In simple terms
A friendly intro before the formal notes — no formulas yet.
The Guitar String's Secret Dance
A standing wave is a stationary pattern created when two identical waves travelling in opposite directions overlap. Between fixed points called nodes, the medium simply swings up and down in place. Resonance is what happens when you drive such a system at one of these special 'magic' frequencies: the amplitude builds to a maximum.
Imagine you and a friend holding a long skipping rope. If you both send waves down it they just pass through each other. But tie one end to a wall and shake the other, and your wave reflects and interferes with the one still arriving. At most frequencies the result is a mess. At certain 'magic' frequencies the rope settles into stable loops that stay in the same place and only oscillate up and down. Those loops are a standing wave, and hitting a magic frequency to grow the biggest loops is resonance.
- 1
Identify the boundary conditions: a string fixed at both ends forces a node at each end; a pipe forces a node at any closed end and an antinode at any open end. This fixes the shape of every allowed pattern.
- 2
Sketch the harmonic you need: draw the fundamental () first, then add loops for higher harmonics, keeping the correct feature (node or antinode) at each boundary.
- 3
Relate the wavelength to the length: read off from your sketch how many quarter- or half-wavelengths fit into . For a string's fundamental, one loop spans half a wavelength, so .
- 4
Apply the wave equation: combine with the length–wavelength relation to solve for frequency, speed or wavelength.
Explore the concept
Use the live diagram, PhET or GeoGebra sim, and synced steps — play it, drag controls, or tap a step.
Step-synced diagram — highlights what to look for in the simulation above.
Step 1
Identify the boundary conditions: a string fixed at both ends forces a node at each end; a pipe forces a node at any closed end and an antinode at any open end. This fixes the shape of every allowed pattern.
26 more simulations for this topic — run them in the Simulations section below
Simulations
Every simulation here runs the real model — try the steps on a card, then check what you see against the notes.
26 simulations · 5 to start with
Start herein this order — each one shows a different piece of the topic
- The Physics ClassroomStart here · 19702 8.1 · IB C.4
Standing Wave Maker
Two waves travel in opposite directions; use the pre-set conditions or set your own and view the standing wave
Why this one: Send two identical waves through each other and watch nodes stay still while antinodes swing.
Try this
- Pick a pre-set and locate the nodes.
- Set your own frequency to reach the next harmonic.
- Count the nodes for each harmonic.
Look for Adjacent nodes are half a wavelength apart.
Physics Interactives by The Physics Classroom · Licensed to MarkScheme (site terms otherwise permit linking only)
- 3JCN PhysicsStart here · 29702 8.1 · IB C.4
Standing Wave
Drive a string at chosen frequency; see nodes and antinodes form
Why this one: Drive the string and step through the frequencies where whole harmonics fit: f_n = nv/2L.
Try this
- Drive the string at the fundamental frequency and count the nodes.
- Double the frequency and count again.
- Triple it and compare the node spacing with the wavelength.
Look for Standing waves form at whole multiples of the fundamental and adjacent nodes are half a wavelength apart.
3JCN Physics Simulation by Thomas Nguyen · CC BY 4.0
- oPhysicsStart here · 39702 8.1 · IB C.4
Air Column Resonance
Sound resonance in open/closed air columns: see incident and reflected waves build a standing wave
Why this one: Watch incident and reflected sound build the pattern: open pipes give every harmonic, closed pipes only odd ones.
Try this
- Watch the incident and reflected waves build in the open column.
- Switch to the closed column.
- Count the nodes in each case.
Look for The open column resonates at every harmonic while the closed column resonates at odd harmonics only.
Simulation by Tom Walsh, oPhysics.com — made with GeoGebra · Licensed to MarkScheme (site: free for non-profit educational use; applets made with GeoGebra)
- oPhysicsStart here · 49702 17.3 · IB C.4 · IB C.1
Resonance
Tutorial with applet: driving a swing at its natural frequency; resonance and amplitude build-up
Why this one: Drive the swing at its natural frequency and the amplitude builds; drive faster or slower and it stays small.
Try this
- Drive the swing at its natural frequency.
- Drive it faster, then slower.
- Watch how the amplitude builds in each case.
Look for Amplitude builds largest when the driving frequency matches the natural frequency.
Simulation by Tom Walsh, oPhysics.com — made with GeoGebra · Licensed to MarkScheme (site: free for non-profit educational use; applets made with GeoGebra)
- The Physics ClassroomStart here · 59702 8.1 · IB C.3 · IB C.4
Boundary Behavior of Waves
View incident, reflected and transmitted pulses and see how their speed, wavelength and amplitude depend on the density of each medium
Why this one: Watch a pulse invert at a fixed end; that phase flip is why a node sits there.
Try this
- Send a pulse from the less dense into the denser medium.
- Compare the reflected pulse with the incident one.
- Reverse the media and repeat.
Look for A pulse reflects inverted from a denser medium and upright from a less dense one.
Physics Interactives by The Physics Classroom · Licensed to MarkScheme (site terms otherwise permit linking only)
More simulations21 more on this topic — core ones first
- oPhysicsCore9702 7.2 · 9702 8.1 · IB C.2
Longitudinal Waves
Longitudinal travelling or standing wave; adjust speed and amplitude; see compressions and rarefactions
Try this
- Run the travelling wave and follow one compression.
- Increase the amplitude and look at the compressions.
- Switch to the standing wave and find the particles that never move.
Look for Particles oscillate along the direction of travel; compressions move at the wave speed while each particle stays near its rest position.
Simulation by Tom Walsh, oPhysics.com — made with GeoGebra · Licensed to MarkScheme (site: free for non-profit educational use; applets made with GeoGebra)
- oPhysicsCore9702 8.1 · 9702 8.3 · IB C.4
Superposition of Transverse Waves
Two transverse waves travelling in opposite directions; adjust speed/wavelength and see the resulting standing wave
Try this
- Run two waves of equal wavelength towards each other.
- Change the wavelength and count the nodes.
- Change the speed and watch the nodes.
Look for Nodes stay fixed half a wavelength apart whatever the speed; a shorter wavelength gives more nodes.
Simulation by Tom Walsh, oPhysics.com — made with GeoGebra · Licensed to MarkScheme (site: free for non-profit educational use; applets made with GeoGebra)
- oPhysicsCore9702 8.1 · IB C.4
Standing Waves
Standing waves on a string or in open/closed air columns; show the two travelling components and particle motion
Try this
- Show the two travelling components under the standing wave.
- Switch from the string to an open air column.
- Switch to a closed column and look at the closed end.
Look for Nodes form where the travelling components always cancel; a closed end is a displacement node, an open end an antinode.
Simulation by Tom Walsh, oPhysics.com — made with GeoGebra · Licensed to MarkScheme (site: free for non-profit educational use; applets made with GeoGebra)
- oPhysicsCore9702 8.1 · IB C.4
Standing Waves on Strings
Standing waves on a string: vary driving frequency, linear density and tension to hit harmonics
Try this
- Raise the driving frequency until the first harmonic appears.
- Keep going to find the second and third harmonics.
- Increase the tension and find the first harmonic again.
Look for Harmonic frequencies are whole-number multiples of the fundamental, which rises with tension and falls with linear density.
Simulation by Tom Walsh, oPhysics.com — made with GeoGebra · Licensed to MarkScheme (site: free for non-profit educational use; applets made with GeoGebra)
- oPhysicsCore9702 8.1 · 9702 7.2 · IB C.4
Air Column Resonance with Longitudinal Waves
Air-column resonance shown with longitudinal particle displacement and pressure representations
Try this
- Switch between the displacement and pressure representations.
- Find the displacement node at the closed end.
- Compare the pressure at an open end.
Look for A displacement node is a pressure antinode, so the closed end has the largest pressure variation and the open end none.
Simulation by Tom Walsh, oPhysics.com — made with GeoGebra · Licensed to MarkScheme (site: free for non-profit educational use; applets made with GeoGebra)
- oPhysicsCore9702 8.1 · 9702 8.3 · IB C.4
Interference & Standing Waves
Tutorial with applet: oppositely travelling waves of equal wavelength form a standing wave
Try this
- Run the two waves and watch the sum.
- Pause and locate the nodes.
- Measure the node spacing against the wavelength.
Look for Nodes are half a wavelength apart and never move.
Simulation by Tom Walsh, oPhysics.com — made with GeoGebra · Licensed to MarkScheme (site: free for non-profit educational use; applets made with GeoGebra)
Key formulas
Tap any symbol to reveal exactly what it means and its units.
Full topic notes
Formal explanation with the rigour you need for the exam.
How a standing wave forms
A standing wave is produced by the superposition of two identical progressive waves — same frequency, same amplitude, same speed — travelling in opposite directions through the same medium. In practice the second wave is usually the reflection of the first from a boundary. Where the two waves are permanently in step they reinforce, and where they are permanently in opposition they cancel. Because the two waves keep passing through each other, those regions of reinforcement and cancellation stay put, and the pattern appears frozen in space: the crests no longer march along, they just grow and shrink on the spot.
Nodes: points where the two waves always cancel, so the displacement is zero at every instant.
Antinodes: points where the two waves always reinforce, so the oscillation reaches maximum amplitude.
Consecutive nodes (or consecutive antinodes) are separated by half a wavelength, .
A node and its neighbouring antinode are separated by a quarter of a wavelength, .
Standing waves versus travelling waves
It is worth being precise about how a standing wave differs from the travelling waves that build it, because examiners test exactly this distinction. A travelling wave transports energy along the medium; a standing wave does not — its energy stays put, sloshing between kinetic and potential within each loop but never flowing past a node. The phase behaviour differs too. In a travelling wave, neighbouring points lag one another, so a disturbance sweeps along. In a standing wave, all the points within one loop (between two adjacent nodes) move in phase — they reach their peaks together — while points in adjacent loops move in exact antiphase. Finally, in a travelling wave every point has the same amplitude; in a standing wave the amplitude depends on position, running from zero at the nodes to a maximum at the antinodes.
Energy: travelling wave transfers energy along the medium; standing wave transfers no net energy.
Amplitude: constant for a travelling wave; varies with position for a standing wave (zero at nodes, maximum at antinodes).
Phase: continuous phase lag along a travelling wave; points within one loop of a standing wave are all in phase, and adjacent loops are in antiphase.
Wavelength: for a standing wave, is still twice the node-to-node distance — it is the wavelength of the two travelling waves that formed it.
Standing waves on a string fixed at both ends
Consider a string of length fixed at both ends, as on a guitar or violin. Because the ends are clamped they cannot move, so each end must be a node. This boundary condition allows only certain wavelengths to fit: the string length must contain a whole number of half-wavelengths. The fundamental (first harmonic) is a single loop, so ; the second harmonic is two loops, ; and so on. These allowed frequencies are the natural frequencies, or harmonics, of the string.
For a string of length fixed at both ends carrying waves of speed : The case is the fundamental (first harmonic); ALL integer harmonics are allowed.
Standing waves in air columns (pipes)
Standing sound waves form in the air inside pipes, which is how wind instruments make their notes. The boundary conditions come from how freely the air can move at each end. A closed end forces the air to be stationary, so it is a displacement node. An open end lets the air move most freely, so it is a displacement antinode. Two cases matter for C.4.
Open pipe (antinode–antinode): Closed pipe (node at the closed end, antinode at the open end):
Pipe open at both ends: an antinode at each end. The allowed frequencies are for — the same set as a string, with ALL harmonics present.
Pipe closed at one end: a node at the closed end and an antinode at the open end. The fundamental fits only a quarter of a wavelength (), and only ODD harmonics exist: for
For the same length and speed of sound , a closed pipe's fundamental is HALF that of an open pipe — it sounds an octave lower.
Identifying nodes and antinodes
Many exam marks are earned simply by reading a standing-wave pattern correctly: counting loops, locating nodes and antinodes, and turning that count into a wavelength. The key facts are that boundaries fix the pattern (node at a fixed or closed end, antinode at an open end) and that the node-to-node distance is always .
Natural frequency, forced oscillations and resonance
Left to itself after a disturbance, any bounded oscillator vibrates at one of its natural frequencies — for a string or air column, these are exactly the harmonics above. If instead a periodic driving force is applied, the system undergoes forced oscillations at the driving frequency. When the driving frequency matches a natural frequency, energy is transferred from driver to system with maximum efficiency and the amplitude builds to a large value: this is resonance. It is why a wine glass shatters at the right sung note, why pushing a swing in time makes it soar, and why an instrument's body amplifies the string's vibration.
Natural frequency: a frequency at which a system oscillates freely once disturbed; a bounded system has a whole set of them (its harmonics).
Forced oscillation: oscillation at the frequency of an external periodic driver, not at the system's own natural frequency.
Resonance: the maximum-amplitude response when the driving frequency equals a natural frequency; energy transfer from driver to system is most efficient there.
Damping
Real oscillators lose energy to resistive forces — friction, air resistance, or the radiation of sound — and this loss is called damping. The amount of damping controls both how a free oscillation dies away and how sharply a driven system resonates. Three regimes are named. With light damping the amplitude falls gradually over many cycles, so the system oscillates many times before stopping; its resonance peak is tall and narrow. With critical damping the system returns to equilibrium in the shortest possible time WITHOUT overshooting or oscillating — the target behaviour for car suspensions, analogue meters and door closers. With heavy (over) damping the resistive force is so large that the system creeps back to equilibrium slowly, again without oscillating, taking longer than the critical case. Increasing damping always lowers the height of the resonance peak and broadens it.
Light damping: slow decay over many oscillations; a tall, narrow resonance peak.
Critical damping: returns to equilibrium in the shortest time with NO oscillation.
Heavy (over) damping: no oscillation, but a slow return to equilibrium — slower than critical.
More damping → lower and broader resonance peak; the resonant amplitude never becomes infinite in a real, damped system.
Common mistakes examiners penalise
Saying a standing wave transfers energy — it does not transfer net energy along the medium. Energy is stored and merely converts between kinetic and potential within each loop.
Treating a node as 'small displacement' rather than zero — a node has ZERO displacement at all times; only an antinode reaches maximum amplitude.
Getting the pipe boundary conditions backwards — a closed end is a displacement NODE and an open end is a displacement ANTINODE. Reversing these gives every wavelength and frequency wrong.
Claiming a closed pipe has all harmonics — a pipe closed at one end supports only the ODD harmonics (1st, 3rd, 5th, ...); its first overtone is the third harmonic, not the second.
Confusing and fundamentals — a string or open pipe has ; a closed pipe has . Using the wrong one halves or doubles every frequency.
Mixing up harmonic and overtone — the th harmonic is times the fundamental, but the 'first overtone' is simply the next allowed frequency above the fundamental (the 3rd harmonic for a closed pipe).
Confusing critical and heavy damping — critical damping returns to equilibrium in the SHORTEST time without oscillating; heavy damping also does not oscillate but is slower.
Dropping units or over-rounding mid-calculation — carry extra figures through the working and round only the final answer, always with the correct unit.
Model answer — marked the way our engine marks it
In Paper 2 the marks are analytic: each is tied to a specific line of working — a method mark (M) or an answer mark (A) — and error-carried-forward (ECF) means a wrong number early on need not cost you the marks that follow. That protection only exists if your method is on the page. Study how each mark below is earned by a specific line, especially where the second harmonic is built from the first.
Where this leads
Standing waves are superposition made visible, and the same boundary-condition thinking reappears throughout physics: in the resonance of electrical circuits, in the modes of a laser cavity, and — in quantum theory — in the standing electron waves that fix the allowed energy levels of an atom. Master the routine here — identify the boundary conditions, sketch the harmonic, read off from the length, then apply , showing every line — and these later, more abstract standing-wave problems become variations on a method you already own.
Worked examples
See the formulas applied — reveal one step at a time, like the exam.
A string of length 0.80 m fixed at both ends carries transverse waves at a speed of 240 m s⁻¹. Calculate the frequency of the first harmonic and of the third harmonic. [4]
- 1
List what you know. m, m s⁻¹, string fixed at both ends so both ends are nodes.
An organ pipe of length 0.85 m is open at both ends. The speed of sound in air is 340 m s⁻¹. (a) Calculate the fundamental frequency of the pipe. [2] (b) The same pipe is now closed at one end. Calculate its new fundamental frequency and state which harmonics it can now produce. [2]
- 1
(a) Open pipe, fundamental (). With an antinode at each end the fundamental fits half a wavelength, so and . [M1: correct relation for an open pipe] Hz. [A1: answer with unit]
A standing wave is set up on a string of length 1.2 m fixed at both ends. The pattern shows the string vibrating in three loops (the third harmonic). Determine (a) the number of nodes and antinodes on the string, and (b) the wavelength of the wave. [3]
- 1
(a) Count the features. In the third harmonic the string vibrates in loops. Each loop has one antinode at its centre, so there are 3 antinodes. Nodes sit at both ends and between every pair of loops, giving 4 nodes in total. [M1: correct counting method] [A1: 4 nodes and 3 antinodes]
A string of length 0.80 m fixed at both ends has a wave speed of 240 m s⁻¹. Calculate the frequency of the first harmonic and the third harmonic. [4]
- 1
Model answer — full working.
How it all connects
The big idea sits in the middle — tap a linked idea to explore the link.
Tap a linked idea to see how it connects back to the main topic — that connection is what examiners reward.
Glossary
Key terms for this topic — skim now; the Check step will test them.
- Standing (stationary) wave
A wave pattern formed by the superposition of two identical progressive waves travelling in opposite directions. The pattern does not move along the medium and there is no net transfer of energy through it.
- Node vs antinode
A node is a point of zero displacement at all times (permanent destructive interference). An antinode is a point that oscillates with maximum amplitude (constructive interference). Nodes and antinodes alternate along the wave.
- Spacing of nodes and antinodes
Consecutive nodes (or consecutive antinodes) are half a wavelength apart, . A node and its neighbouring antinode are a quarter of a wavelength apart, .
- Standing vs travelling wave: energy
A travelling wave transfers energy from one place to another. A standing wave transfers no net energy along the medium; energy is stored, sloshing between kinetic and potential within each loop.
- Standing vs travelling wave: phase
In a travelling wave adjacent points reach maximum displacement at slightly different times (a continuous phase lag). In a standing wave every point within one loop (between two nodes) oscillates in phase; points in adjacent loops are exactly out of phase (antiphase).
- String fixed at both ends: harmonics
Both ends must be nodes. Allowed wavelengths are and frequencies for — ALL harmonics are present.
- Pipe open at both ends
An antinode at each end. Allowed frequencies are for — the same set as a string, with all harmonics present.
- Pipe closed at one end
A node at the closed end and an antinode at the open end. The fundamental wavelength is , and only odd harmonics exist: for
- Natural frequency
A frequency at which a system will oscillate freely once disturbed. A bounded system such as a string or air column has a whole set of natural frequencies — its harmonics.
- Resonance
The large-amplitude response that occurs when a system is driven by a periodic force at (or very near) one of its natural frequencies. Energy is transferred most efficiently from driver to system, so the amplitude grows to a maximum.
- Damping: light, critical, heavy
Damping is the loss of oscillation energy to resistive forces. Light damping: amplitude decays slowly over many oscillations. Critical damping: returns to equilibrium in the shortest time WITHOUT oscillating. Heavy (over) damping: returns slowly, without oscillating, taking longer than critical.
Name it
Read the meaning, then pick which of this lesson’s terms it describes. Miss one and you see what your choice really means.
Both ends must be nodes. Allowed wavelengths are and frequencies for — ALL harmonics are present.
Quick check
Write your answer first, then compare it with the model one — the gap is what you would have lost.
Teach it back
If you can explain it simply, you own it — gaps here are marks you’d lose.
Teach it back
Explain this topic as if teaching a friend. We name the gaps an examiner would still dock.
Revision flashcards
Guess first, then flip — retrieval beats re-reading.
Key takeaways
Review these before you close the topic — retrieval beats re-reading.
Nodes: points where the two waves always cancel, so the displacement is zero at every instant.
Antinodes: points where the two waves always reinforce, so the oscillation reaches maximum amplitude.
Consecutive nodes (or consecutive antinodes) are separated by half a wavelength, .
A node and its neighbouring antinode are separated by a quarter of a wavelength, .
Practice — then mark it
The whole point: a real Cambridge question, marked mark-by-mark.
Get a Paper 2 calculation marked: solve a standing-waves problem with full working
Get a Paper 2 calculation marked: solve a standing-waves problem with full working
Extra simulations & links
PhET, GeoGebra and other curated tools — open in a new tab.
Frequently asked
Checkpoint
One marked question is worth ten re-reads — close the loop before you move on.
Reading it isn’t knowing it — prove it.
Before you move on: do Get a Paper 2 calculation marked: solve a standing-waves problem with full working on paper, snap a photo, and get examiner-style feedback on exactly where you win and lose marks.
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