Standing waves — nodes and antinodes — Physics

Before this

This leads to

Harmonics on a string
Standing waves in pipes
de Broglie wavelength · coming soon

A wave that goes nowhere

Pluck a guitar string and watch it closely. The two ends do not move at all. The middle swings up and down so fast that it looks like a blur. And the blur stays where it is. Nothing travels along the string.

That is strange. Every wave you have met so far moves. A ripple crosses a pond. A sound travels from a speaker to your ear. The wave lectures defined a wave as a disturbance that travels and carries energy. So what is this shape that sits still and swings in place?

It is called a standing wave. It is made of two ordinary travelling waves, moving in opposite directions through the same string. Neither of them stands still. Their sum does. This lecture shows how that happens, and why some points on the string never move at all.

Two waves, one string

Where does the second wave come from? Usually from a reflection. A wave runs along the string, reaches the fixed end and bounces back. Now the string carries two waves at once: the one going out and the one coming back. They have the same frequency and the same wavelength, because they are the same wave. If little energy is lost, they also have almost the same amplitude.

The superposition lecture gave the rule for what happens when waves meet. At every point, the displacement of the string is the sum of the two separate displacements, added with their signs. The waves pass through each other unchanged. So the shape of the string at any moment is simply wave 1 plus wave 2.

Standing wave

A standing wave forms when two waves with the same frequency, the same wavelength and the same amplitude travel in opposite directions through the same medium and superpose.

Watching the sum, one eighth of a period at a time

This is the part that needs careful thinking. Spend time here.

Start at a moment when the two waves line up exactly, crest on crest. Their sum has crests twice as tall. Now let time run on. In one eighth of a period, $T /8$ , each wave moves one eighth of a wavelength. But they move in opposite directions. So they slide apart by $λ /8 + λ /8 = λ /4$ . Now they only partly add up.

After a quarter of a period, $T /4$ , each wave has moved $λ /4$ , so they have slid apart by $λ /2$ . Half a wavelength apart means every crest of one wave sits on a trough of the other. They cancel everywhere. For that one instant, the whole string is flat.

After half a period, $T /2$ , they have slid apart by a whole wavelength. They line up again, and the sum is twice as tall. But it is upside down: where there was a crest, there is now a trough. Then the cycle repeats.

Use the simulation to watch this happen.

Simulation — two waves make a standing wave

Press Step to move on by one eighth of a period, or Play to run it. The dashed lines are the two travelling waves. The thick line is their sum. Watch the dark dots on the axis. Then turn on the envelope.

Five snapshots, one eighth of a period apart. The two dashed waves move in opposite directions. Their sum, the thick line, never moves along the string. The dark dots never move at all.

What should you have seen? The two dashed waves kept moving, one each way. The thick line never moved along the string. It only grew, shrank, flattened and turned upside down, in the same place. And the dark dots stayed exactly still, at every moment. With the envelope on, you can see the whole range of the string's motion: a row of loops, pinched to nothing at the dots.

Nodes and antinodes

The points that never move are called nodes (N). At a node, the two waves always have equal and opposite displacements. When one wave pushes the string up, the other pushes it down by the same amount. So the sum there is zero at every moment.

Halfway between two nodes are the antinodes (A). There, the two waves are always in step, so the string swings with the largest amplitude. That amplitude is twice the amplitude of each travelling wave, because the two add up. Between a node and an antinode, the amplitude grows smoothly from zero to its largest value.

A standing wave with three loops, drawn at two moments half a period apart (solid and dashed). The nodes N never move. The antinodes A swing the most. Adjacent nodes are half a wavelength apart, and a node is a quarter of a wavelength from the next antinode.

Why are the nodes exactly half a wavelength apart? Think about what makes a point a node: the two waves arrive there in antiphase. Now move along the string by $λ /2$ . The wave going right reaches that point half a cycle later. The wave going left reaches it half a cycle earlier. So their timing changes by one whole cycle compared with each other. One whole cycle makes no difference. The two waves are still in antiphase, so this point is a node too.

That is it. That is the whole idea. Every node and antinode comes from the two waves sliding past each other in opposite directions. The antinodes sit halfway between the nodes, a quarter of a wavelength from each.

Spacing in a standing wave

Distance between adjacent nodes $= λ /2$ . Distance between adjacent antinodes $= λ /2$ . Distance from a node to the nearest antinode $= λ /4$ .

Here $λ$ is the wavelength of the two travelling waves that make the standing wave.

This gives a very useful method. Measure the distance between two nodes, double it, and you have the wavelength. The speed of the travelling waves still obeys the wave equation, $v = f λ$ .

Before you read on, try the step that students most often get wrong.

Predict — from node spacing to wavelength

A standing wave on a string has nodes 20 cm apart. What is the wavelength of the waves that made it? Choose an answer, then read the explanation.

Four common answers for nodes 20 cm apart.

10 cm. Not right. This halves the spacing instead of doubling it.
20 cm. Not right. Adjacent nodes are half a wavelength apart, not a whole wavelength.
40 cm. Right. The node spacing is $λ /2$ , so $λ = 2 \times 20 = 40$ cm.
80 cm. Not right. This uses $λ /4$ , the node-to-antinode distance.

How the string moves: phase

Look again at the thick line in the simulation. Every point between two neighbouring nodes moves up together and down together. They all reach their highest point at the same moment. So all the points in one loop are in phase, even though their amplitudes are different.

Points in neighbouring loops are different. When one loop is up, the next loop is down. So points on opposite sides of a node are in antiphase. This is very different from a travelling wave, where every point has the same amplitude and each point is a little behind its neighbour.

Stop and think

At $t = T /4$ the whole string is flat. Is the string still at that moment?

Show the answer

No. It is moving at its fastest. Think of a swing passing through its lowest point: it is in the middle of its motion, not at rest. In the same way, every point of the string (except the nodes) is rushing through the middle position. Each point is momentarily still at the other moments, $t = 0$ and $t = T /2$ , when the string is most bent. A flat string at one instant is not a still string.

Energy: carried or kept?

A travelling wave carries energy from one place to another. In a standing wave, the two travelling waves carry equal amounts of energy in opposite directions. So no energy is carried along the string overall. The energy stays in each loop.

But it does not stay still. When the string is most bent, it is momentarily at rest, so its energy is stored as elastic potential energy in the stretched string. A quarter of a period later, the string is flat and moving fastest, so the energy is kinetic. The energy swaps between these two stores twice in every period, like a swinging pendulum.

Feature	Travelling wave	Standing wave
Does the pattern move along?	Yes	No
Amplitude of the points	The same everywhere	Zero at nodes, largest at antinodes
Phase of neighbouring points	Each a little behind the next	In phase within a loop; antiphase across a node
Energy	Carried along	No overall transfer; kept in each loop

Using node spacing

Worked Example 1 — Counting nodes and loops

A string 60 cm long is fixed at both ends. It vibrates in a standing wave with 3 loops. How many nodes and antinodes are there, and what is the wavelength?

The fixed ends cannot move, so they are nodes. Between the 3 loops there are 2 more nodes. That makes 4 nodes. Each loop has one antinode at its centre, so there are 3 antinodes.

Each loop is half a wavelength long. So $3 \times λ /2 = 60$ cm, and $λ = 40$ cm.

Check: nodes at 0, 20, 40 and 60 cm are 20 cm $= λ /2$ apart. ✓

Worked Example 2 — Finding the wave speed

A standing wave on a string has nodes every 25 cm. The vibrator driving it has a frequency of 400 Hz. Find the speed of the waves on the string.

Node spacing $= λ /2 = 0.25$ m, so $λ = 0.50$ m.

$v = f λ = 400 \times 0.50 = 200$ m/s.

Check: in one period, $1/400$ s, a wave travels $200 \times 0.0025 = 0.50$ m, exactly one wavelength. ✓

Worked Example 3 — A standing wave of sound

Sound in a closed tube forms a standing wave. A small microphone moved along the tube finds quiet points (nodes) at 0, 20, 40 and 60 cm from the closed end. The speed of sound in the air is 343 m/s. Find the frequency.

Node spacing $= 0.20$ m $= λ /2$ , so $λ = 0.40$ m.

$f = v / λ = 343/0.40 \approx 858$ Hz.

Notice what the factor of 2 does. If you took the node spacing as the wavelength, you would get $343/0.20 \approx 1715$ Hz, which is twice the true frequency. Check: $858 \times 0.40 \approx 343$ m/s. ✓

Worked Example 4 — Sound bouncing off a wall (invented)

Kemal stands a loudspeaker in front of a hard wall. It plays one steady note. He walks slowly between the speaker and the wall, and he notices quiet places every 25 cm. Explain this, and find the frequency of the note. Take the speed of sound as 343 m/s.

The wall reflects the sound back towards the speaker. The wave going out and the wave coming back have the same frequency and wavelength, so they make a standing wave. The quiet places are its nodes.

Node spacing $= λ /2 = 0.25$ m, so $λ = 0.50$ m, and $f = 343/0.50 = 686$ Hz.

Check: the loud places, the antinodes, should be halfway between the quiet ones, 12.5 cm from each. In a real room the quiet places are not perfectly silent, because the reflected wave is weaker than the outgoing one and other surfaces reflect sound too.

Three classic mistakes

Trap 1 — Node spacing as the wavelength

Adjacent nodes are $λ /2$ apart, not $λ$ . One whole wavelength covers two loops. Forgetting this gives a wavelength half the true value and a frequency twice the true value.

Trap 2 — "A node is where the string is at zero"

At $t = T /4$ , every point of the string is at zero, but most of them are not nodes. A node is a point where the displacement is zero at all times. An antinode is a point with the largest amplitude, not a point that is at its largest at one moment.

Trap 3 — "A standing wave has no speed"

The pattern does not move, but the two waves that make it do. When you use $v = f λ$ , $v$ is the speed of those travelling waves and $λ$ is their wavelength, found from twice the node spacing.

Try it yourself

Work each one on paper first. Open a hint only when you are stuck.

A rope 1.2 m long is fixed at both ends. It vibrates with 4 loops. How many nodes and antinodes are there? What is the wavelength?

Hint 1
Both fixed ends are nodes. Count the nodes between the loops too.

Hint 2
Each loop is $λ /2$ long, so $4 \times λ /2 = 1.2$ m.

Full answer
5 nodes (2 ends + 3 between the loops) and 4 antinodes. $2 λ = 1.2$ m, so $λ = 0.60$ m. Check: nodes 0.30 m apart, which is $λ /2$ . ✓
A standing wave on a wire has nodes 15 cm apart when the frequency is 1100 Hz. Find the speed of the waves on the wire.

Hint 1
Find the wavelength from the node spacing first.

Hint 2
$λ = 2 \times 0.15$ m. Then use $v = f λ$ .

Full answer
$λ = 0.30$ m, so $v = 1100 \times 0.30 = 330$ m/s. Check: $330/1100 = 0.30$ m. ✓
In a standing sound wave in air, a node is 8.5 cm from the nearest antinode. The speed of sound is 343 m/s. Find the frequency.

Hint 1
Which fraction of a wavelength is the node-to-antinode distance?

Hint 2
$λ /4 = 0.085$ m.

Full answer
$λ = 4 \times 0.085 = 0.34$ m. $f = 343/0.34 \approx 1009$ Hz. Check: $1009 \times 0.34 \approx 343$ m/s. ✓
Microwaves bounce off the metal walls inside a microwave oven and form standing waves. If you remove the turntable and heat a flat bar of chocolate for a short time, it melts in spots at the antinodes. Most ovens use microwaves of frequency 2.45 GHz. (a) Melted spots are found 6.1 cm apart. Use them to estimate the speed of light. (b) Why must the turntable be removed? (c) Another student measures 6.0 cm. By what percentage does her value differ from the accepted speed of light, 3.00 × 10⁸ m/s?

Hint 1
Adjacent antinodes are $λ /2$ apart, just like adjacent nodes.

Hint 2
$λ = 2 \times 0.061$ m. Microwaves are light, so their speed is $c = f λ$ , with $f = 2.45 \times 1 0^{9}$ Hz.

Full answer
(a) $λ = 0.122$ m, so $c = 2.45 \times 1 0^{9} \times 0.122 \approx 2.99 \times 1 0^{8}$ m/s. (b) The standing-wave pattern stays fixed inside the oven. A turntable would carry the chocolate through the hot and cold spots, so it would melt evenly and no spots would show. (That is exactly why ovens have turntables: to heat food evenly.) (c) $λ = 0.120$ m gives $c = 2.94 \times 1 0^{8}$ m/s, which is about 2% below $3.00 \times 1 0^{8}$ m/s. A 1 mm error in the spacing gives about a 2% error in $c$ . Check (a): $2.99 \times 1 0^{8} \div 2.45 \times 1 0^{9} \approx 0.122$ m. ✓

The bigger idea

Standing waves appear wherever waves are trapped and reflected back on themselves. They are inside every stringed instrument, every flute and every organ pipe. They are inside the metal box of a microwave oven, which is why food heats unevenly without a turntable.

A trapped wave cannot have any wavelength it likes. On a guitar string, both ends must be nodes, so only a whole number of loops can fit. That is why a string plays particular notes and not every possible note. The next lecture works out exactly which wavelengths fit. The same idea reached deep into physics. In 1924, Louis de Broglie suggested that an electron in an atom behaves like a wave that must fit around its orbit, like a standing wave. Modern quantum theory replaced that simple picture, but kept its central idea: a wave held in a small space can only take certain patterns, and that is why atoms have only certain energies.

Key concepts in this lesson

A standing wave forms when two waves with the same frequency, wavelength and amplitude travel in opposite directions through the same medium and superpose.
Nodes are points with zero displacement at all times, where the two waves are always in antiphase, and antinodes are points of largest amplitude, twice the amplitude of each wave.
Adjacent nodes, and adjacent antinodes, are half a wavelength apart, and a node is a quarter of a wavelength from the nearest antinode.
All points in one loop move in phase with different amplitudes, and points on opposite sides of a node move in antiphase.
A standing wave carries no energy along the medium overall, and its energy swaps between kinetic and potential stores twice in each period.