Principle of superposition of waves and formation of stationary waves — AQA A-Level Physics
Teaching note checked against the official specification.
The principle of superposition states that when two or more waves meet at a point, the resultant displacement is the vector sum of the individual displacements. Stationary waves form when two progressive waves of the same frequency, wavelength and amplitude travel in opposite directions through each other, typically produced by reflection, creating nodes and antinodes at fixed positions.
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Principle of superposition of waves and formation of stationary waves explained
Principle of superposition of waves and formation of stationary waves covers 7 specification points, including Stationary waves.
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and Nodes and antinodes on strings.. Open a point below for its explanation and revision guidance.
Stationary waves.
A stationary wave forms when two progressive waves of the same frequency and similar amplitude travel in opposite directions through the same medium and superpose. The superposition principle means displacements add algebraically at every point. In a string fixed at both ends, or a microwave reflection, the result has nodes where displacements always cancel and antinodes where they reinforce. Adjacent nodes are separated by half a wavelength, and the distance between a node and the next antinode is a quarter wavelength. Energy is not transferred along a stationary wave; energy is stored and exchanged between kinetic and potential forms. You should be able to describe experiments producing stationary waves and explain the formation in terms of superposition.
Nodes and antinodes on strings.
When a string is fixed at both ends and driven at a suitable frequency, waves reflect at the boundaries and travel back through each other. Superposition of the two travelling waves produces a stationary wave: a pattern that stores energy rather than transporting it along the string. A node is a point of permanently zero displacement, where the two waves always cancel because they arrive in antiphase. An antinode is a point of maximum displacement, where the waves always reinforce because they arrive in phase. Adjacent nodes are separated by half a wavelength, and the same is true of adjacent antinodes. For a string of length L fixed at both ends, the ends must be nodes, so L = nλ/2 and the allowed wavelengths are λ = 2L/n, where n is a positive integer. The simplest pattern, n = 1, has one antinode at the centre.
2l μ The formation of stationary waves by two waves of the same frequency travelling in opposite directions.
A stationary wave forms when two progressive waves of the same frequency and similar amplitude travel in opposite directions through the same region and superpose. On a stretched string, a wave travelling towards a fixed end reflects and travels back, so the incident and reflected waves meet. Where the waves are in phase they reinforce to give an antinode; where they are in antiphase they cancel to give a node. The pattern does not travel along the string: the nodes and antinodes stay in fixed positions, and energy is stored rather than transferred. The frequency is the same for both waves, so the pattern has that frequency, but the distance between adjacent nodes is half the wavelength. The speed of transverse waves on a string is given by v = √(T/μ), where T is the tension and μ is the mass per unit length, so the frequency of a given mode depends on tension, length and μ.
A graphical explanation of formation of stationary waves will be expected.
Stationary waves form when two progressive waves of the same frequency and similar amplitude travel in opposite directions through the same medium. The principle of superposition means displacements add at every point. A graphical explanation shows two travelling waves, often one incident and one reflected, drawn at successive instants; adding their displacements point by point produces a pattern that no longer travels. At nodes, displacements always cancel because the waves are in antiphase; at antinodes, they reinforce in phase. The distance between adjacent nodes is half a wavelength. For example, a string fixed at one end reflects a wave so incident and reflected waves interfere, giving nodes at fixed points and antinodes between them.
Stationary waves formed on a string and those produced with microwaves and sound waves should be considered.
Stationary waves appear in several practical situations. On a stretched string, a wave reflects at a fixed end and interferes with the incident wave; nodes occur at fixed ends and antinodes at free positions, and the string can vibrate in harmonics. With microwaves, a transmitter directs waves at a metal reflector; the reflected wave superposes with the incident wave, producing nodes and antinodes that can be detected with a probe. With sound waves, a loudspeaker facing a hard surface or a resonance tube creates stationary waves; nodes and antinodes in air can be located using a microphone or by resonance. In each case, the same physics applies: same frequency, opposite directions, superposition, and a pattern with fixed nodes and antinodes.
Stationary waves on strings will be described in terms of harmonics. The terms fundamental (for first harmonic) and overtone will not be used.
A stationary wave on a string forms when two identical progressive waves travelling in opposite directions superpose. For a string fixed at both ends, the ends must be nodes, so only certain wavelengths fit: L = nλₙ/2, giving λₙ = 2L/n. The allowed frequencies are fₙ = nv/(2L), where v = √(T/μ), T is tension and μ is mass per unit length. AQA requires these modes to be described as harmonics: first harmonic (n = 1), second harmonic (n = 2), third harmonic (n = 3), and so on. The labels fundamental and overtone must not be used. For example, a 0.80 m string with v = 240 m s⁻¹ has first-harmonic frequency 150 Hz, second harmonic 300 Hz and third harmonic 450 Hz.
Required practical 1: Investigation into the variation of the frequency of stationary waves on a string with length, tension and mass per unit length of the string.
In this required practical, a stretched string is driven by a vibration generator connected to a signal generator. A stationary wave forms when the driving frequency matches a harmonic of the string. For fixed ends, L = nλₙ/2 and v = fₙλₙ, so fₙ = (n/2L)√(T/μ). You investigate how fₙ varies with length L, tension T and mass per unit length μ. Vary one independent variable at a time, keep others constant, and identify the resonant frequency by observing the largest-amplitude stationary wave with clear nodes and antinodes. Plot fₙ against 1/L, against √T and against 1/√μ to obtain straight-line relationships. Assess uncertainty in frequency, length, tension and μ, and evaluate whether the string behaves ideally.
Your focus
- Describe how two progressive waves superpose to form a stationary wave.
- Identify nodes and antinodes and relate their spacing to wavelength.
- Explain why a stationary wave does not transfer energy along the medium.
Show all 21 objectives
- Define a node and an antinode in terms of the displacement of the string at those points.
- Relate the separation of adjacent nodes and antinodes to the wavelength of the waves forming the stationary wave.
- Apply the condition L = nλ/2 to determine allowed wavelengths and frequencies for a string fixed at both ends.
- Explain how two waves of the same frequency travelling in opposite directions superpose to form a stationary wave.
- Describe how nodes and antinodes arise from points of permanent cancellation and reinforcement.
- Use v = √(T/μ) together with the allowed wavelengths on a string to determine frequencies of stationary waves.
- Draw two progressive waves travelling in opposite directions and construct their resultant displacement graph.
- Distinguish nodes from antinodes using displacement and phase.
- Explain how superposition of incident and reflected waves produces a stationary wave pattern.
- Describe how stationary waves are produced on a string, with microwaves and with sound waves.
- Identify nodes and antinodes in each practical arrangement.
- Use node or antinode spacing to determine the wavelength of the waves.
- Describe stationary waves on a string in terms of harmonics, identifying the first, second and third harmonics.
- Derive and apply the relationships L = nλₙ/2 and fₙ = nv/(2L) to string modes.
- Use v = √(T/μ) to determine how tension and mass per unit length affect harmonic frequencies.
- Carry out an investigation into how the frequency of stationary waves on a string depends on length, tension and mass per unit length.
- Identify resonant frequencies from the stationary-wave pattern and record reliable measurements.
- Analyse data graphically to test the relationships fₙ ∝ 1/L, fₙ ∝ √T and fₙ ∝ 1/√μ, and evaluate sources of uncertainty.
Which command word is this?
A concise, factual answer with no explanation required. For example, 'State the principle of superposition' requires: 'The resultant displacement at a point is the vector sum of the individual displacements of the waves meeting at that point.'
Principle of superposition of waves and formation of stationary waves exam tips
Topic Overview
This topic explores how waves interact when they meet, focusing on the principle of superposition and the formation of stationary (standing) waves. You will learn how to add displacements vectorially, predict points of constructive and destructive interference, and describe the key features of stationary waves such as nodes, antinodes, and the relationship between string length and wavelength for different harmonics.
Understanding superposition is fundamental to many areas of physics, including interference phenomena, diffraction gratings, and musical instruments. It also underpins more advanced topics like quantum mechanics and electromagnetic wave behaviour. In AQA A-Level Physics, this topic is assessed in both theoretical explanations and practical calculations, often linking to required practical activities involving strings and microwaves.
Key Concepts
- →Superposition: When two or more waves meet at a point, the resultant displacement is the vector sum of the individual displacements.
- →Stationary waves are formed by the superposition of two progressive waves of the same frequency, wavelength, and amplitude travelling in opposite directions.
- →Nodes are points of zero displacement (destructive interference); antinodes are points of maximum displacement (constructive interference).
- →The distance between adjacent nodes (or antinodes) is half a wavelength (lambda/2).
- →For a string fixed at both ends, stationary waves form when the length L = n x (lambda/2), where n is a positive integer (harmonic number).
Marking Points
- A stationary wave is formed by the superposition of two progressive waves of the same frequency travelling in opposite directions.
- At a node, the two waves always arrive in antiphase, so their displacements cancel and the resultant displacement is always zero.
- At an antinode, the two waves arrive in phase, so their displacements reinforce and the resultant displacement is maximum.
- The distance between adjacent nodes is half a wavelength, and the distance between a node and the next antinode is a quarter wavelength.
- Stationary waves do not transfer energy along the wave; energy is stored and exchanged between kinetic and potential energy.
- A stationary wave can be produced on a stretched string, in a microwave reflection experiment, or using a loudspeaker and reflecting surface.
- A node is a point of zero displacement at all times, produced where the two travelling waves are always in antiphase and cancel by superposition.
- An antinode is a point of maximum displacement, produced where the two travelling waves are always in phase and reinforce by superposition.
- The distance between adjacent nodes, or between adjacent antinodes, is λ/2, so the distance from a node to the next antinode is λ/4.
- For a string fixed at both ends, the fixed ends must be nodes, giving L = nλ/2 and allowed wavelengths λ = 2L/n for n = 1, 2, 3, ...
- The amplitude of the stationary wave varies with position along the string, unlike a travelling wave, which has the same amplitude everywhere.
- Energy in a stationary wave is stored between the nodes rather than transferred along the string, because the two travelling waves carry energy in opposite directions.
- Stationary waves form when two waves of the same frequency travel in opposite directions and superpose.
- The waves must have the same frequency; if the frequencies differ, the pattern of nodes and antinodes moves and is not stationary.
- Reinforcement occurs where the waves arrive in phase, giving an antinode; cancellation occurs where they arrive in antiphase, giving a node.
- The nodes and antinodes remain in fixed positions, so the stationary wave does not transfer energy along the string.
- On a string, reflection at a fixed end produces the oppositely travelling wave, and the fixed end is always a node.
- The wave speed on a string is v = √(T/μ), where T is the tension and μ is the mass per unit length, linking the mode frequency to the physical properties of the string.
- State that two progressive waves of the same frequency travel in opposite directions through the same medium.
- Apply the principle of superposition by adding displacements of the two waves at each point.
- Identify nodes as points of zero resultant displacement where the waves are in antiphase.
- Identify antinodes as points of maximum resultant displacement where the waves are in phase.
- Use a graph or diagram to show incident and reflected waves and their resultant at a chosen instant.
- Relate the node spacing to half a wavelength and note that the stationary wave pattern does not transfer energy.
- Describe how a stationary wave forms on a stretched string by reflection at a fixed end and superposition with the incident wave.
- State that nodes occur at fixed ends of a string and antinodes occur at positions of maximum vibration.
- Describe how microwaves from a transmitter reflect from a metal plate and form a stationary wave detected by a probe.
- Describe how sound waves from a loudspeaker reflect from a hard surface or within a tube to form a stationary wave.
- Identify that in all three cases the stationary wave has the same frequency as the source and a node spacing of half a wavelength.
- Explain how measurements of node or antinode positions can be used to determine wavelength.
- A stationary wave on a string is produced by the superposition of two progressive waves of the same frequency and similar amplitude travelling in opposite directions along the string.
- For a string fixed at both ends, each end is a node, so the length L contains a whole number n of half-wavelengths: L = nλₙ/2, hence λₙ = 2L/n.
- The harmonic number n equals the number of half-wavelengths on the string and also the number of antinodes present.
- The frequency of the nth harmonic is fₙ = nv/(2L), where v is the wave speed on the string.
- The wave speed is given by v = √(T/μ), where T is the tension and μ is the mass per unit length, so fₙ = (n/2L)√(T/μ).
- The first harmonic is the lowest-frequency stationary-wave mode; the second harmonic has twice that frequency and the third harmonic three times it, for an ideal string.
- Answers must use harmonic terminology: first harmonic, second harmonic, third harmonic; the terms fundamental and overtone are not used in this specification.
- Set up a stretched string with a vibration generator near one end and a fixed support or pulley at the other, adjusting tension using hanging masses.
- Keep the harmonic number n constant, usually n = 1, while varying the independent variable so that comparisons are valid.
- For the length investigation, measure the vibrating length L between the fixed end and the vibration generator, and determine the resonant frequency for each L.
- For the tension investigation, calculate T from the hanging mass using T = mg, and measure the resonant frequency for each tension while keeping L and μ constant.
- For the mass per unit length investigation, use strings of different material or thickness, determine μ by weighing a measured length, and measure the resonant frequency while keeping L and T constant.
- Identify resonance by the formation of a stable stationary wave with clearly defined nodes and antinodes and maximum amplitude.
- Process data graphically: fₙ against 1/L, fₙ against √T and fₙ against 1/√μ should each give a straight line through the origin if the theory holds.
- Assess uncertainties in frequency, length, tension and μ, and comment on sources of error such as the mass of the string, friction at the pulley and the finite width of nodes.
Examiner Tips
- 💡Sketch the stationary wave pattern and label nodes and antinodes clearly before explaining the formation.
- 💡Use the phrase ‘two progressive waves travelling in opposite directions’ when defining a stationary wave.
- 💡When asked about distance, relate node spacing to half a wavelength and node-to-antinode spacing to a quarter wavelength.
- 💡Sketch the string pattern and label nodes and antinodes before writing any explanation, so your description matches the diagram.
- 💡When asked to find a wavelength or frequency, first count the number of half-wavelengths that fit between the fixed ends.
- 💡Use the phrase 'superposition of two waves travelling in opposite directions' to show you understand the mechanism, not just the pattern.
- 💡State explicitly that the two waves have the same frequency and travel in opposite directions; both conditions are needed for a stationary wave.
- 💡Link any calculation to the boundary condition that a fixed end is a node, then use L = nλ/2 before applying v = fλ.
- 💡If a question gives tension and mass per unit length, calculate v = √(T/μ) first, then combine it with the allowed wavelengths to find frequencies.
- 💡Draw two separate travelling waves with the same wavelength and amplitude, then add their displacements at several points to show the resultant.
- 💡Label nodes and antinodes clearly and mark one wavelength and the node spacing on the diagram.
- 💡Use the phrase 'superposition of two waves travelling in opposite directions' when explaining formation.
- 💡For each example, state the wave source, the reflector, and how nodes and antinodes are detected or observed.
- 💡Use the node spacing to calculate wavelength, remembering that adjacent nodes are separated by half a wavelength.
- 💡Compare the three examples in a table to show the common physics and the different detection methods.
- 💡Sketch the string for n = 1, 2 and 3, labelling nodes N and antinodes A, then count half-wavelengths to choose the correct relationship.
- 💡When a question gives tension and mass per unit length, calculate v = √(T/μ) first, then substitute into fₙ = nv/(2L).
- 💡Use harmonic vocabulary throughout; avoid the words fundamental and overtone even if they appear in general reading.
- 💡Check unit consistency: convert grams per metre to kilograms per metre and centimetres to metres before calculating.
- 💡State clearly which variable is independent, which is dependent and which are controlled in each part of the investigation.
- 💡Record repeated readings of resonant frequency and take a mean to reduce random uncertainty.
- 💡Plot a graph rather than relying on a single pair of values, and use the gradient or straight-line form to test the predicted relationship.
- 💡Describe how you judged resonance, for example by maximum amplitude and the clearest node pattern, so the method is reproducible.
- 💡Always define superposition in terms of vector addition of displacements, not just 'adding waves'.
- 💡When describing stationary wave formation, explicitly state that the two waves have the same frequency, wavelength, and amplitude, and travel in opposite directions.
- 💡In calculations, sketch the stationary wave to identify nodes and antinodes, and remember that node-to-node distance is lambda/2.
Common Mistakes
- Confusing nodes with points of maximum displacement; the correction is that nodes have zero resultant displacement and antinodes have maximum resultant displacement.
- Thinking that a stationary wave transfers energy from one end to the other; the correction is that energy is stored and exchanged, not propagated.
- Assuming that any two waves superpose to give a stationary wave; the correction is that the waves must have the same frequency and travel in opposite directions.
- Treating a node as a point of minimum but non-zero displacement: correct this by stating that at a node the displacement is exactly zero at every instant.
- Confusing the node-to-node distance with a full wavelength: correct this by using λ/2 between adjacent nodes and λ/4 between a node and the next antinode.
- Assuming any driving frequency produces a stationary wave on a fixed string: correct this by requiring the wavelength to satisfy λ = 2L/n so that both fixed ends are nodes.
- Saying that a stationary wave is produced by a single wave bouncing back and forth: correct this by describing two waves of the same frequency travelling in opposite directions that superpose.
- Believing that energy travels along a stationary wave from one end to the other: correct this by stating that energy is stored between the nodes and there is no net transfer along the string.
- Using v = fλ with the node-to-node distance as λ: correct this by using the full wavelength, which is twice the node-to-node distance.
- Treating a stationary wave as a single travelling wave: correct this by showing two opposite-travelling waves and their superposition.
- Confusing nodes with points of maximum displacement: correct this by defining nodes as zero-displacement points and antinodes as maximum-displacement points.
- Assuming all points on a stationary wave oscillate in phase: correct this by noting that points between adjacent nodes are in phase, but adjacent loops are in antiphase.
- Believing that a stationary wave transfers energy along the string or through the air: correct this by stating that energy is stored in the oscillating system and not propagated.
- Thinking that nodes are always at the ends of every medium: correct this by noting that nodes occur at fixed boundaries, while free ends or open tube ends may be antinodes.
- Assuming microwaves and sound waves cannot form stationary waves because they are not visible: correct this by explaining that any wave reflecting and superposing can form stationary waves.
- Writing that the string length equals one wavelength for the first harmonic; the correction is that the length equals one half-wavelength, so λ₁ = 2L.
- Calling the lowest mode the fundamental or labelling higher modes as overtones; the correction is to use first harmonic, second harmonic and third harmonic.
- Assuming frequency is directly proportional to tension; the correction is that fₙ ∝ √T, because v = √(T/μ).
- Treating the harmonic number as the number of nodes; the correction is that n is the number of half-wavelengths and the number of antinodes, while the number of nodes is n + 1.
- Changing more than one variable at once; the correction is to control all but the independent variable so that any trend can be attributed to that variable.
- Measuring the total string length rather than the vibrating length between the fixed end and the vibration generator; the correction is to measure the length that actually contains the stationary wave.
- Using the mass in grams directly to find tension; the correction is to convert to kilograms and use T = mg with g in m s⁻².
- Assuming the string is massless or that μ is negligible; the correction is to measure μ by weighing a known length and to consider its effect on tension and wave speed.
- Misconception: Stationary waves transfer energy from one point to another. Correction: Stationary waves store energy in the oscillations; there is no net transfer of energy along the wave.
- Misconception: All points on a stationary wave vibrate in phase. Correction: Points within the same loop vibrate in phase, but adjacent loops vibrate in antiphase (180 degrees out of phase).
- Misconception: The distance between two nodes is one wavelength. Correction: The distance between adjacent nodes is half a wavelength (lambda/2).
Revision Plan
- 1Day 1-2: Review the definition of superposition and practice vector addition of displacements using simple diagrams. Use the wave equation to solve basic problems.
- 2Day 3-4: Study stationary wave formation. Watch videos or use simulations to visualise nodes and antinodes. Learn the relationship L = n x (lambda/2) for strings and pipes.
- 3Day 5-6: Work through exam-style questions on stationary waves, including calculations of wavelength, frequency, and harmonic number. Check your answers against mark schemes.
- 4Day 7-8: Complete a practical activity or virtual experiment to measure the wavelength of a stationary wave on a string. Record and analyse your results.
- 5Day 9-10: Create a summary sheet with key definitions, formulas, and common pitfalls. Test yourself with active recall prompts and past paper questions.
Exam Question Types
- 📋Definition and explanation questions: e.g., 'State the principle of superposition' or 'Explain how a stationary wave is formed.' Advice: Use precise terminology and mention vector addition and opposite directions.
- 📋Calculation questions: e.g., 'Calculate the wavelength of the stationary wave' or 'Determine the frequency of the third harmonic.' Advice: Draw a diagram, identify nodes/antinodes, and use L = n x (lambda/2).
- 📋Practical and data analysis questions: e.g., 'Describe an experiment to measure the speed of sound using stationary waves' or 'Analyse the graph of frequency against harmonic number.' Advice: Include details of measurements, apparatus, and how to reduce uncertainty.
- 📋Comparison questions: e.g., 'Compare progressive and stationary waves.' Advice: Make a table of similarities and differences, focusing on energy transfer, amplitude, and phase.
Command Word Expectations (AQA)
A concise, factual answer with no explanation required. For example, 'State the principle of superposition' requires: 'The resultant displacement at a point is the vector sum of the individual displacements of the waves meeting at that point.'
A detailed answer that gives reasons or mechanisms. For example, 'Explain how a stationary wave is formed' requires: 'Two progressive waves of the same frequency, wavelength and amplitude travel in opposite directions through each other. They superpose, and points of constructive interference form antinodes, while points of destructive interference form nodes.'
A numerical answer with working and correct units. For example, 'Calculate the wavelength' requires: 'Use the formula lambda = 2L/n, substitute values, and give the answer in metres.'
How Students Lose Marks (Examiner Pitfalls)
Step-by-Step Worked Solutions
Question: A stationary wave is formed on a string of length 1.2 m fixed at both ends. The string vibrates in its third harmonic. Calculate the wavelength of the wave and the frequency if the wave speed is 48 m/s.
- 1.Step 1: Identify given facts: length L = 1.2 m, harmonic number n = 3, wave speed v = 48 m/s. For a string fixed at both ends, the length of the string equals n x (lambda/2).
- 2.Step 2: Apply the formula: L = n x (lambda/2). Rearranging gives lambda = 2L/n = (2 x 1.2)/3 = 0.80 m.
- 3.Step 3: Use the wave equation v = f x lambda to find frequency: f = v/lambda = 48/0.80 = 60 Hz.
- 4.Step 4: State final conclusion with units: The wavelength is 0.80 m and the frequency is 60 Hz.
Question: Two loudspeakers emit sound waves of frequency 680 Hz in phase. A microphone is moved along a line parallel to the line joining the speakers. The speed of sound in air is 340 m/s. Calculate the distance between successive maxima detected by the microphone.
- 1.Step 1: Identify given facts: frequency f = 680 Hz, speed of sound v = 340 m/s. The distance between successive maxima (constructive interference) is one wavelength.
- 2.Step 2: Calculate wavelength using v = f x lambda: lambda = v/f = 340/680 = 0.50 m.
- 3.Step 3: The distance between successive maxima is equal to the wavelength, so distance = 0.50 m.
- 4.Step 4: State final conclusion with units: The distance between successive maxima is 0.50 m.