Flick a rope tied to a wall and the wave races away from you, bounces back, and races through the wave you are still making. Get the timing right and something strange happens: the wave stops going anywhere. The rope still moves — furiously — but the pattern stands perfectly still. Nothing travels. Nothing is carried away. The energy just sloshes up and down on the spot. That is a standing wave, and it is really two ordinary waves caught in the act of passing through each other.
📘 What you need to know
A standing wave forms when two waves travel in opposite directions along the same line and superpose
The two waves must have the same frequency, the same wavelength and similar amplitudes
In practice the second wave is usually the reflection of the first
The wave pattern does not travel — crests and troughs only move up and down
Standing waves store energy; progressive (travelling) waves transfer energy
Superposition applies to all waves — transverse and longitudinal
What a standing wave looks like
Take a long spring, fix one end to a wall and shake the other end at just the right rate. The spring settles into a pattern of fat loops that seem frozen in place.
Certain points on the spring never move at all, no matter how long you shake it
Halfway between them, points swing up and down wildly, with the biggest amplitude of all
Every point is still oscillating with the same frequency — but each one has its own amplitude
The loops themselves stay put. Nothing slides along the spring
Half a period later the whole pattern has flipped over — it has not slid sideways. Crests become troughs on the spot.
Look at what does not happen in that picture. A normal wave marches across the room. This one just breathes in and out. If you photographed it twice, the crests would be in exactly the same places both times — only their height would differ. That single observation is the whole topic.
How a standing wave is made
Standing waves come straight from the principle of superposition: when two waves overlap, add their displacements at every point.
Send two identical waves along the same string in opposite directions. As they slide through one another, the sum of the two keeps changing shape — and it changes in a very particular way.
At t = 0 and t = T/2 the red and blue waves lie exactly on top of one another, so you see the dashes sitting on the red line. Watch the black curve: it grows, vanishes, and grows back upside down — always crossing zero in the same places. Those fixed crossings are what makes the wave look like it is standing still.
Add the two waves algebraically and the result splits neatly into a part that depends only on position and a part that depends only on time:
The standing wavey = 2A sin(2πx/λ) × cos(2πft)shape fixed in space × amplitude breathing in time
The bracket on the left picks out where the string can move and by how much. The bracket on the right makes the whole shape swell and shrink in step. There is no x and t locked together in a single bracket — and that is the mathematical fingerprint of a wave that goes nowhere.
The conditions
Not any two waves will do it. The pair must have:
the same frequency (so the pattern repeats in step)
the same wavelength (so the cancelling points stay in one place)
similar amplitudes (so the cancelling points actually cancel)
motion along the same line, in opposite directions
“Similar”, not “identical”. If one wave is a little weaker, you still get a standing wave — it is just a slightly scruffy one, because the points that should be dead still now wobble a bit. Exam mark schemes want the word similar for amplitude and same for frequency and wavelength. Do not swap them.
Where the second wave comes from
You rarely need two wave machines. Send a wave at a boundary and it reflects straight back at you, giving you the second wave for free.
At a fixed end the reflected wave comes back inverted (a crest returns as a trough)
The reflection travels back through the wave still arriving, and the two superpose
Shake at the right frequency and the pattern locks into place — a standing wave
The wall cannot move, so the string must be still there. The only way to keep it still is to send back a pulse that is the exact opposite of the one arriving.
🧪 Will these two waves make a standing wave?
Are they on the same line, going opposite ways? If not, stop. No standing wave.
Same frequency? Check. Different frequencies means the pattern drifts instead of standing.
Same wavelength? On one string this follows from the frequency, since v is fixed.
Similar amplitude? If yes, the still points are properly still and the pattern is crisp.
Standing waves store energy
This is the difference that examiners love. A travelling wave is a delivery van: it picks energy up at the source and drops it somewhere else. A standing wave is a child on a trampoline: energy shuttles between kinetic and potential, but it never leaves.
Progressive wave energy moves along
versus
Standing wave energy stays put
Feature
Progressive wave
Standing wave
Energy
Transferred along the wave
Stored, not transferred
Amplitude
Every point reaches the same amplitude in turn
Each point has its own fixed amplitude
Phase
Points within one wavelength differ by anything from 0 to 2π
Points are either exactly in phase or exactly in anti-phase
Still points
None — every point eventually moves
Some points never move at all
The pattern
Travels through the medium at speed v
Stays exactly where it is
The phase rule: on a progressive wave, phase difference varies smoothly with distance. On a standing wave, two points can only be in phase (0) or in anti-phase (π). Nothing in between. Ever.
WE 1
Two identical waves of amplitude 3.0 mm and wavelength 0.80 m travel in opposite directions along a string, where the wave speed is 12 m s⁻¹. Calculate the frequency of the resulting standing wave and its maximum amplitude, and state how fast the pattern travels along the string.
Step 1 — find the frequency from the wave equationf = v / λ = 12 / 0.80f = 15 HzStep 2 — the two waves add where they meet crest on crestmaximum amplitude = 2A = 2 × 3.0amplitude = 6.0 mmStep 3 — how fast does the pattern move?
It does not. speed of the pattern = 0Careful: the two waves that build it still race along at 12 m s⁻¹. It is only the pattern they make that stands still.
WE 2
A wave of amplitude 4.0 mm meets its reflection, which has been weakened to 2.5 mm. Determine the largest and smallest displacement any point on the string can reach, and comment on the pattern produced.
Step 1 — crest meets crest (constructive)largest = 4.0 + 2.5 = 6.5 mmStep 2 — crest meets trough (destructive)smallest = 4.0 − 2.5 = 1.5 mm6.5 mm and 1.5 mmStep 3 — comment
The still points are not fully still — they wobble with amplitude 1.5 mm.
This is exactly why the condition says similar amplitudes. Equal amplitudes give perfect cancellation; unequal ones give a blurred pattern.
WE 3
Two waves travel in opposite directions along the same string, on which the wave speed is 12 m s⁻¹. One has wavelength 0.60 m, the other 0.45 m. Explain whether a standing wave is formed.
Step 1 — find each frequencyf₁ = 12 / 0.60 = 20 Hzf₂ = 12 / 0.45 = 26.7 HzStep 2 — test against the conditions
Different wavelengths, so different frequencies.
Step 3 — concludeNo standing waveThe points where the waves cancel would keep sliding along the string, so no fixed pattern of still points can settle down.
💡 Top tips
Say the definition as a recipe: two waves, same line, opposite directions, same frequency and wavelength, similar amplitude.
Standing waves store energy, progressive waves transfer it. One word each, worth a mark each.
If asked for the speed of a standing wave, the pattern’s speed is zero — but the waves making it still travel at v.
Superposition works for longitudinal waves too. Sound in a pipe does exactly this.
Sketching one? Draw the shape and its mirror image as a dashed curve. That shows it flips, not slides.
⚠ Common mistakes
Saying the two waves need identical amplitudes — they need similar amplitudes
Claiming a standing wave transfers energy along the medium — it stores it
Thinking every point oscillates with the same amplitude, as on a progressive wave
Forgetting that the pattern flips over rather than sliding sideways
Assuming standing waves are only transverse — sound waves in pipes are longitudinal standing waves
Writing that the waves “cancel out everywhere” — they only cancel at certain fixed points
Quick recap: A standing wave is the superposition of two waves of the same frequency and wavelength and similar amplitude, travelling in opposite directions along the same line — usually a wave and its own reflection. The pattern does not move: it swells, flattens, and flips over on the spot. Energy is stored, not transferred, and each point on the wave keeps its own fixed amplitude.
You spotted them already in that first diagram: the points that never move, and the points that swing the furthest. They have proper names — nodes and antinodes — and once you can count them, you can work out wavelengths straight off a picture. That is the next page.
Standing waves not standing still for you?
Book a free meeting and we’ll go through superposition, reflection and past-paper standing wave questions together.
