Doppler effect

Physics SL· Topic 9.5 (Wave Phenomena) - Doppler effect for sound and light· 12 min read

1. Physical Origin of the Doppler Effect★★☆☆☆⏱ 3 min

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📘 Definition

Doppler Effect

The apparent change in observed frequency of a wave caused by relative motion along the line connecting the source and observer. When the source and observer move closer, observed frequency increases; when they move apart, observed frequency decreases.

For a stationary source, wavefronts spread out evenly in all directions with constant separation equal to the rest wavelength. If the source moves towards the observer, each successive wavefront is emitted closer to the previous one, compressing the wavelength in the direction of travel. If the observer moves towards a stationary source, they encounter wavefronts at a faster rate than if they were at rest, even though the wavelength remains unchanged.

📐 Worked Example

A stationary fire truck emits a siren of rest frequency 1000 Hz. A pedestrian runs towards the siren at 5 m/s, while a second pedestrian runs away from the siren at 5 m/s. Compare the observed frequencies for the two pedestrians.

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Step 1: Confirm the source is stationary, so wavelength is unchanged at $λ = v / f_{s}$
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Step 2: The observer moving towards the source encounters wavefronts faster, so observed frequency is higher than 1000 Hz
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Step 3: The observer moving away from the source encounters wavefronts slower, so observed frequency is lower than 1000 Hz
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Step 4: The magnitude of the frequency shift is identical for both observers, only the direction of shift differs.

2. Doppler Shift Formulae for Sound Waves★★★☆☆⏱ 4 min

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🔬 Derivation

Goal:

Derive the Doppler formula for a moving source

Starting from:

Stationary observer, source moving at speed $v_{s}$ towards the observer in a medium where sound travels at speed $v$

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$Time between successive wavefront emissions from source is $T_s = 1/f_s$$
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$Distance source travels between emissions is $v_s T_s$$
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$Compressed observed wavelength is $\lambda_o = \frac{v}{f_s} - v_s T_s = \frac{v - v_s}{f_s}$$
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$Observed frequency is $f_o = \frac{v}{\lambda_o} = f_s \frac{v}{v - v_s}$$

Result:

When the source moves towards the observer, the denominator is smaller than $v$ , so $f_{o} > f_{s}$ as expected.

f_o = f_s \frac{v + v_o}{v - v_s}

📐 Worked Example

A car travels towards a stationary listener at 30 m/s, emitting a horn of frequency 250 Hz. The speed of sound in air is 340 m/s. Calculate the frequency observed by the listener.

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Step 1: Identify known values: $f_{s} = 250$ Hz, $v_{s} = 30$ m/s, $v_{o} = 0$ , $v = 340$ m/s
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Step 2: Source moves towards observer, so use $- v_{s}$ in denominator, $+ 0$ in numerator
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$f_o = 250 \times \frac{340 + 0}{340 - 30} = 250 \times \frac{340}{310}$
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$f_o \approx 274 \text{ Hz}$

✓ Quick check

Test your sign convention understanding:

If an observer moves away from a stationary sound source, what is the correct sign for $v_{o}$ ?
- Positive
- Negative
- Zero
- Equal to $v_{s}$
Reveal answer
Negative —
When moving away, the observer encounters wavefronts slower, so you subtract $v_{o}$ from $v$ in the numerator.

3. Doppler Effect for Electromagnetic Waves★★★☆☆⏱ 3 min

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For electromagnetic waves like light, no propagation medium exists, so the Doppler shift only depends on the relative radial speed $v_{r}$ between source and observer. At IB SL, you only need the non-relativistic approximation that applies when $v_{r} ≪ c$ , the speed of light in vacuum.

\frac{\Delta f}{f_s} \approx \frac{v_r}{c}, \quad \frac{\Delta \lambda}{\lambda_s} \approx \frac{v_r}{c}

📐 Worked Example

A distant galaxy recedes from Earth at 1.2% of the speed of light. Calculate the fractional Doppler shift of its visible 500 nm spectral line.

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Step 1: Identify $v_{r} = 0.012 c$ , source moves away so shift is positive for wavelength
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$\frac{\Delta \lambda}{\lambda_s} = \frac{v_r}{c} = 0.012$
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$\Delta \lambda = 0.012 \times 500 = 6 \text{ nm}$

4. Standard IB Exam Applications★★★★☆⏱ 2 min

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Radar speed guns: Use the double Doppler shift for reflected microwaves to calculate the speed of moving vehicles
Ultrasound medical imaging: Measure blood flow speed using Doppler shift of reflected sound waves
Astronomical redshift: Calculate recession speed of distant stars and galaxies from shifted spectral lines

5. Common Pitfalls

Wrong move:

Swapping the sign convention for $v_{s}$ and using $v + v_{s}$ in the denominator when the source moves towards the observer

Why:

This incorrectly increases the denominator, leading to a lower observed frequency than the rest frequency, which contradicts the physical compression of wavelength

Correct move:

Always apply the rule: source moving towards observer reduces wavelength, so subtract $v_{s}$ from $v$ in the denominator

Wrong move:

Using the full sound Doppler formula for light wave problems

Why:

Light has no propagation medium, so separate speeds for source and observer are undefined, leading to incorrect results

Correct move:

Use the $Δ f / f = v_{r} / c$ non-relativistic approximation for all EM wave Doppler problems at SL

Wrong move:

Forgetting that radar speed guns produce a double Doppler shift (source to moving object, then moving object to detector)

Why:

This leads to calculating half the actual speed of the target object

Correct move:

Multiply the relative speed by 2 in the $Δ f / f$ expression for reflected wave problems

Wrong move:

Stating that Doppler shift is caused by a change in the wave's speed in the medium for a moving source

Why:

Wave speed in a stationary medium is constant, independent of source motion, so this is a common mark-deducting misconception

Correct move:

Explain that the shift comes from compression or stretching of the wavelength between emitted wavefronts

Wrong move:

Using relativistic Doppler shift equations for sound problems

Why:

IB SL syllabus explicitly only requires non-relativistic formulae for sound, and no Lorentz transform derivation is expected

Correct move:

Stick strictly to the IB provided formula sheet expressions for all exam calculations

6. Quick Reference Cheatsheet

Scenario	IB Approved Formula	Key Condition
Observer moving towards stationary sound source	$f_{o} = f_{s} \frac{v + v _{o}}{v}$	Wavelength unchanged, observer encounters wavefronts faster
Observer moving away from stationary sound source	$f_{o} = f_{s} \frac{v - v _{o}}{v}$	Wavelength unchanged, observer encounters wavefronts slower
Source moving towards stationary sound observer	$f_{o} = f_{s} \frac{v}{v - v _{s}}$	Wavelength compressed in direction of travel
Source moving away from stationary sound observer	$f_{o} = f_{s} \frac{v}{v + v _{s}}$	Wavelength stretched in direction of travel
Non-relativistic EM Doppler shift	$\frac{Δ f}{f _{s}} = \frac{v _{r}}{c}$	Only valid for $v_{r} ≪ c$

7. Frequently Asked

Does the Doppler effect work for all types of waves?

Yes, it applies to all wave types including sound, water, and electromagnetic waves like light. The only difference is that for EM waves, no medium is required, so only relative speed between source and observer matters.

Why do we not use the same formula for sound and light Doppler effect at SL?

For sound, motion relative to the air medium changes the observed wavelength and wave speed separately. For non-relativistic light at IB SL, we use the approximate $\frac{Δ f}{f} = \frac{v}{c}$ formula that only depends on relative line-of-sight speed.

When this came up on past exams

AI-estimated based on syllabus patterns — cross-check with official past papers for accuracy. Use only as revision-focus signals.

2023 · Paper 2
Sound Doppler shift calculation
2022 · Paper 1
Qualitative shift direction question
2021 · Paper 2
Radar speed gun application

What's Next

Mastering the Doppler effect gives you a critical foundation for IB SL wave behaviour and astrophysics topics, which together make up ~20% of your total exam marks. You will now be able to solve all standard Doppler shift multiple choice and paper 2 calculation questions, and correctly explain the physical mechanism to earn full method marks. This concept directly extends to cosmological redshift, one of the key pieces of evidence for the expanding universe covered in the astrophysics option. It also connects to wave interference and standing wave problems where relative motion of wave sources can produce beat frequencies. Follow the links below to continue your progress through related unit topics.

→Wave Superposition and Interference