Doppler effect

Lesson C.5 Doppler Effect

Lesson Notes, Worked Examples and Exam Practice

The Doppler effect explains why a passing siren changes pitch and why a spectral line from a moving star can appear at a different wavelength. This lesson develops the wavefront explanation, the low-speed light approximation and the additional HL equations for sound.

Lesson map

C.5.1 — SL and HL: Doppler effect, sound and wavefront diagrams.
C.5.2 — SL and HL: Light, redshift, blueshift and spectral lines.
C.5.3 — Additional HL: Moving-source and moving-observer sound equations and derivations.
Optional extension: Both source and observer moving.

Learning objectives

Explain an observed frequency change using wavefronts.
Distinguish a moving source from a moving observer.
Use the approximate light Doppler relation when the radial speed is much smaller than the speed of light.
For HL, choose signs and calculate observed frequency or source/observer speed.

Data, symbols and assumptions

For sound calculations, use $v=340\,\mathrm{m\,s^{-1}}$ unless another value is given. Use $c=3.00\times10^8\,\mathrm{m\,s^{-1}}$ for light. Give numerical results to 2–3 significant figures and retain extra digits during working.

$f$ is emitted frequency and $f\prime$ is observed frequency. In the sound equations, $v$ is the wave speed, $u_s$ is source speed and $u_o$ is observer speed. These speeds are measured relative to the medium. In the light approximation, $v$ instead denotes the relative radial speed along the line of sight.

For the sound model, assume still air, motion along the source–observer line and speeds below the sound speed. The emitted frequency stays constant. The separate moving-source and moving-observer cases form the assessed HL sound calculations; combined motion is included here as an optional extension.

C.5.1 Understanding the Doppler Effect — SL and HL

Definition

The Doppler effect is a change in the frequency observed or received because of motion of the source or observer. For sound, the motion must be considered relative to the medium.

A passing siren

As a siren approaches a stationary listener, its observed frequency and perceived pitch are higher than the emitted frequency.
As it recedes, the observed frequency and perceived pitch are lower.
The siren can emit a constant frequency throughout: the change occurs in what the listener receives.

Moving source: wavefront spacing changes

Successive wavefronts are emitted from successive source positions. In front of an approaching source they are closer together; behind the source they are farther apart. Sketch wavefronts as circles centred on the successive emission positions, rather than drawing every circle about the source’s current position.

Wavefront explanation

In front: smaller wavelength; a stationary observer receives wavefronts more frequently.
Behind: larger wavelength; a stationary observer receives wavefronts less frequently.
The wave speed through the same still air remains $v$ . Source motion changes spacing, not the sound speed.

$\displaystyle v=f_{\mathrm{medium}}\lambda$

Here $f_{\mathrm{medium}}$ means the frequency measured at a fixed point in the medium. When the source moves, that frequency can differ from its emitted frequency $f$ .

Moving observer: encounter rate changes

For a stationary source, observer motion does not change the wavefront spacing in the air. An observer moving towards the source meets incoming wavefronts more often; an observer moving away meets them less often.

Keep frequency, wavelength and speed distinct

For a moving observer, do not use $f\prime=v/\lambda$ with the unchanged medium-frame sound speed. Use the speed of the wavefronts relative to the observer. A change in observed frequency does not automatically mean that wavelength in the air changes.

C.5.2 Light, Redshift and Blueshift — SL and HL

Light needs no material medium. For radial approach, a spectral line shifts to a shorter wavelength and higher frequency: blueshift. For radial recession, it shifts to a longer wavelength and lower frequency: redshift.

Approximate light Doppler relation

$\displaystyle \frac{|\Delta f|}{f}\approx\frac{|\Delta\lambda|}{\lambda}\approx\frac{v}{c}$

Use emitted or rest values for $f$ and $\lambda$ . Here $v\ll c$ is the magnitude of relative radial speed. The absolute values describe shift magnitudes; signed frequency and wavelength changes have opposite signs.

For a Doppler interpretation, compare a known rest spectral line with the observed line. This determines motion towards or away from the observer along the line of sight. The approximation is not suitable when speed is comparable to $c$ .

Worked example 1 — A shifted spectral line (SL and HL)

A spectral line has rest wavelength $500.0\,\mathrm{nm}$ and is observed at $501.0\,\mathrm{nm}$ . Determine the direction and approximate radial speed.

Show answer and explanation

The wavelength is longer, so this is a redshift: the source is receding in the Doppler model.

$\displaystyle \Delta\lambda=501.0-500.0=1.0\,\mathrm{nm}$

$\displaystyle v\approx c\frac{|\Delta\lambda|}{\lambda}=(3.00\times10^8)\frac{1.0}{500.0}=6.0\times10^5\,\mathrm{m\,s^{-1}}$

Answer: Receding at approximately $600\,\mathrm{km\,s^{-1}}$ . This is $0.0020c$ , consistent with the low-speed approximation.

Applications

Spectral-line shifts help determine the radial motion of stars.
Doppler ultrasound uses shifted reflected sound to investigate blood flow.
Radar uses shifted reflected electromagnetic waves to determine motion. Reflected-wave problems can involve shifts on both the outward and return journeys.

C.5.3 Sound Equations and Derivations — Additional HL

Case A: Moving observer, stationary source

IB moving-observer equation

$\displaystyle f\prime=f\left(\frac{v\pm u_o}{v}\right)$

Observer towards source: use $v+u_o$ .
Observer away from source: use $v-u_o$ .

The stationary source produces wavelength $\lambda=v/f$ . Divide the relative wavefront speed by that wavelength:

$\displaystyle f\prime=\frac{v\pm u_o}{\lambda}=\frac{v\pm u_o}{v/f}=f\left(\frac{v\pm u_o}{v}\right)$

Worked example 2 — Moving observer (HL)

A stationary siren emits $f=520\,\mathrm{Hz}$ . A cyclist moves towards it at $u_o=8.0\,\mathrm{m\,s^{-1}}$ . Take $v=340\,\mathrm{m\,s^{-1}}$ . Calculate the observed frequency.

Show answer and explanation

The observer approaches, so use the plus sign in the numerator.

$\displaystyle f\prime=f\left(\frac{v+u_o}{v}\right)=520\left(\frac{340+8.0}{340}\right)$

$\displaystyle f\prime=532.24\,\mathrm{Hz}\approx532\,\mathrm{Hz}$

Check: $f\prime>f$ , as expected.

Case B: Moving source, stationary observer

IB moving-source equation

$\displaystyle f\prime=f\left(\frac{v}{v\pm u_s}\right)$

Source approaching observer: use $v-u_s$ in the denominator.
Source receding from observer: use $v+u_s$ in the denominator.

Derivation: Wavelength in front of the source

The time between emitted crests is $T=1/f$ . In one period, the earlier crest travels $vT$ while the source advances $u_sT$ . The next crest starts closer to the earlier one, giving:

$\displaystyle \lambda_{\mathrm{front}}=vT-u_sT=(v-u_s)T=\frac{v-u_s}{f}$

A stationary observer receives those wavefronts at the sound speed, so:

$\displaystyle f\prime=\frac{v}{\lambda_{\mathrm{front}}}=\frac{v}{(v-u_s)/f}=f\left(\frac{v}{v-u_s}\right)$

Derivation: Wavelength behind the source

Behind a receding source, source motion increases the separation of consecutive wavefronts:

$\displaystyle \lambda_{\mathrm{behind}}=(v+u_s)T=\frac{v+u_s}{f}$

$\displaystyle f\prime=\frac{v}{\lambda_{\mathrm{behind}}}=f\left(\frac{v}{v+u_s}\right)$

Worked example 3 — Moving source (HL)

A car horn emits $f=400\,\mathrm{Hz}$ . The car approaches a stationary observer at $u_s=20\,\mathrm{m\,s^{-1}}$ . Take $v=340\,\mathrm{m\,s^{-1}}$ . Calculate the observed frequency.

Show answer and explanation

An approaching source requires subtraction in the denominator.

$\displaystyle f\prime=f\left(\frac{v}{v-u_s}\right)=400\left(\frac{340}{340-20}\right)$

$\displaystyle f\prime=400\left(\frac{340}{320}\right)=425\,\mathrm{Hz}$

Answer: $425\,\mathrm{Hz}$ , higher than the emitted frequency.

Choosing signs

A reliable sign check

First decide whether the source or observer moves relative to the medium.
For a single approaching source or observer, the observed frequency must increase.
For a single receding source or observer, the observed frequency must decrease.
Observer motion changes the numerator; source motion changes the denominator.
Use a sketch and retain extra digits before rounding.

Worked example 4 — Finding source speed (HL)

A source emits $1000\,\mathrm{Hz}$ . A stationary observer hears $950\,\mathrm{Hz}$ . Take $v=340\,\mathrm{m\,s^{-1}}$ . Determine the source’s direction and speed.

Show answer and explanation

The observed frequency is lower, so the source recedes.

$\displaystyle f\prime=f\frac{v}{v+u_s}$

$\displaystyle u_s=v\left(\frac{f}{f\prime}-1\right)=340\left(\frac{1000}{950}-1\right)=17.9\,\mathrm{m\,s^{-1}}$

Answer: Receding at approximately $18\,\mathrm{m\,s^{-1}}$ .

Optional Extension — Both Source and Observer Moving

This section retains the combined-motion material from the original lesson. It extends the separate sound equations; IB C.5 exam problems do not require simultaneous source and observer motion.

Combining the two effects

$\displaystyle f\prime=f\left(\frac{v\pm u_o}{v\mp u_s}\right)$

Choose numerator and denominator signs independently from the directions relative to the still medium. For a source and observer moving towards one another, use the plus sign in the numerator and the minus sign in the denominator.

Worked example 5 — Combined approach (extension)

A siren emits $700\,\mathrm{Hz}$ . The source moves towards the observer at $u_s=25\,\mathrm{m\,s^{-1}}$ , and the observer moves towards the source at $u_o=10\,\mathrm{m\,s^{-1}}$ . Take $v=340\,\mathrm{m\,s^{-1}}$ . Calculate the observed frequency.

Show answer and explanation

$\displaystyle f\prime=f\left(\frac{v+u_o}{v-u_s}\right)=700\left(\frac{340+10}{340-25}\right)$

$\displaystyle f\prime=700\left(\frac{350}{315}\right)=777.78\,\mathrm{Hz}\approx778\,\mathrm{Hz}$

Both motions increase the received frequency, so $f\prime>f$ , as expected.

Practice Questions — Expandable Answers

Attempt each question before opening its answer. The questions are original practice material, rather than official IB examination questions.

Question 1 — Explaining a passing siren (SL and HL)

A siren emits a constant frequency. Explain why a stationary listener hears a higher pitch during approach and a lower pitch during recession.

Show answer and explanation

During approach, successive wavefronts in front of the source are closer together and reach the listener more often. During recession, wavefronts are farther apart and arrive less often. The source frequency and sound speed through unchanged air remain constant; the received frequency changes.

Question 2 — A moving listener (HL)

A stationary source emits $500\,\mathrm{Hz}$ . A listener moves away at $12\,\mathrm{m\,s^{-1}}$ . Take $v=340\,\mathrm{m\,s^{-1}}$ . Calculate the observed frequency.

Show answer and explanation

$\displaystyle f\prime=500\left(\frac{340-12}{340}\right)=482.35\,\mathrm{Hz}\approx482\,\mathrm{Hz}$

Moving away decreases the encounter rate, so the observed frequency is lower.

Question 3 — A receding source (HL)

A source emits $700\,\mathrm{Hz}$ and moves away from a stationary observer at $30\,\mathrm{m\,s^{-1}}$ . Take $v=340\,\mathrm{m\,s^{-1}}$ . Calculate the observed frequency.

Show answer and explanation

$\displaystyle f\prime=700\left(\frac{340}{340+30}\right)=643.24\,\mathrm{Hz}\approx643\,\mathrm{Hz}$

Question 4 — Moving observer and wavelength (SL and HL)

A listener runs towards a stationary loudspeaker. State what happens to the frequency they hear and to the wavelength of the sound in the still air.

Show answer and explanation

The heard frequency increases because the listener meets incoming wavefronts more frequently. The wavelength in the air remains unchanged because the source and medium are unchanged.

Question 5 — Light blueshift (SL and HL)

A spectral line has rest wavelength $600.0\,\mathrm{nm}$ and is observed at $599.4\,\mathrm{nm}$ . Determine the Doppler direction and approximate radial speed.

Show answer and explanation

The wavelength is shorter, so the source approaches in the Doppler model.

$\displaystyle |\Delta\lambda|=|599.4-600.0|=0.6\,\mathrm{nm}$

$\displaystyle v\approx c\frac{|\Delta\lambda|}{\lambda}=(3.00\times10^8)\frac{0.6}{600.0}=3.0\times10^5\,\mathrm{m\,s^{-1}}$

Answer: Approaching at approximately $300\,\mathrm{km\,s^{-1}}$ .

Question 6 — Combined motion (optional extension)

A source emits $900\,\mathrm{Hz}$ and moves away from the observer at $30\,\mathrm{m\,s^{-1}}$ . The observer moves towards the source at $15\,\mathrm{m\,s^{-1}}$ . Both speeds are relative to still air. Take $v=340\,\mathrm{m\,s^{-1}}$ . Calculate $f\prime$ .

Show answer and explanation

The receding source gives a plus sign in the denominator. The observer moving towards the source gives a plus sign in the numerator.

$\displaystyle f\prime=900\left(\frac{340+15}{340+30}\right)=900\left(\frac{355}{370}\right)=863.51\,\mathrm{Hz}\approx864\,\mathrm{Hz}$

Exam readiness checklist

I can explain a Doppler shift using wavefronts.
I can distinguish source motion from observer motion and keep medium-frame wavelength separate from observed frequency.
I can interpret redshift and blueshift and use the light approximation only for $v\ll c$ .
For HL, I can select the correct sound equation and rearrange it to find a speed.
I can convert units and check whether the frequency change is physically sensible.

Listening, Truth and Hope

Memory verse — Psalm 119:105 (KJV):
“Thy word is a lamp unto my feet, and a light unto my path.”

Creation: Listening with wonder

Sound and light help us communicate, observe and learn about the world. Their behaviour invites careful study of God’s creation. Give thanks for the ability to hear, see, reason and investigate. Use these gifts to appreciate His works and to care for others.

The Fall: Our need for reliable guidance

Sin can distort our choices and our relationships. We may be tempted to hide mistakes, dismiss another person or seek success without honesty. God’s Word guides us towards truth. In our studies, humility means checking assumptions, admitting errors and listening patiently to others.

Redemption: Trusting Jesus Christ

Jesus Christ died for our sins and rose again to offer forgiveness and new life. Your worth is not determined by examination results. Turn to Him in repentance and faith, and receive Him as your personal God, Saviour and Lord. Ask Him to guide your learning and shape your character.

Restoration: Growing and serving

God promises complete restoration to His people. Grow in your relationship with Him by reading and studying the Bible, spending time in prayer and connecting with other believers in a local church. Let your learning become a way to serve: explain patiently, encourage classmates and use knowledge responsibly.

Reflection: How can you become a better listener to God’s Word and to the people around you?

Prayer: Lord Jesus, thank You for Your creation and Your saving love. Forgive my sins and guide me by Your Word. Give me understanding, patience and honesty in my studies. Help me grow in faith through Scripture, prayer and fellowship, and use what I learn to serve others. Amen.

IB Physics HL Book 2025; Papers 1 & 2 Practice 5 Sections65 Lessons54 Weeks

Curriculum

Doppler effect

Lesson C.5 Doppler Effect

Lesson map

Learning objectives

Data, symbols and assumptions

C.5.1 Understanding the Doppler Effect — SL and HL

Definition

A passing siren

Moving source: wavefront spacing changes

Wavefront explanation

Moving observer: encounter rate changes

Keep frequency, wavelength and speed distinct

C.5.2 Light, Redshift and Blueshift — SL and HL

Approximate light Doppler relation

Worked example 1 — A shifted spectral line (SL and HL)

Applications

C.5.3 Sound Equations and Derivations — Additional HL

Case A: Moving observer, stationary source

IB moving-observer equation

Worked example 2 — Moving observer (HL)

Case B: Moving source, stationary observer

IB moving-source equation

Derivation: Wavelength in front of the source

Derivation: Wavelength behind the source

Worked example 3 — Moving source (HL)

Choosing signs

A reliable sign check

Worked example 4 — Finding source speed (HL)

Optional Extension — Both Source and Observer Moving

Combining the two effects

Worked example 5 — Combined approach (extension)

Practice Questions — Expandable Answers

Question 1 — Explaining a passing siren (SL and HL)

Question 2 — A moving listener (HL)

Question 3 — A receding source (HL)

Question 4 — Moving observer and wavelength (SL and HL)

Question 5 — Light blueshift (SL and HL)

Question 6 — Combined motion (optional extension)

Exam readiness checklist

Listening, Truth and Hope

Creation: Listening with wonder

The Fall: Our need for reliable guidance

Redemption: Trusting Jesus Christ

Restoration: Growing and serving

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