You have probably noticed that the pitch of a siren changes as an ambulance rushes past. This is the Doppler effect — a fundamental property of all waves. In this chapter you will quantify the frequency shift for sound and explore how astronomers use the same principle with light to map the expansion of the universe.
5.1 The Doppler Effect with Sound
Definition
Doppler effect
The Doppler effect is the change in the observed frequency (or pitch) of a wave when the source of the wave and the observer are moving relative to each other. When source and observer approach each other, the observed frequency is higher than the emitted frequency. When they move apart, the observed frequency is lower. This applies to all waves including sound and light.
When a sound source moves toward a listener, it 'catches up' with the sound waves it emits, compressing them. Compressed waves have shorter wavelengths and therefore higher frequency — the pitch sounds higher. When the source moves away, waves are stretched, wavelength increases and pitch decreases. The same effect occurs when the listener moves: moving toward the source means the listener encounters wavefronts more frequently; moving away means less frequently. Speed of sound in air at 20 °C ≈ 340 m·s⁻¹.
Formula
Doppler formula
fL = observed frequency at listener (Hz), fs = frequency emitted by source (Hz), v = speed of sound in medium (m·s⁻¹), vL = speed of listener (m·s⁻¹), vs = speed of source (m·s⁻¹). Upper signs (+vL, −vs) when source/listener move TOWARD each other; lower signs (−vL, +vs) when moving APART.
SI unit: Hz
Watch Out
Sign convention: the upper signs (+ in numerator, − in denominator) apply when source and listener approach each other. The lower signs apply when they move apart. Draw a diagram to decide which applies — do not guess.
Worked Example
An ambulance siren emits sound at 850 Hz and moves toward a stationary observer at 30 m·s⁻¹. The speed of sound is 340 m·s⁻¹. Calculate the frequency heard by the observer.
Given
- fs = 850 Hz
- vs = 30 m·s⁻¹ (toward observer)
- vL = 0 (stationary)
- v = 340 m·s⁻¹
Find
fL
Solution
- 1Source moves toward stationary listener → use upper sign in denominator: (v − vs)
- 2fL = fs × (v + vL) / (v − vs)
- 3fL = 850 × (340 + 0) / (340 − 30)
- 4fL = 850 × 340 / 310
- 5fL = 850 × 1,0968
- 6fL ≈ 932,3 Hz
Worked Example
The same ambulance (fs = 850 Hz, vs = 30 m·s⁻¹) has now passed the observer and is moving away. Calculate the new frequency heard.
Given
- fs = 850 Hz
- vs = 30 m·s⁻¹ (away from observer)
- vL = 0
- v = 340 m·s⁻¹
Find
fL
Solution
- 1Source moves away → lower signs: (v + vs)
- 2fL = 850 × (340 − 0) / (340 + 30)
- 3fL = 850 × 340 / 370
- 4fL ≈ 780,5 Hz
Practice Question
A stationary police siren emits 960 Hz. A motorist drives toward the siren at 20 m·s⁻¹. Speed of sound = 340 m·s⁻¹. Calculate the frequency heard by the motorist.
(4 marks)
Practice Question
A train whistle emits 800 Hz. The train moves away from a stationary listener at 25 m·s⁻¹. Speed of sound = 340 m·s⁻¹. Calculate fL.
(4 marks)
Exam Tip
Exam tip: always check your answer makes physical sense. If source approaches, fL > fs. If source moves away, fL < fs. An answer that violates this indicates a sign error.
5.2 Light: Red-shift, Blue-shift and the Expanding Universe
The Doppler effect applies to light as well as sound. However, for light there is no medium, so it is the relative velocity between source and observer that matters. When a star or galaxy moves toward Earth, its light is blue-shifted (wavelengths compressed → shorter → shifted toward the blue end of the spectrum). When it moves away, the light is red-shifted (wavelengths stretched → longer → shifted toward the red end). Astronomers measure the positions of spectral lines in starlight and compare them to laboratory standards to determine how fast and in which direction the object is moving.
Red-shift vs Blue-shift
| Property | Red-shift | Blue-shift |
|---|---|---|
| Relative motion | Source moves away from observer | Source moves toward observer |
| Wavelength change | Wavelength increases (longer) | Wavelength decreases (shorter) |
| Frequency change | Frequency decreases | Frequency increases |
| Observed spectrum | Spectral lines shift toward red end | Spectral lines shift toward blue end |
| Cosmological evidence | Distant galaxies all red-shifted → universe expanding | Andromeda galaxy (approaching Milky Way) |
Edwin Hubble (1929) discovered that virtually all distant galaxies show red-shift, and the further away the galaxy, the greater the red-shift. This means all galaxies are moving away from each other — the universe is expanding. This is one of the key pieces of evidence for the Big Bang theory. The magnitude of the red-shift allows astronomers to estimate the recession speed of distant galaxies.
Practical Applications of the Doppler Effect
- Radar speed guns: police and traffic cameras emit microwave pulses; the Doppler shift of the reflected signal gives vehicle speed.
- Sonar: submarines and depth sounders emit ultrasound pulses; the frequency shift of the echo reveals the speed and direction of moving underwater objects.
- Medical ultrasound (Doppler echocardiography): doctors measure the Doppler shift of ultrasound reflected off moving red blood cells to map blood flow and detect blockages.
- Astronomy: red-shift measurements determine recession velocities of stars and galaxies.
- Weather radar: Doppler radar detects the speed and direction of raindrops to identify tornado rotation.
Worked Example
A galaxy emits light at a wavelength of 486 nm (blue hydrogen line). On Earth the same line is detected at 500 nm. State whether this galaxy is approaching or receding, and explain your reasoning.
Given
- Emitted wavelength λ_s = 486 nm
- Observed wavelength λ_L = 500 nm
Find
Direction of motion
Solution
- 1λ_L > λ_s → observed wavelength is longer than emitted wavelength
- 2Longer wavelength = shifted toward red = red-shift
- 3Red-shift indicates the galaxy is moving AWAY from Earth
Practice Question
Explain how Doppler ultrasound is used in medicine to measure blood-flow speed. Your answer must refer to the Doppler effect.
(5 marks)
Practice Question
All distant galaxies show red-shifted spectra. What does this tell us about the universe?
(3 marks)
Exam Tip
Exam tip: for the light Doppler questions you are NOT expected to calculate frequencies — only explain blue-shift/red-shift qualitatively and describe the evidence for the expanding universe.
5.3 Doppler Calculations — Mixed Practice
Many examination questions combine a moving source AND a moving listener. The principle is the same: apply upper signs throughout if both are moving toward each other, lower signs if both are moving apart. If they move in opposite directions (one toward, one away) treat each independently using the appropriate sign.
Worked Example
A factory siren emits 1 200 Hz. A worker on a bicycle approaches the siren at 5 m·s⁻¹ while the siren is simultaneously mounted on a vehicle moving toward the worker at 10 m·s⁻¹. Speed of sound = 340 m·s⁻¹. Calculate the frequency heard by the worker.
Given
- fs = 1 200 Hz
- vL = 5 m·s⁻¹ (toward source)
- vs = 10 m·s⁻¹ (toward listener)
- v = 340 m·s⁻¹
Find
fL
Solution
- 1Both source and listener move toward each other → use upper signs throughout
- 2fL = fs × (v + vL) / (v − vs)
- 3fL = 1 200 × (340 + 5) / (340 − 10)
- 4fL = 1 200 × 345 / 330
- 5fL ≈ 1 254,5 Hz
Note
The Doppler formula for sound assumes the medium (air) is stationary. Wind affects the effective speed of sound and would require adjustments, but this is beyond the Grade 12 CAPS syllabus.
Practice Question
A bus horn emits 700 Hz and the bus travels away from a stationary observer at 15 m·s⁻¹. Speed of sound = 340 m·s⁻¹. Calculate fL.
(4 marks)
Practice Question
A stationary speaker emits 500 Hz. A student runs away from the speaker at 4 m·s⁻¹. Speed of sound = 340 m·s⁻¹. Calculate the frequency heard by the student.
(4 marks)