3. Doppler Effect for Sound

Doppler Effect for Sound

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

A familiar example is an emergency vehicle passing you with its siren sounding. As it approaches, the siren seems to have a higher pitch. After it passes and moves away, the pitch becomes lower.

The siren does not need to change the frequency it produces. The change you hear occurs because the motion of the source changes how frequently the sound wavefronts reach you.

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Sound Waves and Frequency

Sound is a mechanical longitudinal wave.

In air, sound travels as alternating regions of:

• compression
• rarefaction

The frequency of a sound is the number of complete wave cycles passing a point each second.

Frequency is measured in:

hertz (Hz)

For example:

500 Hz = 500 cycles per second

Frequency is strongly related to the pitch we hear.

higher frequency → higher pitch

lower frequency → lower pitch

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Pitch and Frequency

Pitch is the perception of how high or low a sound seems.

A high-frequency sound is generally perceived as having a high pitch.

Examples include:

• whistle
• small bell
• high piano note

A low-frequency sound is generally perceived as having a low pitch.

Examples include:

• bass guitar
• large drum
• low piano note

The Doppler effect changes the observed frequency, which causes the observed pitch to change.

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The Siren Example

Imagine standing beside a road.

An ambulance approaches with its siren producing a constant frequency.

As it approaches:

you hear a higher pitch

As it passes:

the pitch drops

As it moves away:

you hear a lower pitch

This characteristic change is caused by the:

Doppler effect

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Why the Siren's Pitch Changes

The siren produces sound waves continuously while the ambulance moves.

Between producing one wavefront and the next, the ambulance changes position.

In front of the ambulance:

wavefronts become closer together

Behind the ambulance:

wavefronts become farther apart

Therefore, observers in front and behind receive different wave patterns.

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In Front of the Moving Siren

The ambulance moves toward the wavefronts it has already produced.

Each new wavefront begins slightly closer to the previous one.

Therefore:

wavefront spacing decreases

This means:

wavelength decreases

Since sound speed through the same air remains approximately constant:

shorter wavelength → higher observed frequency

Therefore:

approaching siren → higher pitch

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Behind the Moving Siren

Behind the ambulance, each successive wavefront begins farther forward than the previous one.

The wavefronts behind become:

farther apart

Therefore:

wavelength increases

With approximately constant sound speed:

longer wavelength → lower observed frequency

Therefore:

receding siren → lower pitch

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The Essential Doppler Rule

For sound, remember:

approaching → higher observed frequency → higher pitch

receding → lower observed frequency → lower pitch

This rule solves many conceptual Doppler questions without any calculation.

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Emitted and Observed Frequency

It is important to distinguish between two frequencies.

Emitted frequency

The frequency actually produced by the source.

Symbol:

fₛ

Observed frequency

The frequency detected by the observer.

Symbol:

fₒ

During a Doppler shift:

fₒ may be different from fₛ

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Example of Emitted and Observed Frequency

Suppose a siren continuously emits:

700 Hz

When the ambulance approaches, an observer might detect a frequency greater than:

700 Hz

After the ambulance passes and moves away, the observer might detect a frequency below:

700 Hz

The siren itself can continue emitting:

700 Hz

The observed frequency changes because of relative motion.

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Wavefront Compression

A wavefront represents points on a wave that are at the same stage of oscillation.

For a moving sound source, the wavefronts in front become compressed.

Compressed wavefronts mean:

shorter wavelength

Using:

v = fλ

if v remains approximately constant:

smaller λ → larger f

Therefore, an observer in front detects a higher frequency.

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Wavefront Stretching

Behind the source, wavefronts are stretched farther apart.

Stretched wavefronts mean:

longer wavelength

If sound speed remains approximately constant:

larger λ → smaller f

Therefore, an observer behind the moving source detects a lower frequency.

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Comparing the Wavefronts

This table summarizes the most important relationship for a moving sound source.

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The Speed of Sound Does Not Increase

A common mistake is to think that sound travels faster because the ambulance is moving toward the observer.

This is not what causes the Doppler effect.

The speed of sound through still air is determined mainly by the properties of the air, especially temperature.

The moving source changes:

wavefront spacing

rather than simply adding its speed to the speed of sound.

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The Wave Equation

Sound waves obey:

v = fλ

where:

v = wave speed

f = frequency

λ = wavelength

For sound travelling through the same air, v remains approximately constant.

Therefore:

shorter wavelength → higher frequency

longer wavelength → lower frequency

This helps explain the Doppler effect for a moving source.

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Conceptual Example: Approaching Car

A car horn approaches a stationary observer.

Question:

Does the observer hear a higher or lower pitch?

Reasoning:

1. The car approaches.
2. Wavefronts ahead are compressed.
3. Wavelength is shorter.
4. Wavefronts reach the observer more frequently.
5. Observed frequency increases.

Answer:

higher pitch

No numerical calculation is required.

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Conceptual Example: Receding Train

A train moves away while sounding its whistle.

Question:

How does the observed pitch compare with the emitted pitch?

Reasoning:

1. Train is moving away.
2. Wavefronts reaching the observer are farther apart.
3. Observed wavelength is longer.
4. Fewer wave cycles reach the observer each second.
5. Observed frequency is lower.

Answer:

lower pitch

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What Happens When the Vehicle Passes?

Before a vehicle passes:

source approaching → higher observed frequency

After it passes:

source receding → lower observed frequency

Therefore, the listener hears a noticeable drop in pitch.

The change can sound sudden because the component of the vehicle's velocity toward the observer changes to a component away from the observer as it passes.

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Source Moving Toward an Observer

For a moving sound source:

source approaches

→ wavefront compression

→ shorter wavelength

→ higher observed frequency

→ higher pitch

This is one of the most important cause-and-effect chains to remember.

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Source Moving Away from an Observer

Similarly:

source recedes

→ wavefront stretching

→ longer wavelength

→ lower observed frequency

→ lower pitch

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A Moving Observer

The observer can also move.

Suppose the sound source is stationary.

Its wavefronts remain evenly spaced in the air.

If the observer moves toward the source, the observer encounters wavefronts more frequently.

Therefore:

observer moving toward source → higher observed frequency

If the observer moves away:

observer moving away from source → lower observed frequency

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Moving Source vs Moving Observer

There is an important difference.

Moving source

Changes the spacing of wavefronts in the medium.

Moving observer

Does not change the spacing of the existing wavefronts. Instead, the observer changes how rapidly they encounter them.

Both situations can produce a Doppler shift.

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Both Moving

The source and observer can both move.

The simplest way to make a conceptual prediction is to ask:

Is the distance between them decreasing or increasing?

If decreasing:

higher observed frequency

If increasing:

lower observed frequency

This is often more useful than focusing only on the individual directions of motion.

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Same Direction Does Not Necessarily Mean Moving Apart

Suppose a siren and observer both travel to the right.

The siren is behind.

If the siren travels faster than the observer, it catches up.

Their separation decreases.

Therefore:

observed frequency increases

If the observer travels faster and pulls away:

separation increases

Therefore:

observed frequency decreases

Always consider the change in distance between source and observer.

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Everyday Doppler Examples

The Doppler effect can be heard from many moving sound sources:

• ambulance sirens
• police sirens
• fire engines
• train whistles
• car horns
• motorcycles
• racing cars
• aircraft
• fast-moving machinery

The effect is most noticeable when the source moves quickly compared with the speed of sound.

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Train Whistle

Imagine standing at a railway station.

A train approaches while sounding its whistle.

Before the train passes:

pitch is higher

As the train passes:

pitch drops

After the train passes:

pitch is lower

The whistle does not need to change its emitted frequency.

The changing pitch results from the Doppler effect.

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Racing Cars

The Doppler effect is particularly noticeable with racing vehicles.

As a race car approaches:

engine frequencies are shifted upward

After it passes:

frequencies are shifted downward

The familiar change in engine sound is partly a Doppler effect.

The engine's actual operating frequency may also change as the driver accelerates or changes gears, so real vehicle sounds can be more complicated than the idealized model.

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Aircraft

Aircraft can also produce noticeable Doppler shifts.

As an aircraft approaches:

observed sound frequencies increase

As it recedes:

observed frequencies decrease

The effect can be combined with delays caused by the finite speed of sound, particularly for fast aircraft.

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Doppler Equation for a Moving Source

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

fₒ = fₛv/(v ∓ vₛ)

where:

fₒ = observed frequency

fₛ = emitted frequency

v = speed of sound

vₛ = speed of source

For an approaching source:

fₒ = fₛv/(v − vₛ)

For a receding source:

fₒ = fₛv/(v + vₛ)

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Worked Example: Approaching Siren

An ambulance emits:

fₛ = 800 Hz

It approaches a stationary observer at:

vₛ = 20 m/s

Take the speed of sound as:

v = 340 m/s

Use:

fₒ = fₛv/(v − vₛ)

fₒ = 800 × 340/(340 − 20)

fₒ = 272000/320

fₒ = 850 Hz

The observer hears:

850 Hz

This is higher than the emitted frequency.

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Worked Example: Receding Siren

The same ambulance moves away at:

20 m/s.

Use:

fₒ = fₛv/(v + vₛ)

fₒ = 800 × 340/(340 + 20)

fₒ = 272000/360

fₒ ≈ 756 Hz

Therefore:

approaching → 850 Hz

emitted → 800 Hz

receding → 756 Hz

This matches our conceptual prediction.

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Predict Before Calculating

Before using an equation, always make a prediction.

Ask:

Is the source approaching or receding?

If approaching:

your calculated answer must satisfy:

fₒ > fₛ

If receding:

your answer must satisfy:

fₒ < fₛ

This is an excellent way to catch calculation or sign errors.

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Conceptual Doppler Problems

Many Doppler problems do not require an equation.

For example:

A motorcycle approaches a pedestrian.

Prediction:

higher observed frequency.

A train moves away from a station.

Prediction:

lower observed frequency.

A cyclist moves toward a stationary bell.

Prediction:

higher observed frequency.

A car and observer remain the same distance apart.

Prediction:

no ordinary Doppler shift due to radial relative motion.

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Which Observer Hears the Highest Frequency?

Suppose a moving siren has three observers.

Observer A is directly ahead.

Observer B is directly behind.

Observer C is mostly to the side.

Observer A experiences the strongest approaching radial motion and therefore hears a higher frequency.

Observer B experiences receding motion and hears a lower frequency.

Observer C experiences a smaller radial component of the source's velocity and therefore usually experiences a smaller Doppler shift at that instant.

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Radial Motion

The ordinary Doppler effect depends primarily on motion along the line connecting the source and observer.

This is called:

radial motion

Motion directly toward the observer produces a strong positive frequency shift.

Motion directly away produces a strong negative frequency shift.

Purely sideways motion has no instantaneous classical radial Doppler shift at the moment when the motion is exactly perpendicular to the source-observer line.

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Doppler Effect and Loudness

An approaching siren usually becomes:

• louder
• higher in pitch

A receding siren usually becomes:

• quieter
• lower in pitch

However, these are caused by different effects.

Pitch change

is caused by the Doppler frequency shift.

Loudness change

is mainly caused by changing distance and sound intensity.

Therefore:

louder does not mean higher frequency.

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Doppler Effect and Amplitude

The Doppler effect concerns:

frequency

It does not fundamentally require a change in amplitude.

A sound could theoretically undergo a Doppler frequency shift while maintaining similar amplitude.

In everyday situations, amplitude often changes because the distance between source and observer changes.

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Doppler Ultrasound

The Doppler effect for sound has important medical applications.

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

The waves can reflect from moving blood cells.

Because the reflecting cells are moving, the frequency of the returning ultrasound changes.

Doctors can analyze this frequency shift to obtain information about blood flow.

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Measuring Blood Flow

Doppler ultrasound can provide information about:

• direction of blood flow
• relative blood-flow speed
• circulation
• heart function
• blood vessels

Blood moving toward the ultrasound transducer produces a different frequency shift from blood moving away.

The Doppler effect therefore allows motion inside the body to be studied without surgery.

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Ultrasound Uses Reflected Sound

Doppler ultrasound combines more than one wave behaviour.

The ultrasound first:

travels into the body

It then:

reflects from moving structures

The motion produces:

a Doppler frequency shift

The returning sound is detected and analyzed.

This is an excellent example of several wave concepts working together in one technology.

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Bats and Echolocation

Bats produce high-frequency sounds and detect returning echoes.

This process is called:

echolocation

Reflection provides information about distance.

Doppler shifts can also provide information about relative motion in species that make use of such frequency changes.

This can help an animal obtain information about moving prey or surroundings.

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Speed Measurement

The Doppler principle can be used to determine motion by measuring frequency changes.

For sound-based systems, a known frequency is transmitted.

If the sound reflects from a moving object, the returned frequency may be shifted.

The size and direction of the shift can provide information about motion.

Similar principles are used with electromagnetic waves in radar.

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Sound from Rotating Objects

The Doppler effect can also occur when parts of a rotating object move alternately toward and away from an observer.

For example, rotating machinery can produce complicated frequency patterns.

Engineers can analyze sound and vibration frequencies to investigate:

• machine speed
• vibration
• faults
• rotating components

Wave analysis is therefore useful in engineering diagnostics.

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Siren Passing at Constant Speed

Consider a vehicle moving past an observer at constant speed.

Even though the vehicle's speed remains constant, the observed frequency changes.

Why?

Because the radial component of its velocity changes.

While approaching:

radial motion is toward the observer.

After passing:

radial motion is away.

Therefore, constant vehicle speed does not mean constant observed frequency.

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Does the Pitch Change Exactly at the Passing Point?

In an idealized geometry, the source's radial velocity relative to an observer beside its path becomes zero at the instant of closest approach.

The observed frequency changes continuously with geometry.

In everyday listening, the transition can sound like a rapid drop because the source moves quickly from approaching to receding.

This is more accurate than imagining that the siren itself suddenly changes frequency.

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Sound Speed and Temperature

The speed of sound in air depends on conditions, particularly temperature.

A commonly used classroom value is:

340 m/s

although the actual value varies.

For example, around room temperature it is often closer to:

343 m/s

Doppler calculations normally provide or specify the sound speed to use.

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A Complete Siren Example

An ambulance emits a constant:

750 Hz

Before passing an observer, it approaches.

Therefore:

• wavefronts ahead are compressed
• wavelength decreases
• observed frequency is greater than 750 Hz
• observed pitch is higher

After passing:

• wavefronts reaching the observer are stretched apart
• wavelength increases
• observed frequency is below 750 Hz
• observed pitch is lower

The emitted frequency remains:

750 Hz

The observed change is caused by relative motion.

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A Complete Conceptual Comparison

Consider four situations.

A. Source approaches stationary observer

Observed frequency:

increases

B. Source recedes from stationary observer

Observed frequency:

decreases

C. Observer approaches stationary source

Observed frequency:

increases

D. Observer recedes from stationary source

Observed frequency:

decreases

Therefore, the simplest conceptual rule remains:

distance decreasing → frequency increasing

distance increasing → frequency decreasing

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Interpreting Doppler Diagrams

When a diagram shows circular sound wavefronts around a moving source, examine their spacing.

If wavefronts are closer together:

shorter wavelength

→ higher observed frequency

→ higher pitch

If wavefronts are farther apart:

longer wavelength

→ lower observed frequency

→ lower pitch

The source is moving toward the compressed side.

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A Reliable Strategy for Conceptual Doppler Problems

When solving a Doppler question:

1. Identify the sound source.
2. Identify the observer.
3. Determine which one is moving.
4. Decide whether the distance between them is decreasing or increasing.
5. Decreasing distance means approaching.
6. Increasing distance means receding.
7. Approaching means higher observed frequency.
8. Receding means lower observed frequency.
9. For sound, convert frequency prediction into pitch.
10. If a moving source is shown, inspect wavefront spacing.
11. Compressed wavefronts mean shorter wavelength and higher frequency.
12. Stretched wavefronts mean longer wavelength and lower frequency.
13. Distinguish emitted frequency from observed frequency.
14. Do not confuse frequency changes with loudness changes.
15. Check that your explanation matches the physical situation.

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A Reliable Strategy for Numerical Problems

If a calculation is required:

1. Write the emitted frequency fₛ.
2. Write the speed of sound v.
3. Write the source speed vₛ if given.
4. Determine whether the source approaches or recedes.
5. Predict whether fₒ should be higher or lower than fₛ.
6. Select the appropriate equation.
7. Substitute the values.
8. Calculate the observed frequency.
9. Include Hz.
10. Compare the result with your prediction.

For example:

approaching 700 Hz source

→ answer must be greater than 700 Hz

receding 700 Hz source

→ answer must be less than 700 Hz

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Common Mistakes

Thinking the siren itself changes frequency as it passes

The emitted frequency can remain constant while the observed frequency changes.

Thinking an approaching source produces a lower pitch

Approaching means higher observed frequency and higher pitch.

Thinking a receding source produces a higher pitch

Receding means lower observed frequency and lower pitch.

Thinking compressed wavefronts mean lower frequency

Compressed wavefronts have shorter wavelength and correspond to higher observed frequency for the same sound speed.

Thinking the source makes sound travel faster

The source changes wavefront spacing rather than simply adding its velocity to the sound speed.

Confusing pitch with loudness

Pitch is strongly related to frequency. Loudness is related to intensity and perception.

Using loudness as evidence of the Doppler effect

The Doppler effect specifically concerns frequency.

Thinking only the source can move

Motion of the observer can also produce a Doppler shift.

Looking only at direction instead of relative motion

Two objects can move in the same direction while still approaching one another.

Forgetting to predict before calculating

A quick higher/lower prediction is one of the easiest ways to detect a wrong numerical answer.

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Key Terms

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

Sound wave — mechanical longitudinal wave produced by vibrations and transmitted through a medium.

Source — object producing the sound.

Observer — person or detector receiving the sound.

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

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

Pitch — perception associated strongly with sound frequency.

Frequency — number of wave cycles passing a point each second.

Hertz (Hz) — unit of frequency equal to one cycle per second.

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

Wavefront — representation of points on a wave that are at the same phase.

Wavefront compression — reduction in spacing between successive wavefronts ahead of a moving source.

Wavefront stretching — increase in spacing between successive wavefronts behind a moving source.

Approaching — relative motion that decreases source-observer separation.

Receding — relative motion that increases source-observer separation.

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

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

Doppler shift — difference between emitted and observed frequencies.

Doppler ultrasound — medical technique that uses frequency shifts in ultrasound to investigate moving structures such as blood.

Echolocation — use of reflected sound to obtain information about objects and surroundings.

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Key Takeaways

• The Doppler effect explains why a moving siren appears to change pitch.
• Sound is a mechanical longitudinal wave.
• Frequency is measured in hertz.
• Higher frequency is generally perceived as higher pitch.
• Lower frequency is generally perceived as lower pitch.
• An approaching source produces a higher observed frequency.
• A receding source produces a lower observed frequency.
• The emitted frequency can remain constant while the observed frequency changes.
• A moving source compresses wavefronts ahead of it.
• Compressed wavefronts have shorter wavelength.
• Shorter wavelength corresponds to higher observed frequency when sound speed is unchanged.
• A moving source stretches wavefronts behind it.
• Stretched wavefronts have longer wavelength.
• Longer wavelength corresponds to lower observed frequency.
• The moving source does not simply make sound travel faster.
• A moving observer can also experience a Doppler shift.
• An observer approaching a source detects a higher frequency.
• An observer moving away detects a lower frequency.
• Relative motion toward or away is more important than simply asking which object moves.
• Changes in loudness and changes in pitch have different causes.
• Sirens, train whistles, car horns, motorcycles, racing vehicles, and aircraft provide everyday examples.
• Doppler ultrasound uses frequency shifts to study blood flow.
• The size of the Doppler shift can provide information about relative motion.
• Conceptual Doppler problems can often be solved simply by deciding whether the source and observer are approaching or receding.

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Check Your Understanding

1. Define the Doppler effect for sound.
2. Explain why an approaching ambulance siren sounds higher in pitch than its emitted pitch.
3. Explain why the same siren sounds lower in pitch after the ambulance passes and moves away.
4. Distinguish between emitted frequency and observed frequency.
5. A siren emits a constant frequency of 700 Hz. An observer hears 750 Hz. Is the source most likely approaching or receding? Explain.
6. Another observer hears the same 700 Hz siren at 660 Hz. Is the source approaching or receding?
7. Describe what happens to the wavelength of sound in front of a moving source.
8. Describe what happens to the wavelength behind a moving source.
9. Explain why the speed of sound does not simply increase because a sound source moves toward an observer.
10. A stationary source produces a 500 Hz sound. An observer moves toward the source. Predict whether the observed frequency is greater than, equal to, or less than 500 Hz and explain why.
11. An ambulance emits an 800 Hz siren and approaches a stationary observer at 20 m/s. Using a sound speed of 340 m/s, calculate the observed frequency.
12. Calculate the observed frequency when the same ambulance moves away at 20 m/s.
13. An approaching siren becomes both louder and higher in pitch. Explain why only the pitch change is specifically the Doppler effect.
14. Describe how Doppler ultrasound uses changes in sound frequency to provide information about blood flow.
15. A student says, "When an ambulance drives toward me, its siren sounds higher because the ambulance pushes the sound forward and makes the sound travel faster." Evaluate this statement. Explain the actual changes in wave speed, wavelength, observed frequency, and pitch.