The Doppler Effect: Grade 12 Sound, Light and Red Shift
A siren sounds higher as it comes towards you and drops the moment it passes. That drop is the Doppler effect, and it is the same physics that tells astronomers the universe is expanding.
This page covers the definition, the formula and all four cases, worked examples in both directions, the demonstration you can run in a classroom with no special apparatus, ultrasound and medicine, and red shift. Plus the one sign error that costs more marks in this chapter than everything else put together.
What the Doppler effect is
The Doppler effect is the apparent change in the frequency of a wave when the source and the observer are moving relative to each other.
Three words in that sentence carry marks.
| Word | Why it matters |
|---|---|
| Apparent | The source has not changed. A siren set to 780 Hz emits 780 Hz the whole way past you. What changes is what arrives |
| Frequency | Not volume. A sound also gets louder as it approaches, but that is not the Doppler effect |
| Relative | It does not matter which one is moving. A moving source and a moving observer produce the same effect |
Christian Doppler explained it in 1842, and it applies to every kind of wave: sound, light, radio, ultrasound.
Why the pitch changes
A stationary source sends out wavefronts as a set of evenly spaced circles. Everyone standing around it hears the same pitch, because the wavefronts arrive at the same rate in every direction.
Now let the source move. Each new wavefront leaves from a point slightly further along the path than the one before it. The circles stop being concentric.
| Where you stand | What happens to the wavefronts | What you hear |
|---|---|---|
| In front of the moving source | They bunch up. Shorter wavelength | Higher frequency, higher pitch |
| Behind the moving source | They spread out. Longer wavelength | Lower frequency, lower pitch |
The picture that makes it click is a swan swimming across a pond. The ripples pile up against its chest and trail out behind it in a long wake. Sound does exactly the same thing, and you cannot see it, which is the whole reason this topic needs a diagram.
The speed of the sound never changes. That is set by the air, not by the source. Only the wavelength and the frequency change, and they change together, because v = fλ and v is fixed.
The Doppler effect formula
One equation covers every case:
fL = ((v ± vL) ÷ (v ± vs)) × fs
| Symbol | Means | Unit |
|---|---|---|
| fL | Frequency the listener actually hears | Hz |
| fs | Frequency the source emits | Hz |
| v | Speed of sound in air, normally given as 340 m·s-1 | m·s-1 |
| vL | Speed of the listener | m·s-1 |
| vs | Speed of the source | m·s-1 |
The listener goes on top, the source goes on the bottom. Mixing those two up is the second most common error in this chapter.
The four cases
| What is moving | Direction | Equation | Pitch |
|---|---|---|---|
| Source, listener still | Towards the listener | fL = (v ÷ (v − vs)) fs | Up |
| Source, listener still | Away from the listener | fL = (v ÷ (v + vs)) fs | Down |
| Listener, source still | Towards the source | fL = ((v + vL) ÷ v) fs | Up |
| Listener, source still | Away from the source | fL = ((v − vL) ÷ v) fs | Down |
Do not memorise that table
There is one rule that replaces all four rows, and it works every single time.
If the gap between the source and the listener is closing, the fraction must come out bigger than 1. If the gap is opening, the fraction must come out smaller than 1.
That is it. Write the fraction down, look at it, and ask whether it is above or below 1. If the gap is closing and your fraction is 340 ÷ 370, you have the sign the wrong way round and you know it before you touch a calculator.
Work out whether the pitch goes up or down before you substitute anything. Your answer then has to agree with what your own ears already told you, and a sign error cannot survive that check.
Worked examples
A source moving past you
A delivery van sounds its hooter at 400 Hz and drives past a learner standing at the kerb at 25 m·s-1. Take the speed of sound as 340 m·s-1.
As it approaches. The gap is closing, so the fraction must be bigger than 1.
- fL = (v ÷ (v − vs)) fs
- fL = (340 ÷ (340 − 25)) × 400
- fL = (340 ÷ 315) × 400 = 1,0794 × 400
- fL = 431,7 Hz
Once it has passed. The gap is opening, so the fraction must be smaller than 1.
- fL = (340 ÷ (340 + 25)) × 400
- fL = (340 ÷ 365) × 400 = 0,9315 × 400
- fL = 372,6 Hz
The pitch drops by 59,1 Hz in the instant the van goes past, and the hooter itself never changed. That drop is what you hear at a robot every day of your life.
You moving past the source
A learner cycles at 8 m·s-1 straight towards a school bell ringing steadily at 512 Hz.
- fL = ((v + vL) ÷ v) fs
- fL = ((340 + 8) ÷ 340) × 512 = (348 ÷ 340) × 512
- fL = 524,0 Hz
Cycling away from the bell at the same speed:
- fL = ((340 − 8) ÷ 340) × 512 = (332 ÷ 340) × 512
- fL = 500,0 Hz
Notice the listener speed sits on top this time, and the source speed is gone entirely, because the bell is not moving.
Working backwards, which is the hard one
This is the version that appears in the harder exam papers, and it catches people out because it needs two equations at once.
A stationary observer at a level crossing measures a train siren at 560 Hz as the train approaches and 500 Hz after it has passed. Find the speed of the train and the true frequency of the siren.
Write both cases and divide one by the other. The unknown fs cancels.
- Approaching: 560 = (340 ÷ (340 − vs)) fs
- Receding: 500 = (340 ÷ (340 + vs)) fs
- Dividing: 560 ÷ 500 = (340 + vs) ÷ (340 − vs) = 1,12
- 340 + vs = 1,12(340 − vs) = 380,8 − 1,12vs
- 2,12vs = 40,8
- vs = 19,2 m·s-1, which is about 69 km·h-1
Now put that back into either equation:
- 500 = (340 ÷ (340 + 19,2)) × fs
- fs = 528,3 Hz
Always substitute back into the other equation to check. 528,3 Hz through the approaching case gives 560 Hz, which is where you started, so both answers are right.
Where this fits in the curriculum
| Subject | Physical Sciences |
|---|---|
| Grade | 12 |
| Term | 2 |
| Topic | Waves, sound and light |
| Status | Examinable theory. No formal assessment attached |
| Paper | Physics, Paper 1 |
| Teaching time | 4 hours in the annual teaching plan |
There is no prescribed practical for this chapter, which is exactly why it gets skimmed and then costs marks in the finals. Four hours is not much for two units, and the chapter usually carries somewhere between 13 and 17 marks in Paper 1, almost always including a calculation and a red shift explanation.
The syllabus asks for the Doppler effect with sound and ultrasound, and with light, as red shift and the evidence for an expanding universe. Both halves get examined.
The demonstration, and it takes two minutes
You do not need a laboratory for this one. You need something that makes a steady sound and a piece of strong string.
Apparatus
| Item | Qty | Note |
|---|---|---|
| A small buzzer or a tuning fork | 1 | It has to emit one steady note, not a beep |
| Strong string | About 1 m | Not cotton. It carries the whole load |
| A clear space | 3 m radius | Outside, or a cleared classroom |
Method
- Tie the string to the buzzer or the fork securely and test the knot by pulling hard on it. The whole class is standing around the swinging end.
- Switch the buzzer on, or strike the tuning fork on something soft, and hold it still. Let everyone listen to the steady note first. They need the reference.
- Swing it in a wide horizontal circle above your head at a steady speed.
- Ask the class what they hear.
What they should hear
A rising and falling wail, once per revolution. The pitch peaks at the moment the source is moving straight towards them and bottoms out half a turn later when it is moving straight away.
The person swinging it hears nothing change at all, because the source is not moving relative to them. Get them to say so out loud. It is the cleanest possible proof that the word in the definition is relative.
Safety
The knot is the only real risk and it is a serious one. A tuning fork coming off a string at head height in a full classroom is an injury. Check it, then check it again, and keep the swinging radius clear.
Ultrasound, and why this chapter is in medicine
Ultrasound is sound above 20 kHz, too high for human hearing, and it behaves like any other sound wave. Which means it shows the Doppler effect, and that turns out to be extremely useful.
A Doppler flow meter is held against the skin at an angle to a blood vessel. It sends ultrasound in, the red blood cells reflect it, and a detector picks up the echo. The cells are moving, so the echo comes back at a different frequency from the one that was sent.
| Echo comes back | The blood is |
|---|---|
| Higher than the sent frequency | Flowing towards the probe |
| Lower than the sent frequency | Flowing away from the probe |
| The same | Not flowing, which is the finding that matters clinically |
Because the speed of sound in blood is known, the size of the shift gives the speed of the flow. Three uses come up in exams:
- Measuring blood flow speed and finding blockages in arteries
- Echocardiography, imaging the structure and the valves of a beating heart
- Detecting a foetal heartbeat, which is the one everybody has heard of
None of it involves radiation, which is why it is safe to use in pregnancy where an X-ray would not be.
Where else it turns up
| Application | What is moving |
|---|---|
| Traffic speed cameras | Radio waves bounce off the car; the shift gives the speed |
| Weather radar | Rain and hail moving inside a storm cell, which is how rotation is spotted |
| Bats and dolphins | The shift in their own returning call tells them whether the prey is closing or fleeing |
| Astronomy | Whole galaxies, which is the next section |
Red shift and the expanding universe
Light does the same thing sound does. A light source racing away from you stretches its waves out, and a light source racing towards you bunches them up.
| The source is | Wavelength | Spectral lines move towards | Called |
|---|---|---|---|
| Moving away | Longer | The red end of the spectrum | Red shift |
| Moving towards | Shorter | The blue end of the spectrum | Blue shift |
Red shift is the lengthening of the wavelengths of the spectral lines from distant stars and galaxies over time. That is the definition to learn word for word.
Spectral lines are the key to the whole thing
Every element gives off light at its own fixed set of frequencies. Hydrogen in a laboratory in Johannesburg produces exactly the same pattern of lines as hydrogen in a galaxy a billion light years away.
So astronomers know precisely where the lines should be. When the whole pattern arrives shifted towards the red, the pattern has not changed. The galaxy is moving away.
What that told us
In 1868 William Huggins found that the light from Sirius was shifted towards the red, and he was the first to explain it as the Doppler effect on a star moving away from us.
Vesto Slipher then found that most galaxies outside our own show red shift. Between 1922 and 1924, Edwin Hubble proved those distant galaxies really were separate galaxies, and worked out something remarkable: the further away a galaxy is, the faster it is receding. That ratio of speed to distance is Hubble's Law.
Almost everything out there is moving away from us, and the far things are moving away faster. That is the observational evidence that the universe is expanding, and it is where the Big Bang theory comes from.
Whether it expands forever or eventually contracts again is still an open question, and an exam is allowed to ask you what the evidence shows rather than what the answer is.
One thing to be careful about
The equation on your data sheet is for sound. Red shift in Grade 12 is examined as an explanation, not as a substitution into fL = ((v ± vL) ÷ (v ± vs)) fs.
Where a calculation on starlight does come up, it works from the fractional change in wavelength rather than from the sound equation. Do not put the speed of light into the sound formula. It is a guaranteed nought.
Looking up from Southern Africa
Long before anyone measured a spectral line, people here were reading the sky carefully enough to run an agricultural year off it.
| Called | Known internationally as |
|---|---|
| IsiLimela, the Seven Sisters | The Pleiades |
| The Three Zebras | Orion's Belt |
| The Giraffes | The Southern Cross and the Pointers |
| The Lion | Betelgeuse |
The Venda called the Giraffes Thutlwa, rising above the trees. When they climbed above the southern treeline in October, it was the signal to finish planting.
IsiLimela disappears in winter. When one star reappears, then three, then the whole cluster, the year was said to be renewed and it was time to start preparing the soil.
None of that is Doppler physics. It is the same instinct, which is to watch what the sky is doing and work out what it means.
The mistakes that cost the marks
| The mistake | What to do instead |
|---|---|
| Getting the sign the wrong way round. Far and away the biggest one | Decide up or down first. Then check the fraction is above 1 for a closing gap and below 1 for an opening one |
| Putting the source speed on top | Listener on top, source on the bottom. Every time |
| Saying the frequency of the siren changes | It does not. Say apparent or observed frequency. The examiner is looking for that word |
| Saying the speed of the sound changes | Only the medium sets the speed. Wavelength and frequency change, v does not |
| Leaving the speed in km·h-1 | Divide by 3,6 first. It is a free mark and it is thrown away constantly |
| Saying red shift means the star has turned red | It means the spectral lines have moved towards the red end. The star looks the same colour it always did |
| Confusing louder with higher | A siren gets louder as it comes closer because you are nearer to it. That is not the Doppler effect |
If the demonstration does not work
| What happens | Why |
|---|---|
| Nobody hears a change | Not swinging fast enough. The shift scales with speed, and a slow swing gives a shift too small to notice |
| The sound is there but the pitch is flat | The buzzer is beeping rather than holding one note. It has to be continuous |
| The tuning fork goes quiet almost at once | Normal. A struck fork fades in seconds, which is why a buzzer works better for a full class |
| Only the front row hears it | Too much background noise, or the swing radius is too small. Take it outside |
| The learners hear a change but describe it as louder and softer | Fair enough, and it is worth stopping on. The volume genuinely does change too. Ask them to listen for the note rather than the level |
Free worksheet and marking memo
Both free, no sign up, straight to the PDF.
- Learner worksheet, 40 marks, with multiple choice, both directions of calculation, a working-backwards question and red shift
- Marking memorandum, with full working, the mark breakdown line by line and a note on the six places learners drop marks in this chapter
Related pages
- Constructive and destructive interference. The ripple tank practical, and the same wavefront thinking
- Snell's Law and refraction. Grade 11 geometrical optics
- The law of reflection. Grade 11, Term 2
Apparatus
The demonstration needs almost nothing, and what it does need lives in the sound and waves range.
A small buzzer is the better choice for a full class than a tuning fork, because it holds its note for as long as you need it. A struck fork is quieter and it has faded before the back row has worked out what they are listening for.
If you are teaching the whole waves and sound section, the wave form helix shows longitudinal and transverse pulses on the same piece of apparatus, which is worth having in the room when you get to this chapter.