift toward longer wavelengths:

redshift

If they shift toward shorter wavelengths:

blueshift

The shift can reveal motion along our line of sight.

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Measuring Wavelength Shift

The wavelength shift can be written:

Δλ = λ_observed − λ_rest

For relatively low speeds compared with the speed of light:

v_r/c ≈ Δλ/λ_rest

where:

• v_r = radial velocity
• c = speed of light
• Δλ = wavelength shift
• λ_rest = original wavelength

This approximation is useful when:

v << c

At speeds approaching the speed of light, relativistic Doppler equations are required.

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Worked Example: Stellar Redshift

A spectral line normally occurs at:

λ_rest = 500 nm

It is observed at:

λ_observed = 500.5 nm

Shift:

Δλ = 500.5 − 500

Δλ = 0.5 nm

Using:

v_r/c ≈ Δλ/λ_rest

v_r/c ≈ 0.5/500

v_r/c ≈ 0.001

Using:

c = 3.0 × 10⁸ m/s

v_r ≈ 0.001(3.0 × 10⁸)

v_r ≈ 3.0 × 10⁵ m/s

or:

v_r ≈ 300 km/s

Because the wavelength increased, the source is receding.

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Stars and the Doppler Effect

Astronomers use Doppler shifts to study stellar motion.

They can determine whether a star is moving:

• toward Earth
• away from Earth

They can also measure how quickly the star is moving along our line of sight.

This information helps astronomers investigate:

• binary stars
• stellar motion
• galactic rotation
• exoplanets

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Detecting Exoplanets

A planet and its star both orbit their common center of mass.

Although the star moves much less than the planet, it can undergo a small periodic motion.

As the star moves:

toward Earth → slight blueshift

then:

away from Earth → slight redshift

Repeated shifts in the star's spectrum can provide evidence for an orbiting planet.

This technique is called the radial velocity method.

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Binary Stars

Two stars orbiting each other can also produce periodic Doppler shifts.

As one star moves toward Earth, its spectral lines become blueshifted.

As it moves away, they become redshifted.

These patterns can reveal:

• orbital speeds
• orbital periods
• properties of the stellar system

The stars may therefore reveal their motion even when their orbital movement cannot be directly seen.

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Galaxies and Redshift

Many distant galaxies show redshifted spectra.

Their spectral features appear at wavelengths longer than their laboratory values.

This provides evidence that many galaxies are receding from us.

Observations of galactic redshift played a major role in developing our understanding of the expanding universe.

For very distant galaxies, however, cosmological redshift is best understood through the expansion of space, rather than simply treating it as an ordinary sound-like Doppler effect.

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Doppler Shift vs Cosmological Redshift

For nearby objects moving through space, Doppler shift provides a useful description of wavelength changes caused by relative motion.

For very distant galaxies, expansion of the universe stretches the wavelength of travelling light.

Therefore, although both effects can produce redshift:

Doppler redshift → relative motion

cosmological redshift → expansion of space during light travel

At introductory levels they are often discussed together, but the distinction is important.

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Navigation

Doppler measurements can contribute to navigation systems.

If a vehicle measures frequency shifts from known signals or reflected waves, it can obtain information about relative motion.

Applications can include:

• aircraft navigation
• spacecraft navigation
• marine navigation
• satellite tracking
• autonomous systems

Doppler measurements are especially useful because they provide information about velocity, not just position.

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Satellites

Radio signals transmitted between satellites and ground stations can experience Doppler shifts.

As a satellite approaches a receiver:

observed frequency increases

As it moves away:

observed frequency decreases

Tracking these changes can provide information about:

• orbital motion
• relative velocity
• satellite position and trajectory

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Spacecraft Navigation

Space agencies can monitor radio signals exchanged with spacecraft.

Precise measurements of Doppler shift can reveal extremely small changes in radial velocity.

This helps engineers determine:

• spacecraft speed
• trajectory
• orbital changes
• effects of gravitational encounters

Doppler measurements are therefore important far beyond everyday sound.

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Motion Sensors

Some motion sensors transmit electromagnetic waves and analyze returning signals.

If an object moves, the reflected signal can experience a Doppler shift.

This principle can be used in:

• automatic doors
• security systems
• occupancy sensors
• traffic monitoring
• industrial automation

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Sports Technology

Radar systems can use Doppler shifts to measure the speeds of moving objects such as:

• baseballs
• tennis serves
• golf balls
• vehicles
• other sports equipment

The measurement can be made without attaching a sensor directly to the moving object.

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Speed Cameras

Some traffic-speed systems use radar.

A signal is transmitted toward a moving vehicle.

The reflected signal returns with a Doppler shift.

The system uses this frequency difference to determine the vehicle's radial speed.

The underlying sequence is:

transmitted wave → reflection from moving vehicle → frequency shift → speed calculation

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Sonic Booms Are Different

The Doppler effect should not be confused with a sonic boom.

As a source approaches the speed of sound, wavefronts become strongly compressed.

If the source moves faster than sound, it produces a shock-wave pattern.

A sonic boom results from these shock waves.

The Doppler effect occurs at ordinary subsonic speeds as well and does not require the source to exceed the speed of sound.

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Mach Number

The speed of an object relative to the speed of sound can be described using the Mach number:

Mach number = object speed / speed of sound

For example, if:

v_object = 170 m/s

and:

v_sound = 340 m/s

then:

Mach = 170/340

Mach = 0.50

An object travelling at Mach 1 is travelling approximately at the local speed of sound.

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Everyday Examples

Doppler shifts can be observed or applied in:

• ambulance sirens
• train horns
• race cars
• aircraft
• radar
• sonar
• weather monitoring
• ultrasound
• astronomy
• satellite tracking
• speed measurement

The same wave principle connects an enormous range of technologies.

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A Complete Sound Problem

A train horn produces:

f_source = 600 Hz

The train approaches a stationary observer at:

v_source = 30 m/s

Take:

v = 340 m/s

Approaching frequency:

f_observed = 600 × 340/(340 − 30)

f_observed = 600 × 340/310

f_observed ≈ 658 Hz

After passing:

f_observed = 600 × 340/(340 + 30)

f_observed = 600 × 340/370

f_observed ≈ 551 Hz

The observer therefore hears a clear decrease in pitch as the train passes.

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A Complete Astronomy Problem

A spectral line has a laboratory wavelength of:

λ_rest = 600 nm

A star's spectrum shows the line at:

λ_observed = 599.4 nm

Change:

Δλ = 599.4 − 600

Δλ = −0.6 nm

Because the wavelength decreased, this is a blueshift.

Using the magnitude:

|v_r|/c ≈ 0.6/600

|v_r|/c ≈ 0.001

Therefore:

|v_r| ≈ 0.001(3.0 × 10⁸)

|v_r| ≈ 3.0 × 10⁵ m/s

or:

300 km/s

The negative wavelength shift indicates that the star is approaching.

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Choosing the Correct Doppler Relationship

Before calculating, identify the situation.

Ask:

What type of wave is involved?

Sound and electromagnetic waves require somewhat different treatments.

What is moving?

Is it:

• the source
• the observer
• both
• a reflecting target?

Are they approaching or separating?

Approaching should increase observed frequency.

Separating should decrease observed frequency.

Is only radial motion relevant?

Many Doppler technologies measure the velocity component along the line of sight.

Checking these points helps prevent sign errors.

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Common Mistakes

Thinking an approaching sound source makes sound travel faster

The speed of sound through the same stationary medium remains approximately unchanged. The wavelength changes.

Confusing frequency with loudness

Frequency affects pitch. Intensity is more closely related to loudness.

Thinking the source's actual emitted frequency changes

The Doppler effect concerns the frequency observed because of relative motion.

Treating a moving source and moving observer as physically identical

A moving sound source changes wavelength in the medium. A moving observer changes how quickly existing wavefronts are encountered.

Assuming all motion produces the same Doppler shift

Only the velocity component along the line of sight directly contributes to the radial Doppler shift.

Thinking redshift means an object actually turns red

Its spectral wavelengths shift toward longer wavelengths.

Thinking blueshift always means blue visible light

It means wavelengths shift toward shorter values.

Treating every astronomical redshift as a simple Doppler shift

Large-scale cosmological redshift involves expansion of space.

Forgetting the factor of 2 in simple reflected-wave Doppler measurements

Radar and Doppler ultrasound can involve a wave travelling to a moving target and then returning.

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Key Terms

Doppler effect — change in observed frequency caused by relative motion between a source and an observer.

Doppler shift — difference between observed and emitted frequency.

Observed frequency — frequency measured by an observer.

Source frequency — frequency produced by the wave source.

Radial velocity — component of velocity toward or away from an observer.

Wavefront — surface connecting points of a wave in the same phase.

Redshift — shift of electromagnetic radiation toward longer wavelengths.

Blueshift — shift toward shorter wavelengths.

Spectral line — characteristic wavelength in a spectrum associated with particular atomic or molecular processes.

Radar — system using electromagnetic waves to detect and measure objects.

Doppler radar — radar that uses frequency shifts to determine motion.

Sonar — system using sound waves to detect or investigate objects, commonly underwater.

Doppler ultrasound — medical technique using frequency shifts in reflected ultrasound to investigate motion such as blood flow.

Radial velocity method — astronomical method that detects periodic Doppler shifts in a star's spectrum, often used to identify exoplanets.

Cosmological redshift — increase in wavelength associated with the expansion of the universe.

Mach number — ratio of an object's speed to the local speed of sound.

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Key Takeaways

• The Doppler effect is a change in observed frequency caused by relative motion.
• Approaching motion generally produces a higher observed frequency.
• Receding motion generally produces a lower observed frequency.
• For a moving sound source, wavefronts are compressed ahead of the source and spread out behind it.
• A moving sound source changes the wavelength in the medium.
• A moving observer does not change the wavelength already travelling through the medium but encounters wavefronts at a different rate.
• The speed of sound is determined mainly by the medium rather than the speed of the source.
• Doppler shift measures the component of motion along the line of sight.
• Radar uses electromagnetic waves and Doppler shifts to measure motion.
• Sonar can use sound-wave Doppler shifts to investigate underwater motion.
• Doppler weather radar measures motion of precipitation and can help reveal storm movement.
• Doppler ultrasound uses frequency shifts to investigate blood flow.
• Redshift corresponds to increasing wavelength and usually indicates recession in a simple Doppler interpretation.
• Blueshift corresponds to decreasing wavelength and usually indicates approach.
• Astronomers use spectral shifts to measure stellar and galactic motion.
• Periodic Doppler shifts in a star can provide evidence for an orbiting exoplanet.
• Cosmological redshift must be distinguished from a simple local Doppler shift.
• Doppler measurements are used in navigation, satellites, spacecraft, medicine, astronomy, transportation, weather forecasting, and sports.
• Frequency, wavelength, wave speed, and relative motion must be carefully distinguished when analyzing Doppler problems.
• The Doppler effect demonstrates how one fundamental wave principle can be applied across sound, medicine, transportation, astronomy, and modern technology.

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Check Your Understanding

1. Define the Doppler effect.
2. Explain why an approaching ambulance sounds higher in pitch than the same ambulance moving away.
3. Describe what happens to the wavelength in front of and behind a moving sound source.
4. Explain the difference between the Doppler effect produced by a moving sound source and that produced by a moving observer.
5. A siren emits 800 Hz while approaching a stationary observer at 20 m/s. Using a sound speed of 340 m/s, calculate the observed frequency.
6. The same 800 Hz siren moves away at 20 m/s. Calculate the observed frequency.
7. Explain why the speed of the source should not simply be added to the speed of sound when analyzing an approaching sound source.
8. What is radial velocity, and why is it important in radar and astronomical Doppler measurements?
9. Explain how Doppler radar can determine the speed of a moving vehicle.
10. Compare the use of the Doppler effect in radar and sonar.
11. Explain how Doppler ultrasound can be used to determine the direction and speed of blood flow.
12. A spectral line normally occurs at 500 nm but is observed at 501 nm. Identify whether this is redshift or blueshift and calculate the approximate radial speed using c = 3.0 × 10⁸ m/s.
13. Explain how periodic redshifts and blueshifts in a star's spectrum can provide evidence for an orbiting planet.
14. Distinguish between ordinary Doppler redshift and cosmological redshift.
15. Choose two applications from astronomy, medicine, navigation, weather monitoring, traffic measurement, or space exploration. Explain how each uses the Doppler effect and identify what information can be obtained from the measured frequency or wavelength shift.