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Edexcel A Level Physics · Topic 5 Waves and the Particle Nature of Light · Paper 2
Standing Waves (Stationary Waves)
Also on the AQA spec (7408, 3.3.1.3) and the OCR A spec (H556, 4.4.4).
In this lesson you'll learn to: explain how a standing wave forms from a wave and its
reflection, state the three conditions it needs, identify nodes and antinodes and explain
them using phase, interference and path difference, use the λ/2 spacing, compare
progressive and standing waves, draw the harmonics on a string and in open and
closed tubes, and find the speed of light with a microwave oven.
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Prefer to watch? Scroll to the bottom for the video on this topic
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A video walkthrough is coming soon — it will appear at the bottom of this page
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Start here — have a play
Don't read anything yet. Watch the blue wave travel to the fixed end and come back as the orange wave. The bold navy line is their sum: the only thing you would actually see. Then tap the string at a node and at an antinode: are the two waves in phase or in anti-phase there? Tabs 2 and 3 do the same with sound in an open tube and a closed tube: what is at each end? On tab 4, melt some chocolate and measure the speed of light.
↔ Swipe the picture sideways to see all of it
Show
Harmonic
Speed
Probe
or tap the picture
Incident wave here
0
Reflected wave here
0
Resultant (the sum)
0
Phase difference
–
Sound is longitudinal: the air particles (the strip under the graph) move back and forth along the tube. The graph plots their displacement, the way exam questions draw it. A closed end is always a node (the air there can't move); an open end is always an antinode.
Superposition: at every point, the resultant displacement is the sum of the two waves' displacements. Watch the probe: the navy dot is always the blue arrow plus the orange arrow. Displacements are in units of one wave's amplitude, so the most the sum can be is 2.
Show
↔ Swipe the picture sideways to see the whole bar
Why it works: microwaves reflect off the metal walls, so two waves travel in opposite directions and a standing wave forms. The chocolate melts fastest at the antinodes (the hot spots), and adjacent antinodes are λ/2 apart. The frequency is printed on the oven's label.
⭐ Key things to know
Three conditions for a standing wave
- Two waves superpose (coincide).
- They travel in opposite directions, because one is the reflection of the other.
- So they are coherent: same frequency (and wavelength), similar amplitude.
Coherent
Two waves with a constant phase difference (so they must have the same frequency).
Superposition
When two waves meet, the resultant displacement is the sum of the individual displacements. Superposition and interference: use them as the same word.
1st harmonic: string and tubes
- String: node at each end. Length = λ/2.
- Open tube: antinode at each end. Length = λ/2.
- Closed tube: node at the closed end, antinode at the open end. Length = λ/4. Odd harmonics only.
Progressive vs standing
Progressive waves transfer energy.
A standing wave has no net transfer of energy, and its nodes and antinodes stay at fixed points.
ANTINODE
- the two waves are in phase
- constructive interference
- path difference = a whole number of wavelengths, nλ
- oscillates between maximum + and − amplitude
- most energy: brightest, loudest, highest, hottest
NODE
- the two waves are in anti-phase (180° apart)
- destructive interference
- path difference = an odd number of half wavelengths, (n + ½)λ
- no movement: zero displacement all the time
- least energy: darkest, quietest, lowest, coldest
Distances: adjacent nodes are λ/2 apart. Adjacent antinodes (two hot spots, two loudest points) are λ/2 apart. A node to the next antinode is λ/4.
▲ Slowed right down. We let the reflected wave leave at the source end, so the two waves stay easy to see.
Two waves, one standing wave
The notes give you the three conditions, what happens at a node and an antinode, the λ/2 rule, and the microwave oven method, written the way the mark scheme wants it.
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Revision notes
What is a standing wave?
A standing wave (also called a stationary wave: Edexcel uses both names, AQA and OCR say “stationary”) is a wave with no net transfer of energy and a pattern of nodes and antinodes at fixed positions.
How a standing wave forms: three conditions
1
Two waves superpose
Two waves of the same type meet (coincide) and superpose.
2
Opposite directions
They travel in opposite directions, because one is the reflection of the other.
3
So they are coherent
A wave and its own reflection have the same frequency and wavelength, and a similar amplitude.
⭐ Two definitions to learn word for word
Coherent: two waves with a constant phase difference (so they have the same frequency).
Superposition: when two waves meet, the resultant displacement is the sum of the individual displacements.
In these answers, superposition and interference mean the same thing.
Reflection at a fixed end
When a wave on a string hits a fixed end (like the bridge of a guitar), it reflects upside down: a 180° phase change (π rad, half a cycle). At the fixed end the two waves are always equal and opposite, so they cancel, and there is always a node there. The string is fixed, so it couldn't move anyway.
Nodes and antinodes
| Antinode | Node | |
|---|---|---|
| The two waves are | in phase (phase difference 0) | in anti-phase (phase difference 180°) |
| Interference | constructive | destructive |
| Path difference* | a whole number of wavelengths, nλ | an odd number of half wavelengths, (n + ½)λ |
| Displacement | oscillates between maximum + and − amplitude | zero all the time: no movement |
| Energy | most: brightest, loudest, highest, hottest | least: darkest, quietest, lowest, coldest |
* For two waves that set off in phase. On a string, count the half-wavelength flip at the fixed end as part of the path difference.
⭐ The distances
Adjacent nodes are λ/2 apart. Adjacent antinodes (two hot spots, two loudest points) are λ/2 apart. A node to the next antinode is λ/4.
So one “loop” of a standing wave is half a wavelength, and two loops make one wavelength.
💡 Phase in a standing wave (a favourite multiple-choice question)
Every point between two adjacent nodes moves in phase: they all reach their highest point together, just with different amplitudes. Points on either side of a node are in anti-phase (180°). This is different from a progressive wave, where the phase changes all the way along.
Progressive waves vs standing waves
| Progressive wave | Standing wave | |
|---|---|---|
| Energy | transfers energy | no net transfer of energy |
| Pattern | moves along | nodes and antinodes at fixed points |
| Amplitude | the same at every point | varies: zero at nodes, maximum at antinodes |
| Phase | changes continuously along one wavelength | in phase between adjacent nodes; anti-phase across a node |
| Wavelength | distance between adjacent points in phase | twice the distance between adjacent nodes |
Where you meet standing waves
- Stringed instruments. A guitar string is fixed at both ends, so there is a node at each end.
- The microwave oven. Microwaves reflect off the metal walls, giving hot spots (antinodes) and cold spots (nodes).
- Sound in a pipe and wind instruments: the sound reflects at the end of the air column.
Strings, open tubes and closed tubes OCR A 4.4.4 (e)
It is the same physics every time. Only the ends change: a fixed end of a string and a closed end of a tube are always nodes; an open end of a tube is always an antinode. In a tube the wave is sound, which is longitudinal, so the graph shows the displacement of the air particles along the tube.
| The ends | 1st harmonic | Harmonics you get | |
|---|---|---|---|
| String | node at each end | L = λ/2, so λ = 2L | all: f, 2f, 3f, 4f… |
| Open tube | antinode at each end | L = λ/2, so λ = 2L | all: f, 2f, 3f, 4f… |
| Closed tube | node at the closed end, antinode at the open end | L = λ/4, so λ = 4L | odd only: f, 3f, 5f… |
💡 Why a closed tube only has odd harmonics
It must have a node at one end and an antinode at the other, so it can only fit an odd number of quarter wavelengths: λ/4, 3λ/4, 5λ/4… That gives f, 3f, 5f. The next one up from the 1st harmonic is the 3rd, not the 2nd.
Measuring the speed of light with a microwave oven
1
Turntable out
Take out the turntable and heat a bar of chocolate (or cheese) until it melts in spots.
With the turntable in, the bar moves through the hot spots and melts evenly.2
Measure, then double
The melted spots are antinodes. Adjacent spots are λ/2 apart, so double the spacing to get λ.
Measure across several spots and divide, to cut the uncertainty.3
c = f λ
Read the frequency off the oven's label (usually 2.45 GHz) and use c = f λ.
6.1 cm spacing → λ = 0.122 m → c = 2.99 × 10⁸ m s⁻¹.⚠ Don't confuse: a diffraction grating is not a standing wave
A diffraction grating (and Young's double slit) also gives bright and dark points by interference, but the waves spread out from the slits in the same direction, towards the screen. Nothing is reflected back on itself, so there are no nodes and antinodes on a string or in a box: it is not a standing wave.
Covered separately The speed of a wave on a string
v = √(T/μ) (Edexcel 68), and Core Practical 7 (Edexcel 69; AQA Required Practical 1;
OCR A 4.4.4(e)(ii) and (g)) build on this lesson.
How to write it in the exam
“Explain how a standing wave forms on the string.”
The wave reflects at the fixed end, so two coherent waves travel
in opposite directions. They superpose. Where they are
in phase they interfere constructively and form an antinode;
where they are in anti-phase they interfere destructively and form a
node.
the two waves what they do nodes and antinodes
Q1 · A string is fixed at one end and made to vibrate at the other. Explain how a standing wave forms on the string. [4]
- The wave is reflected at the fixed end (1)
- So two coherent waves travel in opposite directions (1)
- They superpose / interfere (1)
- In phase gives constructive interference, an antinode; anti-phase gives destructive interference, a node (1)
Q2 · Explain why a point at a node does not move. [2]
- The two waves arrive in anti-phase, so their displacements are always equal and opposite (1)
- The resultant displacement is their sum, which is zero at all times: destructive interference (1)
Q3 · A student takes the turntable out of a microwave oven and heats a slab of chocolate. Adjacent melted spots are 6.1 cm apart. The oven's frequency is 2.45 GHz. Calculate the speed of the microwaves. [3]
- Adjacent hot spots are λ/2 apart, so λ = 2 × 6.1 = 12.2 cm = 0.122 m (1)
- c = f λ = 2.45 × 10⁹ × 0.122 (1)
- c = 2.99 × 10⁸ m s⁻¹ (1)
Where marks go missing: using 6.1 cm as the wavelength. That gives 1.49 × 10⁸ m s⁻¹, exactly half the right answer. Also: forgetting GHz means × 10⁹, and cm to m.
Q4 · State two differences between a progressive wave and a standing wave. [2]
- A progressive wave transfers energy; a standing wave has no net transfer of energy (1)
- Any one more: in a standing wave the amplitude varies (zero at nodes), but it is the same everywhere on a progressive wave; or the nodes and antinodes are at fixed points; or points between adjacent nodes are all in phase (1)
Q5 · Two points on a standing wave are on either side of a node. State the phase difference between them. [1]
180° (π rad): they are in anti-phase.
Q6 · A tube closed at one end is 0.33 m long. It resonates at its 1st harmonic. The speed of sound is 340 m s⁻¹. Calculate the frequency. [3]
- Closed tube, 1st harmonic: L = λ/4, so λ = 4 × 0.33 = 1.32 m (1)
- f = v ÷ λ = 340 ÷ 1.32 (1)
- f = 260 Hz (258 Hz) (1)
Where marks go missing: using λ = 2L as if the tube were open, which gives 515 Hz, about double the right answer.
✅ Check your answer
For “how does it form”, have all four: reflection, two coherent waves in opposite directions, superpose, then in phase → antinode and anti-phase → node. For any distance question, ask: adjacent nodes or antinodes? Then it is λ/2.
📘 Now practise it
Fill in the standing-waves gap fill (nodes, antinodes, phase, path difference), then try the exam questions, with answers.
Edexcel A Level Physics · Topic 5 Waves · standing wavesThe free practice sheet for this lesson is on its way. Until then, the five exam questions above have full answers.
✅ Can you do it? Tick as you go
0% complete🎉 Nice work! You've ticked off every objective for this spec point.
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Prefer to watch? Here's the whole thing
Video walkthrough — coming soon
Everything above, explained out loud — useful for a last-minute recap, or if you'd rather hear it than read it.
Everything above, explained out loud — useful for a last-minute recap, or if you'd rather hear it than read it. The video is on its way.
🎬
Video coming soon
The walkthrough for this lesson is being filmed and will appear right here. Until then, the animation and the notes above cover every mark.
Now put it into practice
Practise nodes, antinodes, phase and the microwave oven method, with answers.
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