Explain superposition and stationary-wave formation
Add overlapping displacements and explain how opposite-travelling waves form stationary nodes and antinodes.
AS Level; Cambridge International Physics 9702, 2025–2027 syllabus, Topic 8. These original models and calculations are study aids; full timed assessment, supervised practical competence and human subject-teacher review remain outstanding.
Before you begin
- Use wavelength, frequency, phase difference, trigonometry and consistent SI units.
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
What does the principle of superposition say about overlapping waves?
What you will learn
- Explain and use the principle of superposition
- Explain stationary-wave formation with a graphical model
- Identify nodes and antinodes from the stationary pattern
Superposition adds simultaneous displacements
When two waves overlap, the resultant displacement at a point is the algebraic sum of the individual displacements there. A positive and a negative displacement can partly or completely cancel; two displacements of the same sign reinforce.
This rule applies at each position and instant while the waves overlap. After the overlap, each wave continues according to its own propagation; the temporary resultant does not permanently remove either wave.
Opposite-travelling waves can make a stationary pattern
A stationary wave forms when waves of the same frequency and amplitude travel in opposite directions in the same region and superpose, commonly an incident wave and its reflection. Its pattern does not progress along the medium.
Nodes are fixed positions where the resultant displacement is always zero. Antinodes are positions where oscillation amplitude is greatest. A graphical construction adds the two travelling-wave displacements at successive instants: nodes remain fixed while antinodes oscillate between positive and negative maxima.
Pause and explain
What is true of a node on an ideal stationary wave?
Worked example
Two waves at one point have displacements +3.0 mm and −1.0 mm at the same instant. Find the resultant displacement and state whether the overlap is reinforcement or partial cancellation.
Show the worked solution
- Apply the algebraic-sum rule at that point.
- Resultant displacement = +3.0 + (−1.0) mm.
- The opposite signs mean partial cancellation.
Answer +2.0 mm; the waves partially cancel at that instant.
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Test this topic →From guided practice to a new situation
Write your answer and reasoning first. Use a hint only if you are stuck. The model solution is for self-checking; your written work is not automatically marked.
At one instant, two overlapping pulses have displacements −4.0 cm and +1.5 cm at a point. Find the resultant displacement.
Give me a hint
Keep the signs because displacement has a direction from equilibrium.
Compare my reasoning
- Write the signed values: −4.0 cm and +1.5 cm.
- Add them algebraically: −4.0 + 1.5.
- Retain the sign of the resultant.
−2.5 cm.
Look for these in your work
- Both signed displacements used
- Algebraic sum calculated
- Direction stated
Two equal-amplitude travelling waves move in opposite directions and overlap. Describe what happens at a node and at an antinode in the stationary pattern.
Give me a hint
Compare the resultant at fixed positions through time.
Compare my reasoning
- At a node, opposing displacements cancel at every instant.
- At an antinode, the oscillation amplitude is greatest.
- Both positions stay fixed, so the pattern is stationary.
A node has zero resultant displacement throughout; an antinode oscillates with maximum amplitude. Both remain at fixed positions.
Look for these in your work
- Node condition stated
- Antinode condition stated
- Stationary locations distinguished
A learner says a stationary wave is just a progressive wave whose energy travels more slowly. Correct the explanation using the pattern and its formation.
Give me a hint
Separate the motion of an individual travelling disturbance from the fixed nodes and antinodes.
Compare my reasoning
- A stationary pattern has fixed nodes and antinodes rather than a progressing wave profile.
- It forms by superposition of suitable waves travelling in opposite directions.
- The learner's description changes neither the formation nor the defining fixed pattern.
A stationary wave is a fixed interference pattern formed by superposition of opposite-travelling waves; its nodes and antinodes do not progress along the medium.
Look for these in your work
- Fixed pattern identified
- Formation explained
- Incorrect speed claim corrected
Common mistakes
- Adding displacement magnitudes while ignoring their signs
- Calling an antinode a point of permanently maximum displacement
- Describing a stationary pattern as travelling along the medium
What can you explain now?
What does the principle of superposition say about overlapping waves?
Compare with the explanation
Add their displacements algebraically
The resultant at a point is the signed sum of the individual displacements.
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Make it stick
How can two moving waves create a pattern with positions that remain fixed?
If you needed the solution, close it and solve the problem again from a blank page. A correct answer is stronger when you can explain the reason for each step.