Doppler effect for a moving sound source

Physics· 12 min read

1. Physical Origin of the Shift★★☆☆☆⏱ 3 min

When a sound source moves towards a stationary observer, each successive wavefront is emitted at a position closer to the observer than the previous one. This compresses the spacing between wavefronts, reducing the observed wavelength and increasing the detected frequency.

If the source moves away from the observer, each new wavefront is emitted further away than the last, stretching the wavelength and lowering the observed frequency.

📘 Definition

Doppler Effect for Moving Sound Source

The apparent shift in observed sound frequency caused by compression or rarefaction of emitted wavefronts as the source travels along the line connecting it to a stationary observer

✓ Quick check

Test your basic understanding:

A siren moves directly towards you. What happens to the pitch you hear?
- Increases
- Decreases
- Stays identical
Reveal answer
Increases —
Compressed wavefronts raise the observed frequency, so pitch increases.
The source moves perpendicular to your position. What happens to observed frequency?
- Sharply increases
- No measurable shift
- Sharply decreases
Reveal answer
No measurable shift —
No relative motion along the line connecting source and observer means no wavelength change.

2. Derivation of the Frequency Formula★★★★☆⏱ 4 min

🔬 Derivation

Goal:

Derive observed frequency for a moving sound source

Starting from:

Speed of sound in still air = $v$ , source speed = $v_{s}$ , source emitted frequency = $f_{0}$

1
Time period of emitted wave: $T = 1/ f_{0}$
2
In one time period, the source travels a distance $v_{s} T = v_{s} / f_{0}$ towards the observer
3
Original unshifted wavelength: $λ_{0} = v / f_{0}$
4
New compressed wavelength for approaching source: $λ_{o} = λ_{0} - v_{s} / f_{0} = (v - v_{s}) / f_{0}$
5
Observed frequency: $f_{o} = v / λ_{o} = f_{0} \times v / (v - v_{s})$
6
For receding source, add $v_{s} / f_{0}$ to original wavelength to get $f_{o} = f_{0} \times v / (v + v_{s})$

Result:

The sign in the denominator depends only on direction of travel: minus for approaching, plus for receding.

📐 Worked Example

A siren emitting a 500 Hz tone moves towards a stationary observer at 20 m/s. Speed of sound in air is 340 m/s. Calculate the observed frequency.

1
Confirm the source is approaching the observer, so use the minus sign in the denominator
2
$f_o = 500 \times \frac{340}{340 - 20}$
3
Simplify denominator: 340 - 20 = 320 m/s
4
$f_o = 500 \times 1.0625 = 531.25 \text{ Hz}$
5
Round to 3 significant figures: 531 Hz

3. Exam-Focused Problem Solving★★★☆☆⏱ 3 min

📐 Worked Example

A train moving away from a station platform at 35 m/s emits a 1200 Hz whistle. Speed of sound is 330 m/s. Find the observed frequency for a passenger standing on the platform.

1
Confirm the source is receding, so use the plus sign in the denominator
2
$f_o = 1200 \times \frac{330}{330 + 35}$
3
Simplify denominator: 330 + 35 = 365 m/s
4
$f_o = 1200 \times 0.9041 = 1085 \text{ Hz}$
5
Round to 3 significant figures: 1090 Hz

4. Common Sound Doppler Applications★★☆☆☆⏱ 2 min

The moving source Doppler effect for sound is used in multiple real-world systems that appear in CIE context questions.

Ultrasound medical scanners that measure blood flow velocity via frequency shift of reflected sound waves
Speed measurement devices for moving vehicles that use audible sound wave reflection
Emergency siren pitch change that alerts drivers to approaching or receding emergency vehicles

5. Common Pitfalls

Wrong move:

Mixing moving source and moving observer formulas

Why:

The two cases have different physical derivations, so swapping them gives incorrect frequency shift magnitude

Correct move:

Explicitly confirm if the source or observer is moving before selecting your formula

Wrong move:

Using $v + v_{s}$ in the denominator for an approaching source

Why:

An approaching source compresses wavelength, so the denominator must be smaller than $v$ to produce a higher observed frequency

Correct move:

Use $v - v_{s}$ for approaching sources, $v + v_{s}$ for receding sources

Wrong move:

Stating frequency shift comes from a change in speed of sound

Why:

Speed of sound in still air is constant for fixed temperature, shift comes only from changed wavefront spacing

Correct move:

Explicitly reference compressed or stretched wavelength in all explanation answers

Wrong move:

Giving final answers to 1 significant figure

Why:

CIE mark schemes require 2 or 3 significant figures for all calculation questions, 1 s.f. loses the final accuracy mark

Correct move:

Round all final frequency answers to 3 significant figures unless specified otherwise

Wrong move:

Forgetting the Hz unit for final frequency answers

Why:

Missing units lose 1 independent mark in almost all 9702 calculation questions

Correct move:

Add the Hz unit immediately after calculating your final value

6. Quick Reference Cheatsheet

Scenario	Formula	Key Exam Note
Source moving towards stationary observer	$f_{o} = f_{0} \frac{v}{v - v _{s}}$	$f_{o} > f_{0}$ , denominator smaller than $v$
Source moving away from stationary observer	$f_{o} = f_{0} \frac{v}{v + v _{s}}$	$f_{o} < f_{0}$ , denominator larger than $v$
Standard speed of sound value	$v = 330 - 340$ m/s	Use the value given in the question, not a memorised one

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.

2024 · Paper 4
Frequency shift calculation for siren
2022 · Paper 2
Explanation of wavefront compression
2021 · Paper 3
Practical Doppler data analysis

What's Next

Mastering the moving source Doppler effect is a core requirement for CIE 9702 A Level Physics, as this concept forms the foundation for extended A2 topics including electromagnetic Doppler red shift used in astrophysics, and combined source-observer relative motion problems that regularly appear on Paper 4. You will also apply your understanding of frequency shift to practical data analysis questions that test your ability to calculate unknown source speed from measured frequency values. Ensure you can fully distinguish between moving source and moving observer scenarios before progressing, as mixing these two cases is the single most common cause of lost marks on exam papers.