Learn7408Paper 3B · Option: AstrophysicsCosmologyThe Doppler effect

Cosmology · Subtopic 2 of 3

The Doppler Effect

When a sound source moves, its wavefronts bunch up in front and spread out behind. A stationary observer in front receives a shorter wavelength and a higher frequency; behind, a longer wavelength and lower frequency. For a source moving at $v_s$ towards or away from you, $f_o=\dfrac{f_sv}{v\mp v_s}$ .

Cambridge 9702 Paper 1Cambridge 9702 Paper 2Edexcel YPH11 WPH15AQA 7408 Paper 3B≈ 30 minRevision 5 · updated 2026-10-27

Section 1 of 14

Learning objectives

Explain the Doppler effect using wavefronts.
Use $f_o=\dfrac{f_sv}{v\pm v_s}$ for a moving source and stationary observer.
Find the speed of a source from observed frequencies.
Relate the Doppler effect to redshift of light.

Pause and think

An ambulance siren sounds higher-pitched as it approaches and lower as it moves away. Is the siren itself changing pitch?

Section 2 of 14

Core idea

Start hereThe ambulance siren

As an ambulance races past, its siren drops in pitch: "neee-oooow". The siren doesn't change; the waves reaching you do. Ahead of a moving source, wavefronts bunch up; behind, they spread out.

The big idea

$f_o=\frac{f_sv}{v\mp v_s}$ : minus approaching (higher pitch), plus receding (lower).

1. Why the frequency changes

Wavefronts from a source moving to the right

Each circle was emitted from where the source was at that moment. Ahead of the source the wavelength is shorter.

Observed frequency for an approaching source

$\lambda'=\lambda-v_sT$
In one period the source moves $v_sT$ towards you, so wavefronts are closer.
$\lambda'=\frac v{f_s}-\frac{v_s}{f_s}=\frac{v-v_s}{f_s}$
$\lambda=v/f_s$ , $T=1/f_s$ .
$f_o=\frac v{\lambda'}=\frac{f_sv}{v-v_s}$
Waves still travel at $v$ through the air.

Receding:

\lambda'=\lambda+v_sT

, giving

v+v_s

f_o=\frac{f_s\,v}{v\mp v_s}

$f_s$: source frequency.
$v$: wave speed.
$v_s$: source speed.

Observed frequency as the source speed changes (source 700 Hz)

The faster the source, the bigger the shift. Light shows the same effect: galaxies moving away show redshift.

Worked exampleA siren

700 Hz at 30 m s $^{-1}$ ; $v=340$ m s $^{-1}$ .

Goal: Approaching and receding frequencies.
Strategy: Formula, both signs.

$\frac{700\times340}{310}=768\text{ Hz}$
Approaching.
$\frac{700\times340}{370}=643\text{ Hz}$
Receding.

Answer

768

Hz;

643

Hz.

✓ CheckHigher then lower than 700 ✓.

Worked exampleFinding the source speed

500 Hz source heard at 520 Hz approaching.

Goal: $v_s$ .
Strategy: Rearrange.

$520(340-v_s)=500(340)$
Cross-multiply.
$v_s=340-326.9=13.1\text{ m s}^{-1}$
Solve.

Answer

13.1

m s

^{-1}

✓ Check

\frac{500\times340}{326.9}=520

✓.

Common misconceptionWrong sign Approaching must give a higher frequency: use

v-v_s

Pause and think

Light from a receding galaxy?

Summary

Approaching: higher; receding: lower.
$f_o=f_sv/(v\mp v_s)$ .

Doppler effect: Frequency change due to relative motion.

HistoryBuys Ballot's trumpeters. Christian Doppler predicted the effect in

1842

. In

1845

, Christophorus Buys Ballot tested it in the Netherlands by putting trumpeters on an open railway carriage. Musicians with perfect pitch on the platform heard the note rise and fall as the train passed.

Fun factBats use Doppler to hunt. Bats measure the Doppler shift of their own echoes to tell whether an insect is flying towards or away from them, and how fast. Some bats even lower their call frequency as they speed up, so the echo returns at the frequency their ears are most sensitive to.

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