In simple terms
A friendly intro before the formal notes — no formulas yet.
Stationary waves
Cambridge 9702 Paper 2 — Stationary waves (8.1). Senpai Corner diagram-backed pilot with premium structure and live visuals.
- 1
Stationary waves are formed by the principle of superposition.
- 2
Superposition states that when two or more waves meet at a point, the resultant displacement is the vector sum of the individual displacements.
- 3
A stationary wave is produced by the superposition of two progressive waves of the same frequency, wavelength, and amplitude, travelling in opposite directions.
- 4
This is commonly achieved when a wave reflects from a boundary and interferes with the incident wave.
What this topic covers
The official Cambridge syllabus points this lesson works through.
- 8.1.1
Explain and use the principle of superposition
- 8.1.2
Show an understanding of experiments that demonstrate stationary waves using microwaves, stretched strings and air columns (it will be assumed that end corrections are negligible; knowledge of the concept of end corrections is not required)
- 8.1.3
Explain the formation of a stationary wave using a graphical method, and identify nodes and antinodes
- 8.1.4
Understand how wavelength may be determined from the positions of nodes or antinodes of a stationary wave
Explore the concept
Use the live diagram and synced steps — play it or tap a step card to walk through.
Pattern Two waves interfere to give a fixed pattern — scrub time to see it oscillate in place.
23 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.
23 simulations · 5 to start with
Start herein this order — each one shows a different piece of the topic
- PhETStart here · 19702 8.1
Wave on a String
Fix the far end, drive the string and tune the frequency until nodes stand still.
Why this one: Fix the far end and raise the frequency until nodes stand still; each loop is λ/2 and the next harmonic adds one loop.
Try this
- Choose “Oscillate”, “Fixed End”, damping none; raise f slowly until clear nodes appear.
- Count the loops — each loop is λ/2. Find the next frequency that gives one more loop.
- Switch to “Loose End” — the free end becomes an antinode.
Look for Standing waves form only at resonant frequencies; nodes are λ/2 apart.
Simulation by PhET Interactive Simulations, University of Colorado Boulder · Licensed to MarkScheme (public licence CC BY-NC 4.0 since 2026-03-30)
- 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: Double then triple the driving frequency: harmonics appear at whole multiples of the fundamental.
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
- The Physics ClassroomStart here · 39702 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: Two waves travelling opposite ways superpose; adjacent nodes sit half a wavelength apart.
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)
- oPhysicsStart here · 49702 8.1 · IB C.4
Standing Waves on Strings
Standing waves on a string: vary driving frequency, linear density and tension to hit harmonics
Why this one: Raise the tension and the first harmonic climbs; add linear density and it falls: f = (1/2L)√(T/μ).
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)
- oPhysicsStart here · 59702 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)
More simulations18 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 · 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)
- oPhysicsCore9702 8.1 · IB C.4
Standing Waves on Strings (tutorial)
Tutorial page (GIF animations, no applet): first three harmonics on a string, wavelength-length relations
Try this
- Compare the three harmonic animations.
- Count the loops in each harmonic.
- Relate each wavelength to the string length.
Look for The nth harmonic fits n half-wavelengths into the string length.
Open on oPhysicsRuns on their siteSimulation 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.
Tap a symbol — great for exam definitions
Full topic notes
Formal explanation with the rigour you need for the exam.
The Genesis of a Stationary Wave
A stationary wave forms when two progressive waves, possessing the exact same frequency, wavelength, and amplitude, travel through the same medium in opposite directions. They combine or 'superpose' at every point. This often happens when a wave reflects off a boundary and interferes with the incident wave.
Stationary waves are formed by the principle of superposition.
Superposition states that when two or more waves meet at a point, the resultant displacement is the vector sum of the individual displacements.
A stationary wave is produced by the superposition of two progressive waves of the same frequency, wavelength, and amplitude, travelling in opposite directions.
This is commonly achieved when a wave reflects from a boundary and interferes with the incident wave.
Constructive interference (waves in phase) and destructive interference (waves in antiphase) occur at fixed positions.
Nodes and Antinodes: The Fixed Points
At a node, the two progressive waves always interfere destructively, resulting in zero or minimum displacement. Particles at nodes never move from their equilibrium position. Conversely, at an antinode, the waves always interfere constructively, leading to maximum displacement. Particles here oscillate with the largest amplitude.
The distance between an adjacent node and antinode is a quarter of a wavelength (λ/4).
The distance between two adjacent nodes is half a wavelength (λ/2).
The distance between two adjacent antinodes is also half a wavelength (λ/2).
Harmonics on a String
When a string or air column vibrates, it can form various stationary wave patterns called harmonics. Each harmonic corresponds to a specific resonant frequency, which is an integer multiple of the lowest possible frequency, known as the fundamental frequency (or first harmonic).
The fundamental frequency produces a stationary wave with just one antinode between the fixed ends (e.g., a string). The n-th harmonic will feature 'n' antinodes within the vibrating section of the medium.
Formula: Frequency of a Vibrating String
Where:
- is the frequency of vibration (Hz)
- is the length of the vibrating string (m)
- is the tension in the string (N)
- (mu) is the mass per unit length of the string (kg m⁻¹)
Stationary Waves in Air Columns
Stationary waves are also fundamental to the operation of wind instruments. They are formed in a column of air within a pipe, caused by sound waves reflecting from the ends of the pipe. The nature of the stationary wave depends on whether the ends of the pipe are open or closed.
Pipe Closed at One End: For a pipe of length L that is closed at one end and open at the other, the fundamental mode of vibration (first harmonic) has one node at the closed end and one antinode at the open end. This means the length of the pipe is a quarter of a wavelength (). Because of this boundary condition, only odd-numbered harmonics can be produced (, etc.).
Pipe Open at Both Ends: For a pipe of length L that is open at both ends, there must be an antinode at each end. The fundamental mode of vibration has an antinode at each end and one node in the middle. This means the length of the pipe is half a wavelength (). All integer harmonics can be produced (, etc.).
A closed end of a pipe is a boundary where the air particles cannot move, so it is always a displacement node.
An open end of a pipe is a boundary where the air particles are free to move with maximum amplitude, so it is always a displacement antinode.
A common mistake is confusing stationary waves with progressive waves. Remember, stationary waves do NOT transfer net energy, and their nodes and antinodes are fixed in position, unlike the constantly moving crests and troughs of progressive waves. Also, be precise with and distances, and carefully identify the boundary conditions for strings (node-node) versus air columns (node-antinode or antinode-antinode).
Worked examples
See the formulas applied — reveal one step at a time, like the exam.
A string of length 0.80 m has a mass of 4.0 g. When stretched with a tension of 20 N, calculate the fundamental frequency of vibration.
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First, calculate the mass per unit length, . Remember to convert mass to kilograms.
A guitar string of length 75 cm is fixed at both ends. It is plucked and vibrates in its third harmonic mode. The frequency of this sound is found to be 660 Hz. (a) Determine the wavelength of the waves on the string. (b) Calculate the speed of the progressive waves on the string.
- 1
First, relate the string length to the wavelength for the third harmonic. The third harmonic (n=3) has three antinodes ('loops') on the string. The total length L contains three half-wavelengths.
How it all connects
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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.
- node
At a node, the two progressive waves always interfere destructively, resulting in zero or minimum displacement.
- antinode
Conversely, at an antinode, the waves always interfere constructively, leading to maximum displacement.
- fundamental frequency
Each harmonic corresponds to a specific resonant frequency, which is an integer multiple of the lowest possible frequency, known as the fundamental frequency (or first harmonic).
- Pipe Closed at One End
Pipe Closed at One End:
- Pipe Open at Both Ends
Pipe Open at Both Ends:
Quick check
Write your answer first, then compare it with the model one — the gap is what you would have lost.
Teach it back
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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.
Stationary waves are formed by the principle of superposition.
Superposition states that when two or more waves meet at a point, the resultant displacement is the vector sum of the individual displacements.
A stationary wave is produced by the superposition of two progressive waves of the same frequency, wavelength, and amplitude, travelling in opposite directions.
This is commonly achieved when a wave reflects from a boundary and interferes with the incident wave.
Constructive interference (waves in phase) and destructive interference (waves in antiphase) occur at fixed positions.
Practice — then mark it
The whole point: a real Cambridge question, marked mark-by-mark.
The piston is moved to the left. The frequency of the sound wave emitted by the loudspeaker is then changed so that a stationary wave is formed with same number of antinodes as in Fig. 4.1. State and explain the change that is made to the frequency of the sound wave.
The wave is a water wave produced by a dipper S₁ attached to a vibrator in a ripple tank. An identical dipper S₂ is attached to the same vibrator. The two dippers produce an interference pattern on the water in the tank, as shown in Fig. 4.3. The wave crests from each source are represented by solid lines on Fig. 4.3 and the wave troughs are represented by dashed lines. At point P in Fig. 4.3, the wave from S₁ has the same amplitude A as the wave from S₂. Describe and explain the amplitude of the resultant wave at point P.
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 9702/22 · Q4(b)(iii) on paper, snap a photo, and get examiner-style feedback on exactly where you win and lose marks.
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