Moving Sources and Observers

The Doppler effect occurs when there is relative motion between a wave source and an observer. However, a moving source and a moving observer do not affect the wave in exactly the same way.

The key distinction is:

• A moving source changes the spacing of the wavefronts in the medium.
• A moving observer changes the rate at which the observer encounters existing wavefronts.

Both situations can change the observed frequency, but the physical reason for the change is different.

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Source and Observer

The source is the object producing the waves.

Examples include:

• a loudspeaker
• an ambulance siren
• a train whistle
• a vibrating tuning fork
• a star producing light

The observer is the person or detector receiving the waves.

Examples include:

• a person
• a microphone
• an ultrasound detector
• a radar receiver
• a telescope

Either the source, the observer, or both may be moving.

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Relative Motion

The most important first question is:

Is the source-observer distance decreasing or increasing?

If they are approaching each other:

observed frequency increases

If they are moving apart:

observed frequency decreases

This rule works whether the source moves, the observer moves, or both move.

However, the wave pattern itself is different in each situation.

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Four Basic Doppler Situations

The simplest situations are:

A useful shortcut is:

toward → higher frequency

away → lower frequency

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A Stationary Source and Stationary Observer

Begin with the simplest situation.

Suppose a sound source and observer are both stationary relative to the air.

The source produces:

fₛ = 500 Hz

The observer receives:

fₒ = 500 Hz

The wavefronts are evenly spaced around the stationary source.

There is no Doppler shift.

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Moving Source

Now suppose the observer remains stationary while the source moves.

A moving source changes where each successive wavefront is produced.

This changes the spacing of the wavefronts.

Ahead of the source:

wavefronts are compressed

Behind the source:

wavefronts are stretched apart

This is the characteristic wave pattern of a moving source.

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Why a Moving Source Compresses Wavefronts

Imagine a siren moving toward an observer.

The siren emits one wave crest.

Before it emits the next crest, the siren moves forward.

The second crest therefore begins closer to the first crest in the forward direction.

The next crest begins even farther forward.

The result is:

smaller spacing between wavefronts

Therefore:

shorter wavelength

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Moving Source Toward an Observer

For a source moving toward a stationary observer:

wavefronts compress

→ wavelength decreases

→ wavefronts arrive more frequently

→ observed frequency increases

For sound:

higher frequency → higher perceived pitch

Therefore:

approaching moving source → higher observed pitch

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Moving Source Away from an Observer

Behind a moving source, each new wavefront begins farther from the previous one.

Therefore:

wavefronts spread apart

→ wavelength increases

→ wavefronts arrive less frequently

→ observed frequency decreases

For sound:

lower frequency → lower perceived pitch

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The Wave Speed Does Not Change

Suppose an ambulance moves toward you.

It may seem reasonable to think:

"Since the ambulance is moving forward, the sound must travel faster."

That is not what happens.

In still air, the speed of sound is determined mainly by the properties of the air.

The ambulance changes the:

wavelength

not the normal propagation speed of sound through the air.

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Moving Source and Wavelength

The wave equation is:

v = fλ

The source continues producing waves at its emitted frequency.

But because the source changes position between successive emissions, the wavelengths in front of and behind it are different.

Ahead:

shorter λ

Behind:

longer λ

This altered wavefront spacing is one of the easiest ways to recognize a moving source in a Doppler diagram.

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Example: Moving Ambulance

Suppose an ambulance travels to the right.

An observer stands ahead of it.

The wavefronts between the ambulance and observer are compressed.

Therefore, the observer detects:

higher frequency

Another observer stands behind the ambulance.

The wavefronts reaching this observer are stretched apart.

Therefore, this observer detects:

lower frequency

The same source can therefore produce different observed frequencies at different locations.

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Moving Observer

Now consider a different situation.

The source remains stationary, but the observer moves.

The stationary source produces evenly spaced wavefronts.

Therefore, unlike the moving-source situation:

the wavelength in the medium is not changed by the observer's motion.

Instead, the observer changes how rapidly they encounter the wavefronts.

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Observer Moving Toward the Source

Suppose an observer moves toward a stationary sound source.

The observer moves into the approaching wavefronts.

As a result, the observer encounters more wavefronts each second.

Therefore:

observed frequency increases

The wavelength in the air has not changed.

The observer simply encounters the wavefronts at a greater rate.

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Observer Moving Away from the Source

Suppose the observer moves away from a stationary source.

The observer moves in the same general direction as the wavefronts.

The wavefronts take longer to catch the observer.

Therefore, fewer wavefronts reach the observer each second.

The result is:

observed frequency decreases

Again, the wavelength already present in the medium remains unchanged.

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The Critical Difference

This distinction is extremely important.

Moving source:

changes wavefront spacing.

Moving observer:

does not change the wavefront spacing in the medium; instead, it changes how quickly the observer encounters the wavefronts.

Both can change observed frequency.

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Comparing Moving Sources and Moving Observers

The final frequency trend is similar, but the physical mechanism is different.

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Interpreting a Moving-Source Diagram

A moving-source diagram usually shows wavefronts that are:

• crowded together on one side
• spread apart on the opposite side

The source is moving toward the:

compressed wavefronts

and away from the:

stretched wavefronts

Therefore:

compressed side → higher observed frequency

stretched side → lower observed frequency

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Finding the Direction of Source Motion

Suppose a diagram does not show a velocity arrow.

You can still determine the direction of motion.

Look for the side where the wavefronts are:

closest together

The source is moving toward that side.

For example:

wavefronts compressed on right → source moving right

wavefronts compressed on left → source moving left

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Interpreting a Moving-Observer Diagram

For a stationary source, the wavefronts should remain evenly spaced.

Look at the observer's motion arrow.

If the observer moves toward the source:

frequency increases

If the observer moves away:

frequency decreases

Do not redraw the wavefronts as compressed simply because the observer is moving.

The observer's motion does not alter the waves already travelling through the medium.

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Compression and Stretching of Wavefronts

The terms compression and stretching in a Doppler diagram describe the spacing between successive wavefronts.

Compressed wavefronts mean:

shorter wavelength

Stretched wavefronts mean:

longer wavelength

For a moving source:

front → compressed

behind → stretched

provided the source is moving slower than the wave speed.

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Wavefront Spacing and Frequency

For a stationary observer in a stationary medium:

v = fλ

If the wave speed remains constant:

shorter λ → higher observed f

longer λ → lower observed f

Therefore:

compressed wavefronts → higher frequency

stretched wavefronts → lower frequency

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Worked Example: Reading a Diagram

A sound source moves to the right.

The wavelength in front is:

λ = 0.60 m

The speed of sound is:

v = 340 m/s

The frequency detected by an observer ahead is:

f = v/λ

f = 340/0.60

f ≈ 567 Hz

Behind the source, suppose:

λ = 0.80 m

Then:

f = 340/0.80

f = 425 Hz

The observer ahead detects a much higher frequency than the observer behind.

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Doppler Equation for a Moving Source

For a stationary observer and a source moving directly toward or away:

fₒ = fₛv/(v ∓ vₛ)

where:

fₒ = observed frequency

fₛ = emitted frequency

v = wave speed

vₛ = source speed

For an approaching source:

fₒ = fₛv/(v − vₛ)

For a receding source:

fₒ = fₛv/(v + vₛ)

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Why the Signs Work for a Moving Source

When the source approaches:

denominator = v − vₛ

The denominator becomes smaller.

Therefore:

fₒ becomes larger

When the source recedes:

denominator = v + vₛ

The denominator becomes larger.

Therefore:

fₒ becomes smaller

This matches our physical prediction.

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Worked Example: Moving Source Toward Observer

A siren emits:

fₛ = 600 Hz

The source moves toward a stationary observer at:

vₛ = 25 m/s

Assume:

v = 340 m/s

Use:

fₒ = fₛv/(v − vₛ)

fₒ = 600 × 340/(340 − 25)

fₒ = 204000/315

fₒ ≈ 648 Hz

The observed frequency is higher than 600 Hz.

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Worked Example: Moving Source Away

The same source moves away at:

25 m/s.

Use:

fₒ = fₛv/(v + vₛ)

fₒ = 600 × 340/(340 + 25)

fₒ = 204000/365

fₒ ≈ 559 Hz

Therefore:

approaching → approximately 648 Hz

emitted → 600 Hz

receding → approximately 559 Hz

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Doppler Equation for a Moving Observer

For a moving observer and stationary source:

fₒ = fₛ(v ± vₒ)/v

where:

vₒ = observer speed

For an observer moving toward the source:

fₒ = fₛ(v + vₒ)/v

For an observer moving away:

fₒ = fₛ(v − vₒ)/v

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Worked Example: Observer Moving Toward Source

A stationary source produces:

fₛ = 600 Hz

An observer approaches at:

vₒ = 25 m/s

Use:

fₒ = 600(340 + 25)/340

fₒ = 600 × 365/340

fₒ ≈ 644 Hz

The observed frequency increases.

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Worked Example: Observer Moving Away

The observer now moves away at:

25 m/s.

fₒ = 600(340 − 25)/340

fₒ = 600 × 315/340

fₒ ≈ 556 Hz

The observed frequency decreases.

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Same Speed Does Not Give Exactly the Same Shift

Compare the previous examples.

Source approaching at 25 m/s:

fₒ ≈ 648 Hz

Observer approaching at 25 m/s:

fₒ ≈ 644 Hz

These are similar but not exactly equal.

Why?

Because moving sources and moving observers affect sound differently.

A moving source changes:

wavelength

A moving observer changes:

wavefront encounter rate

This is why the equations are different.

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Both Source and Observer Moving

Sometimes both objects move.

For sound in a stationary medium, a commonly used equation is:

fₒ = fₛ(v ± vₒ)/(v ∓ vₛ)

The signs must be chosen according to whether the motions bring the source and observer closer together or farther apart.

A useful physical check is always:

approaching → fₒ should increase

receding → fₒ should decrease

If your calculation gives the opposite result, check the signs.

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Worked Example: Both Approaching

A source emits:

fₛ = 500 Hz

The source moves toward the observer at:

vₛ = 20 m/s

The observer moves toward the source at:

vₒ = 10 m/s

Assume:

v = 340 m/s

Use:

fₒ = fₛ(v + vₒ)/(v − vₛ)

fₒ = 500(340 + 10)/(340 − 20)

fₒ = 500 × 350/320

fₒ ≈ 547 Hz

Since they are approaching:

fₒ > fₛ

as expected.

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Both Moving in the Same Direction

Suppose a source and observer are both moving to the right.

You cannot determine the Doppler shift simply by seeing that they move in the same direction.

You must compare their speeds.

For example:

source behind moving faster than observer → separation decreases

Therefore:

approaching

source behind moving slower than observer → separation increases

Therefore:

receding

Always determine whether their distance is changing.

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Example: Catching Up

A car producing a horn travels at:

30 m/s.

An observer in another vehicle ahead travels in the same direction at:

20 m/s.

The source is gaining on the observer.

Their separation is decreasing.

Therefore, they are approaching relative to one another.

The observer detects a frequency:

higher than the emitted frequency

for the usual sound Doppler situation.

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Example: Pulling Away

Now suppose the observer ahead travels at:

35 m/s

while the source behind travels at:

25 m/s.

The distance between them increases.

Therefore:

they are receding

The observed frequency is:

lower than the emitted frequency

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Sideways Motion

Suppose a source moves across an observer's field of view rather than directly toward or away.

The Doppler shift depends on the component of velocity along the line connecting source and observer.

This is the:

radial component of velocity

Motion directly toward or away gives the largest ordinary Doppler effect for a given speed.

Motion perpendicular to the line of sight has no instantaneous classical radial Doppler shift at the point of closest approach.

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A Vehicle Passing an Observer

Consider a car approaching a person standing beside a road.

Before the car passes:

the car has a velocity component toward the observer.

Therefore:

fₒ > fₛ

After the car passes:

the car has a velocity component away from the observer.

Therefore:

fₒ < fₛ

Near the moment of closest approach, the radial component changes sign.

This produces the familiar change from higher to lower pitch.

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Comparing Doppler Situations

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Moving Source vs Moving Observer Diagrams

A useful way to distinguish the diagrams is to inspect the wavefronts.

Moving source

Wavefront spacing is uneven.

There are compressed wavefronts ahead and stretched wavefronts behind.

Moving observer

If the source is stationary, wavefront spacing remains symmetrical and even.

The observer simply moves through those wavefronts.

This visual distinction is extremely useful in examination questions.

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Interpreting Arrows

Doppler diagrams may contain several types of arrows.

A source arrow shows:

source velocity

An observer arrow shows:

observer velocity

Wave propagation arrows show:

direction of wave travel

Do not confuse the motion of the source or observer with the motion of the waves.

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Source Motion and Wave Motion Are Different

Suppose a source moves right while producing sound.

The sound itself spreads outward in many directions through the medium.

Some waves travel:

• ahead of the source
• behind the source
• sideways from the source

The source motion changes where successive wavefronts originate.

It does not mean every sound wave simply moves in the same direction as the source.

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Everyday Example: Train and Passenger

Imagine a stationary train whistle beside a platform while a passenger moves toward it on another train.

The whistle produces evenly spaced wavefronts.

The moving passenger encounters these wavefronts more frequently.

This is a:

moving-observer Doppler effect

Now imagine the passenger standing still while the whistle itself approaches on a moving train.

The wavefronts are compressed ahead of the train.

This is a:

moving-source Doppler effect

Both produce higher observed frequency, but for different reasons.

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Doppler Ultrasound

Moving-source and moving-observer ideas are also useful when thinking about medical Doppler ultrasound.

Ultrasound is transmitted toward moving blood cells.

The blood cells interact with and reflect the waves.

Their motion causes frequency shifts in the returning ultrasound.

The equipment analyzes these shifts to determine information about:

• direction of blood flow
• relative speed of blood flow

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Radar

Doppler radar sends electromagnetic waves toward moving targets.

The waves reflect from the target.

The frequency of the returning wave differs from the transmitted frequency when the target has motion toward or away from the radar.

This allows radar to measure:

radial velocity

Applications include:

• vehicle speed detection
• aircraft tracking
• weather radar
• navigation

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Astronomy

In astronomy, the observer is often approximately stationary relative to the measurement system while a star or galaxy has motion toward or away from us.

Motion away produces a shift toward longer wavelengths.

Motion toward produces a shift toward shorter wavelengths.

Astronomers use these shifts to investigate the radial motion of astronomical objects.

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What Happens as Source Speed Increases?

As a sound source moves faster toward an observer:

• wavefront compression increases
• forward wavelength decreases
• observed frequency increases

As its speed approaches the speed of sound, the wavefronts ahead become increasingly compressed.

This leads to a more extreme wave phenomenon.

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Approaching the Speed of Sound

When an object moves through air at approximately the speed of sound, successive pressure disturbances can build up into a strong wavefront.

Instead of ordinary separated wavefronts moving ahead of the source, disturbances accumulate.

This is associated with the formation of a:

shock wave

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Supersonic Motion

If an object travels faster than sound:

v_source > v_sound

it outruns the sound disturbances it produces.

The wavefronts form a cone-shaped pattern called a:

Mach cone

When the shock wave passes an observer, it can be heard as a:

sonic boom

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The Sonic Boom Is Not Just an Extreme Pitch Shift

A sonic boom is associated with a shock wave produced by supersonic motion.

It should not simply be described as an extremely high-pitched Doppler sound.

The physics involves the accumulation and geometry of pressure disturbances into a shock front.

It is related to moving-source wave behaviour but represents a different regime from the ordinary Doppler effect.

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A Complete Moving-Source Example

A source moves toward a stationary observer.

The source emits sound at:

600 Hz.

Because the source moves forward between successive emissions:

• wavefronts ahead become compressed
• forward wavelength decreases
• sound speed through the stationary air remains approximately unchanged
• observed frequency increases

The observer therefore detects:

fₒ > 600 Hz

After the source passes:

• wavefronts reaching the observer are more widely spaced
• wavelength is greater
• observed frequency is lower

Therefore:

fₒ < 600 Hz

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A Complete Moving-Observer Example

A stationary source emits:

600 Hz.

The wavefronts in the air are evenly spaced.

An observer moves toward the source.

The observer encounters more than 600 wavefront cycles per second.

Therefore:

fₒ > 600 Hz

The observer then turns around and moves away.

They encounter fewer wavefronts each second.

Therefore:

fₒ < 600 Hz

Throughout this process, the source remains stationary and the wavelength in the air remains unchanged.

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A Reliable Strategy for Doppler Diagrams

When given a Doppler diagram:

1. Locate the source.
2. Locate the observer.
3. Identify which one is moving.
4. Examine the wavefront spacing.
5. Uneven wavefront spacing suggests a moving source.
6. Compressed wavefronts show the direction toward which the source moves.
7. Even wavefront spacing with a moving observer suggests a stationary source.
8. Determine whether source and observer are approaching or separating.
9. Approaching means higher observed frequency.
10. Separating means lower observed frequency.
11. Distinguish wavelength changes from observer encounter-rate changes.
12. Check whether your explanation matches the diagram.

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A Reliable Strategy for Calculations

For numerical Doppler questions:

1. Write the emitted frequency fₛ.
2. Write the wave speed v.
3. Identify source speed vₛ if present.
4. Identify observer speed vₒ if present.
5. Determine whether each motion is toward or away.
6. Predict whether fₒ should be higher or lower than fₛ.
7. Choose the correct Doppler equation.
8. Substitute values carefully.
9. Calculate fₒ.
10. Compare the result with your prediction.
11. If approaching but fₒ < fₛ, recheck the signs.
12. Include Hz in the final answer.

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Common Mistakes

Thinking a moving source and moving observer affect waves identically

They can produce similar frequency shifts, but the physical mechanisms differ.

Thinking a moving observer compresses the wavefronts

The observer does not change the wavelength already present in the medium.

Thinking a moving source changes the speed of sound

It changes wavefront spacing, not the ordinary sound speed through stationary air.

Looking only at direction arrows

Always determine whether the separation between source and observer increases or decreases.

Assuming two objects moving in the same direction are receding

A faster object behind may be catching the object ahead.

Thinking compressed wavefronts mean lower frequency

For the same wave speed, compressed wavefronts mean shorter wavelength and higher frequency.

Thinking stretched wavefronts mean higher frequency

Greater wavelength corresponds to lower frequency when wave speed is unchanged.

Using the moving-source equation for a moving observer

The equations are different because the physical situations are different.

Assuming equal source and observer speeds produce exactly equal Doppler shifts

For sound in a medium, equal source and observer speeds generally do not produce exactly identical shifts.

Ignoring radial motion

The ordinary Doppler shift depends on motion toward or away along the source-observer line.

Thinking a sonic boom is simply a high-pitched Doppler effect

A sonic boom is produced by a shock wave associated with supersonic motion.

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Key Terms

Source — object or system producing waves.

Observer — person or detector receiving waves.

Relative motion — motion of the source and observer compared with one another.

Doppler effect — observed frequency change caused by relative motion.

Emitted frequency (fₛ) — frequency produced by the source.

Observed frequency (fₒ) — frequency detected by the observer.

Moving source — wave source whose position changes relative to the medium and observer.

Moving observer — detector whose position changes relative to the source and wavefronts.

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

Wavefront compression — reduced spacing between successive wavefronts.

Wavefront stretching — increased spacing between successive wavefronts.

Wavelength (λ) — distance between corresponding points on consecutive waves.

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

Approaching — motion that decreases source-observer separation.

Receding — motion that increases source-observer separation.

Shock wave — strong disturbance formed when wave disturbances accumulate due to sufficiently rapid source motion.

Mach cone — cone-shaped shock-wave pattern associated with supersonic motion.

Sonic boom — sound produced when a shock wave from a supersonic object passes an observer.

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Key Takeaways

• Both moving sources and moving observers can produce the Doppler effect.
• Approaching motion increases observed frequency.
• Receding motion decreases observed frequency.
• A moving source changes the spacing of wavefronts in the medium.
• Wavefronts are compressed ahead of an approaching moving source.
• Wavefronts are stretched behind a moving source.
• Compressed wavefronts have shorter wavelength.
• Stretched wavefronts have longer wavelength.
• A moving observer does not change the wavelength already present in the medium.
• A moving observer changes how rapidly they encounter wavefronts.
• An observer moving toward a source encounters wavefronts more frequently.
• An observer moving away encounters them less frequently.
• Moving-source and moving-observer Doppler shifts are similar but not physically identical.
• Their Doppler equations are therefore different for waves such as sound travelling through a medium.
• The source's motion does not simply increase or decrease the normal speed of sound.
• A Doppler diagram with uneven wavefront spacing usually represents a moving source.
• Compressed wavefronts indicate the direction of source motion.
• For two objects moving in the same direction, their relative speeds determine whether they approach or separate.
• The Doppler effect depends primarily on the radial component of relative motion.
• Doppler ultrasound, radar, astronomy, and speed measurement all use frequency shifts caused by motion.
• At sufficiently high source speeds, ordinary moving-source behaviour leads toward shock-wave phenomena.
• Supersonic objects can produce Mach cones and sonic booms.

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Check Your Understanding

1. Distinguish between a moving source and a moving observer.
2. Explain what happens to the wavefronts in front of a sound source moving through still air.
3. Explain what happens to the wavefronts behind the moving source.
4. Why does an observer ahead of an approaching source detect a higher frequency?
5. A stationary source produces evenly spaced wavefronts. An observer moves toward the source. Explain why the observed frequency increases even though the wavelength in the medium remains unchanged.
6. Compare how a moving source and a moving observer produce a Doppler shift.
7. A diagram shows a source with wavefronts tightly packed on its right side and widely spaced on its left side. Determine the direction in which the source is moving and explain your reasoning.
8. A 600 Hz siren moves toward a stationary observer at 25 m/s. If the speed of sound is 340 m/s, calculate the observed frequency.
9. Calculate the observed frequency if the same 600 Hz source moves away at 25 m/s.
10. A stationary 600 Hz source is approached by an observer travelling at 25 m/s. Calculate the observed frequency and compare it with the moving-source result from Question 8.
11. Two vehicles travel in the same direction. A siren source behind travels at 30 m/s while the observer ahead travels at 20 m/s. Are they approaching or receding? Predict whether the observed frequency is higher or lower than the emitted frequency.
12. Explain why the speed of sound through the air does not increase simply because the sound source moves forward.
13. Explain why motion directly toward or away from an observer produces a stronger ordinary Doppler effect than motion perpendicular to the source-observer line.
14. Describe how you could distinguish a moving-source diagram from a moving-observer diagram by examining the wavefronts.
15. A student says, "Moving a source toward an observer and moving an observer toward a source are exactly the same because both produce a higher frequency." Evaluate this statement. Explain what is similar about the two situations, what is physically different, and how the wavefront pattern changes in each case.