Waves and sound · Model
Transverse waves, reflection and superposition
Two waves arrive at the same patch of water and each one wants the surface somewhere different. The surface obeys both at once, and the arithmetic for that is the whole lesson.
Start here
Two stones, one pond, and patches of water that never move.
Drop two stones into still water at the same moment, a metre apart. Two sets of rings spread out and run into each other. Where they cross, some patches of the surface heave twice as far as either ring on its own. Others sit almost dead flat while rings pour through them from both sides.
At a spot where the two sets of ripples are crossing, what is the water doing?
The surface cannot be in two places, so it does the only thing left: it goes to the sum of what each wave was asking for. A 6 mm lift and a 6 mm lift make a 12 mm lift. A 6 mm lift and a 6 mm drop make no movement at all. Adding the two displacements at every point is called superposition, and it is the whole of what happens where waves meet.
A wave on water is transverse: the surface moves up and down, at right angles to the direction the wave is travelling. That up-or-down amount at a point is its displacement, and it is a displacement rather than a distance because it has a direction — up counts as positive, down as negative.
Waves on water also reflect. Put a barrier across a tank and a wave train bounces back off it and travels the other way, still a wave, still the same wavelength. And when two waves are in the same water at the same time, the surface takes the sum of both displacements at every point. Two crests arriving together give a bigger crest — the waves add. A crest arriving with a trough of the same size gives flat water — the waves cancel. Both are superposition; nothing else is happening.
At the bench · two wave trains sent along one channel
Set two waves going. Read the third.
Move a control to begin
Two paddles at the same end of a narrow channel, both laying crests 500 mm apart. Set the height of each, and set whether the second one starts crest-with-crest or crest-with-trough.
Commit first. Two waves of exactly the same height meet crest on trough. What is the water doing where they overlap?
8 mm
6 mm
How B arrives
Wave A alone
—
Wave B alone
—
Where they meet
—
What the two are doing
—
The relationship · a bar, not a triangle
Where two waves meet, the surface takes the sum of both displacements.
Crest on crest: R = a + b
Crest on trough: R = a − b
Every height is measured from the still level, in millimetres.
Worked example · one step at a time
Two ripples meet crest on crest. One is 8 mm high, the other is 5 mm. How high is the water where they meet?
Step 0 of 5
Convert
8 mm stays 8 mm · 5 mm stays 5 mm
Both heights are already in millimetres, so there is nothing to convert.
Formula
R = a + b
Crest on crest, so the two displacements are both upwards and they add.
Insert
R = 8 mm + 5 mm
Both heights are measured from the still level.
Fine-tune
8 + 5 = 13
Millimetres added to millimetres leave millimetres.
Answer
R = 13 mm
Thirteen millimetres above the still level, and only while the two are overlapping.
Worked example · one step at a time
A crest 1.2 cm high meets a crest 7 mm high. How high is the water where they meet?
Step 0 of 5
Convert
1.2 cm × 10 = 12 mm
Two heights cannot be added until they are in the same unit, and a centimetre is ten millimetres.
Formula
R = a + b
Crest on crest, so both displacements are upwards and they add.
Insert
R = 12 mm + 7 mm
The converted height goes in. The 1.2 never does.
Fine-tune
12 + 7 = 19
Millimetres added to millimetres leave millimetres.
Answer
R = 19 mm
Add 1.2 to 7 and you get 8.2 of nothing at all — the units were never the same.
Your turn · the same five steps
Your two waves: 8 mm and 6 mm, arriving crest on crest.
Write all five lines before you check. Both heights are the ones your own channel is showing.
The five lines, marked
Convert
8 mm stays 8 mm · 6 mm stays 6 mm
Both sliders read in millimetres, so there is nothing to convert.
Formula
R = a + b
Crest on crest, so both displacements are upwards and they add.
Insert
R = 8 mm + 6 mm
Both heights come from the sliders above, measured from the still level.
Fine-tune
8 + 6 = 14
Millimetres and millimetres leave millimetres.
Answer
R = 14 mm
14 millimetres from the still level, and only while the two are overlapping.
The five lines above give 14 mm where the two meet.
A crest 0.9 cm high meets a crest 4 mm high, crest on crest.
This one needs the Convert line to do some work.
The five lines, marked
Convert
0.9 cm × 10 = 9 mm
Two heights cannot be added until they share a unit, and a centimetre is ten millimetres.
Formula
R = a + b
Crest on crest, so both are upwards and they add.
Insert
R = 9 mm + 4 mm
The converted height goes in. The 0.9 never does.
Fine-tune
9 + 4 = 13
Millimetres added to millimetres leave millimetres.
Answer
R = 13 mm
Add 0.9 to 4 and you get 4.9 of nothing at all.
The five lines give 13 mm. The whole question turned on the first one.
Key fact
Where two waves overlap, the surface takes the sum of the two displacements at every point. Crest on crest adds and gives a bigger wave; crest on trough of the same size cancels and gives flat water. Both waves then carry on past the overlap exactly as they were.
Think again
“When two waves cancel, they destroy each other.”
Cancelling is something that happens to a place, not to the waves. While the two are on top of one another the surface there is flat, because one wave is asking it to rise by as much as the other is asking it to drop. Keep watching and both waves come out the far side with their original heights, wavelengths and directions, as though nothing had happened. Two ripples crossing a pond do not knock lumps out of each other; they pass straight through.
“If the water is flat, the energy has gone.”
Energy is not stored point by point in the surface, and cancelling does not remove any. Wherever two waves cancel there are other places, half a wavelength away, where the same two waves add — the energy is moved about the pattern, not deleted. That is why the crossing rings from two stones show still patches and violent patches side by side: the total is unchanged, and it has simply been dealt out unevenly.
Mastery ladder
Not started yet.
Rungs 3 and 4 you mark yourself.
Rung 1 · Calculate
Two waves meet crest on crest. One has an amplitude of 6 mm, the other 4 mm. What is the amplitude of the water where they overlap?
Rung 2 · The one that catches people
Two wave trains of amplitude 7 mm meet exactly crest on trough. A student says the two waves have destroyed each other. Which statement is right?
Rung 3 · Explain
Two stones are dropped into a pond a metre apart at the same moment. Explain why some patches of the crossing pattern heave twice as far as either ripple on its own while other patches stay almost flat.
Rung 4 · Take it somewhere new
A harbour wall has two gaps in it. Inside the harbour, boats moored at some spots rock hard on a swell while boats a few metres away barely move, and the pattern is the same every time that swell runs. Explain what is going on, then say what would change if one gap were blocked up.
Key note
Waves on water are transverse: the surface is displaced at right angles to the direction of travel, upwards or downwards. Waves reflect off a barrier and travel back. Where two waves overlap, the surface takes the sum of the two displacements at every point: crest on crest adds to a bigger wave, and crest on trough of equal size cancels to flat water. That is superposition, and both waves leave the overlap unchanged.
Going further
Reflection and superposition together make something worth seeing. Send a wave train down a channel with a barrier at the far end and the reflected train runs back through the train still coming in. Where the two superpose, some points end up permanently still and some heave hard, and — because both trains have the same wavelength — the still points sit half a wavelength apart and never move along. The pattern stops looking like a travelling wave at all and starts looking like water rocking on the spot, which is what a bath sloshing end to end is doing.
The same rule works for waves that are not on water. Two loudspeakers wired to the same note give quiet patches you can walk through, and noise-cancelling headphones work by generating a wave that arrives crest on trough with the sound they want gone. Different material, identical arithmetic — which is the reason superposition is taught on water first, where you can see it.
Before this lesson
Connects to
At GCSE this becomes
- Path difference and phase, constructive and destructive interference, standing waves, and the two-source interference experiment.
Where to next
Ask Mr Badmus AI
Got a crossing pattern you cannot account for?
The bench is a teaching model. It shows the two waves as though they travel along the same line with the same wavelength of 500 mm, which is the simplest case and the only one where a single number describes the overlap; ripples spreading from two stones cross at an angle, and the result then changes from place to place across the pattern. All three traces are drawn to one scale of 2.5 pixels per millimetre. The traces are snapshots rather than animations, and a real overlap is moving. Heights are displacements from the still level, so a crest counts as up and a trough as down.
Lesson content © MrBadmusAI.