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Standing Waves on a Stretched String

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A Level Physics Practical: Standing Waves on a Stretched String

An enhanced A Level Physics practical guide to producing standing waves on a stretched string, identifying resonant modes and investigating how resonant frequency depends on vibrating length, tension and mass per unit length. The guide develops wave-speed calculations, linearised graph analysis, quantitative uncertainty, evaluation, troubleshooting, exam preparation and technician support.

 

student investigating standing waves on a stretched string

 

Teacher note: This resource is provided as a practical support guide to accompany laboratory equipment. Teachers should adapt procedures and risk assessments to suit their curriculum requirements, examination board specifications and local laboratory policies. Switch off the signal and vibration generators before touching the string, changing the hanging mass or adjusting the apparatus.

Level
A Level Physics
Lesson time
Two 60-minute lessons
Risk level
Centre assessment required
Skills
Resonance, graphs and uncertainty
Support
Teacher and technician notes
01
Practical overview 02
Background theory 03
Apparatus 04
Experimental method 05
Extension investigations 06
Graph analysis 07
Uncertainty and evaluation 08
Troubleshooting 09
Risk assessment 10
Exam support

Practical overview

In this experiment, students produce standing - or stationary - waves on a stretched string or wire. They identify resonant modes and investigate how the resonant frequency depends on the vibrating length, L, the tension, T, and the mass per unit length, μ. Students also use the number of loops in the standing-wave pattern to determine wavelength and wave speed.

Core investigation: keep L, T and μ fixed, find at least four clear resonant modes, count the loops to assign the harmonic number n, approach each resonance from below and above, and use a graph of fn against n to determine wave speed from the gradient.

Why this practical matters

Standing waves make superposition and resonance directly observable. Waves travelling in opposite directions superpose to produce a stationary pattern containing nodes and antinodes.

  • Tests mathematical relationships for waves on a stretched string.
  • Develops practical resonance-finding techniques.
  • Builds linearised graph and gradient-analysis skills.
  • Connects ideal wave models with real effects such as damping and imperfect boundaries.

Learning objectives

  • Produce and recognise stable standing-wave patterns.
  • Identify nodes, antinodes and harmonic number correctly.
  • Use λn = 2L/n, v = fλ and v = √(T/μ).
  • Apply fn = (n/2L)√(T/μ).
  • Investigate fn ∝ 1/L, fn2 ∝ T and fn2 ∝ 1/μ.
  • Determine linear mass density from the mass and length of a string sample.
  • Analyse gradients, intercepts and uncertainty quantitatively.

Length

At fixed T, μ and n: fn ∝ 1/L.

Tension

At fixed L, μ and n: fn ∝ √T, so fn2 ∝ T.

Linear density

At fixed L, T and n: fn ∝ 1/√μ, so fn2 ∝ 1/μ.

Harmonic number

At fixed L, T and μ: fn ∝ n.

Background theory

A progressive wave transfers energy along a medium. For any wave, the wave speed is related to frequency and wavelength by v = fλ. Standing waves form by the superposition of two coherent progressive waves of the same frequency and equal amplitude travelling in opposite directions.

Key standing-wave relationships: λn = 2L/n    |    v = fλ    |    v = √(T/μ)    |    fn = (n/2L)√(T/μ)

Nodes

Nodes are positions of zero displacement amplitude. The effective vibrating length is measured between the two end nodes.

Antinodes

Antinodes are positions of maximum amplitude. The number of loops - or antinodes - equals the harmonic number n.

Phase

Points within the same loop oscillate in phase. Adjacent loops oscillate in antiphase, with a phase difference of π rad.

Energy

In an ideal standing wave there is no net time-averaged energy transfer along the string. In the real apparatus, the driver supplies energy and damping dissipates it.

Harmonic Loops / antinodes Wavelength Relationship to L
First, n = 1 1 λ1 = 2L L = λ1/2
Second, n = 2 2 λ2 = L L = λ2
Third, n = 3 3 λ3 = 2L/3 L = 3λ3/2
Fourth, n = 4 4 λ4 = L/2 L = 2λ4

Model assumption: these relationships assume that the effective vibrating section has a node at each end and that end effects are negligible.

Standing Waves: Nodes & Antinodes Explained

Resonance and resonant frequency

Resonance occurs when the driving frequency is close to one of the natural frequencies of the stretched string. Successive cycles of the driver reinforce the selected mode, so its steady-state amplitude becomes locally maximum. In the real apparatus, damping limits the amplitude.

Variable changed Quantities kept constant Prediction
L T, μ, n fn ∝ 1/L
T L, μ, n fn ∝ √T, so fn2 ∝ T
μ L, T, n fn ∝ 1/√μ, so fn2 ∝ 1/μ
n L, T, μ fn ∝ n

Apparatus and setup

Core equipment

  • Signal generator.
  • Low-voltage vibration generator.
  • Light string or wire of known or measurable linear mass density.
  • Pulley and secure pulley support.
  • Mass hanger and suitable slotted masses.
  • Two secure clamp stands or the manufacturer’s mounting system.
  • Metre rule.
  • Digital balance.
  • A long sample of each string for measuring μ.
  • Adjustable bridge, moveable support or another means of changing L.
  • Connecting leads.
  • Padded catch tray or box beneath the mass hanger.

Optional equipment

  • Frequency counter where the generator does not provide a reliable digital display.
  • Force sensor for checking the actual horizontal tension.
  • Stroboscope, only where its use has been separately risk assessed.
  • Micrometer if wire diameter is investigated as an additional extension.

Equipment limit: use the lowest tension and driving amplitude that produce a clear pattern. Do not exceed the rated load of the vibration generator, string, pulley, mass hanger or supports.

Apparatus setup

  1. Secure the vibration generator and pulley supports using heavy bases or bench clamps.
  2. With the signal generator switched off, connect it to the vibration generator using suitable leads.
  3. Attach the string to the vibration generator, pass it horizontally over the top of the pulley and suspend the mass hanger vertically.
  4. Add a centre-approved total mass, including the mass hanger. Position a padded catcher beneath the hanger without allowing it to touch or support the mass.
  5. Align the apparatus so the string is horizontal, the pulley rotates freely and the string does not rub against any support.
  6. Place a metre rule parallel to the vibrating section. Measure L between the two effective end nodes, not between the physical attachments.
  7. If pulley friction is negligible, calculate the nominal tension using T ≈ mtotalg, where mtotal includes the hanger and all added masses and g = 9.81 N kg−1.

 

Standing Waves: Apparatus Setup

 

Experimental method: resonant frequencies at fixed L, T and μ

The core investigation finds several resonant modes while keeping the vibrating length, tension and linear mass density unchanged. The number of loops gives the harmonic number, n.

Step-by-step method

  1. Secure the vibration generator and pulley supports. Align the string horizontally and place a padded catch tray beneath the mass hanger.
  2. With the signal generator switched off, attach the string and add a suitable known mass. Include the hanger when determining mtotal.
  3. Calculate the nominal tension: T ≈ mtotalg. This assumes that the mass is stationary and pulley friction is negligible.
  4. Measure the effective vibrating length L between the two end nodes. Record the metre-rule resolution and estimated uncertainty.
  5. Set the generator output amplitude to a low value. Begin at a low frequency within the safe operating range. Keep the drive-amplitude setting constant for all resonance measurements.
  6. Increase the frequency slowly until a clear, stable standing-wave pattern is produced. Fine-adjust to the local maximum amplitude.
  7. Count the loops. The number of loops is the harmonic number n. Do not assume the first observed resonance is necessarily n = 1.
  8. Approach the resonance peak from below and above. Record both frequencies and calculate their mean.
  9. Record n, fn, L, T and the string identity. Repeat for at least four clear modes while keeping L, T and μ unchanged.
  10. Calculate λn = 2L/n and vn = fnλn.
  11. Plot fn against n. From gradient = v/(2L), calculate v = 2L × gradient.
  12. Switch off the generator before touching the string, changing the mass, changing the length or altering the supports.

Good resonance practice

  • Use a low but clearly visible driving amplitude.
  • Sweep the frequency slowly.
  • Identify the harmonic by counting loops, not nodes.
  • Fine-adjust to the local maximum steady-state amplitude.
  • Approach the peak from both below and above to reduce judgement bias.
  • Keep the drive-amplitude setting constant when comparing resonances.
  • Keep the hanging mass stationary.

Examiner reminder: sketch and count the loops before assigning n. Then check that λn = 2L/n is consistent with the observed pattern.

Observed harmonic n Loops checked f↑ / Hz f↓ / Hz Mean fn / Hz λn = 2L/n / m vn = fnλn / m s−1
             
             
             
             
             

Fixed settings to record

String ID: ________    |    μ = ________ kg m−1    |    L = ________ m    |    mtotal = ________ kg    |    T = ________ N    |    estimated uncertainty in L = ________ m

Step-by-Step Experimental Workflow - Standing Waves on a String

Calculating wavelength and wave speed

1. Wavelength

For the nth harmonic:

λn = 2L/n

2. Wave speed

Use the measured resonant frequency:

vn = fnλn

3. Gradient method

For fn against n:

v = 2L × gradient

Consistency check: for an ideal string at constant tension and linear density, individually calculated wave-speed values should be approximately constant for different harmonics. The guide recommends using the graph gradient as the primary final result and individual vn values as a check.

Calculations: Wavelength & Wave Speed

Extension investigations: effects of L, T and μ

For each extension, select and retain one harmonic number n, use at least five values spanning a safe range, verify the loop count at every setting, approach each resonance from below and above, repeat measurements and calculate a mean frequency, and keep the drive-amplitude setting constant.

Investigation A: vibrating length

Keep T, μ and n constant. Change L using a moveable bridge or suitable adjustable support. Find the resonant frequency at each length.

Plot: fn against 1/L.

Expected relationship: fn ∝ 1/L.

Investigation B: tension

Keep L, μ and n constant. Change the total hanging mass within the approved apparatus range. Use T ≈ mtotalg where pulley friction is negligible.

Plot: fn2 against T.

Expected relationship: fn2 = [n2/(4L2μ)]T.

Investigation C: mass per unit length

Use strings or wires of different linear mass density. Determine each value using a long sample: μ = msample/lsample. Keep L, T and n constant.

Plot: fn2 against 1/μ.

Expected relationship: fn2 = [n2T/(4L2)](1/μ).

Safety and measurement note: record the balance and length-scale resolutions when determining μ. Switch off before changing the string, length or hanging mass. Use only centre-approved mass and frequency ranges within the ratings of the actual vibration generator, string, pulley, hanger and supports.

Graph analysis

Use an unconstrained best-fit line. Although the theoretical relationship may pass through the origin, forcing an experimental line through the origin can hide end effects, pulley friction or a frequency offset.

Graph 1: harmonic sequence

Plot fn against n.

fn = (v/2L)n
Therefore v = 2L × gradient.

Graph 2: varying length

Plot fn against 1/L.

fn = (n/2)√(T/μ)(1/L).

Graph 3: varying tension

Plot fn2 against T.

Expected gradient = n2/(4L2μ).

Graph 4: varying linear density

Plot fn2 against 1/μ.

Expected gradient = n2T/(4L2).

Graph-quality checklist

  • Label axes with quantity and unit.
  • Use a large gradient triangle.
  • Show uncertainty bars where meaningful.
  • Calculate maximum and minimum acceptable gradients where required.
  • Discuss the physical meaning of the intercept.
  • A non-zero intercept in fn against n may indicate that effective node positions do not coincide with the assumed boundaries.

Sources of uncertainty and advanced evaluation

Source Why it matters Improvement / response
Node location The minimum-motion region has a finite width. Estimate uncertainty from the width of the node region, not only the metre-rule scale.
Length Both end-node positions contribute to uncertainty in L. Use a longer vibrating length and include uncertainty from both end positions.
Frequency A resonance peak has a finite width. Average upward and downward sweeps or bracket the peak.
Tension T ≈ mtotalg is only approximate when pulley friction is present. Use a low-friction pulley or a force sensor to reduce or quantify the effect.
Linear density A short string sample gives a large percentage uncertainty. Measure the mass and length of a long sample.
End effects Effective nodes may not coincide with physical attachment points. Measure between effective end nodes and interpret any non-zero intercept.
Drive amplitude Excessive drive can alter boundary behaviour and make the pattern unstable. Use the lowest amplitude that produces a clear pattern and keep it constant.
String uniformity Knots, fraying, stretching and non-uniform diameter change μ. Inspect and replace damaged material; use a uniform sample.

Quantitative uncertainty

For v = 2Lf/n, with n counted exactly:

uv/v ≈ uL/L + uf/f

For v = √(T/μ):

uv/v ≈ ½(uT/T + uμ/μ)

Improvements

  • Use a longer vibrating length to reduce percentage uncertainty.
  • Use several harmonics and determine wave speed from the fn-against-n gradient.
  • Check the specification or calibration of digital displays rather than assuming accuracy.
  • Repeat measurements and average results to improve reliability.
  • Use a low-friction pulley.
  • Mark node positions carefully; use a stroboscope only after a separate flicker-risk assessment.

Advanced evaluation points (A/A*)

  • Use the complete theoretical model: a linearised graph tests the relationship more strongly than agreement between a few individually calculated values.
  • Interpret the intercept: a significant non-zero intercept can indicate an effective-length correction, frequency offset or incorrectly assigned harmonic numbers. Do not automatically remove it by forcing the line through the origin.
  • Separate random and systematic effects: repeated resonance judgements can reduce random uncertainty but do not remove pulley friction, incorrect node positions, non-uniform μ or generator calibration error.
  • Evaluate pulley friction: the horizontal tension may differ from mtotalg.
  • Evaluate end conditions: the vibration generator and pulley are not perfectly fixed boundaries.
  • Evaluate the string model: real strings can stretch, sag, possess stiffness and become non-uniform.
  • Use quantitative uncertainty: compare the experimental gradient with maximum and minimum acceptable values and state whether the theoretical value lies within the experimental uncertainty.

Troubleshooting guide

Problem Possible causes Safe response
No clear standing wave Wrong frequency range; output too low; poor connection; string rubbing. Check connections with power off, check alignment, then sweep frequency slowly.
Pattern unstable Slightly off resonance; drive amplitude too large; loose clamp; mass bouncing. Reduce amplitude, secure the apparatus and fine-adjust frequency.
String motion distorted String rubbing on pulley; pulley misaligned; damaged string. Switch off, realign and inspect the string and pulley.
Hanging mass oscillates Excessive drive amplitude or unsuitable tension. Switch off, support the hanger, reduce drive and check the approved mass.
String or wire breaks Damage, excessive tension or a sharp contact point. Switch off immediately, keep clear of falling masses, support and remove the hanger, replace the string and inspect the apparatus before restarting.
Calculated v changes strongly with n Wrong loop count; wrong L; poor resonance judgement; end effects. Sketch each pattern, remeasure between nodes and use a best-fit graph.

Common misconceptions and student mistakes

Common misconceptions

“Standing waves travel along the string.”
The standing-wave pattern does not travel along the string, although it is formed from progressive component waves travelling in opposite directions.

“Nodes are points of maximum vibration.”
A node is a position of zero amplitude; an antinode is a position of maximum amplitude.

“Increasing frequency always increases wave speed.”
For a given string at fixed T and μ, changing the generator frequency changes the resonant mode and wavelength; it does not significantly change the propagation speed.

“Resonance occurs at any frequency.”
Resonance occurs only near the natural frequencies of the stretched string.

“A standing wave transfers no energy anywhere.”
In an ideal standing wave there is no net time-averaged energy transfer along the string. In the real apparatus the driver supplies energy that is dissipated through damping.

High-impact student mistakes

  • Measuring total string length rather than the effective vibrating length.
  • Counting nodes instead of loops.
  • Using an unstable resonance.
  • Recording frequency inaccurately.
  • Forgetting that wavelength depends on harmonic number.
  • Treating the first observed resonance as n = 1 without counting loops.
  • Forgetting to include the mass hanger in mtotal.
  • Recording mass in kilograms as though it were tension in newtons.
  • Measuring between attachment points rather than effective end nodes.
  • Forcing a graph through the origin.
  • Counting n + 1 nodes instead of n loops.
  • Changing L, T or μ without keeping the other variables and n fixed.

Resonance Explained

Motion and Phase of Points on a Standing Wave

Teacher and technician preparation

Before the lesson

  • Test each signal generator and vibration generator together.
  • Identify a safe frequency, amplitude and mass range for the actual equipment.
  • Record the maximum permitted load for the vibrator, pulley, hanger and string.
  • Inspect strings or wires for fraying, sharp bends, knots and damage.
  • Prepare long samples for measuring μ, labelled by string type.
  • Measure or provide approximate μ values so expected resonant frequencies can be estimated.
  • Provide an adjustable bridge or suitable method for changing L.
  • Secure stands with heavy bases or bench clamps.
  • Position a padded catcher beneath each mass hanger.
  • Check pulleys rotate freely and strings do not rub on supports.
  • Pre-test several clear modes at each station.
  • Mark the generator setting that gives adequate but not excessive amplitude.

High-success technician tips

  • Use light, flexible strings in good condition for clear resonance patterns.
  • Check pulleys spin freely to maintain consistent tension.
  • Prepare identical mass sets for all groups to improve consistency.
  • Set up a working demonstration station before students arrive.
  • Keep spare strings and leads available.
  • Where a stroboscope is supplied, prepare separate flicker-risk controls.

Suggested lesson timing - two-lesson structure

Lesson 1 activity Time Lesson 2 activity Time
Safety, theory and apparatus demonstration 10 min Planning variables and risks 10 min
Setup and checks 10 min Investigate one of L, T or μ 30 min
Find at least four resonant modes 20 min Process and graph data 15 min
Calculate λ and v 10 min Evaluation and conclusions 5 min
Plot fn against n and evaluate 10 min    

Teacher assessment opportunities

Students can demonstrate correct identification of nodes and antinodes, recognition of resonance, wavelength measurement, harmonic relationships, wave-speed calculations, graphical analysis, quantitative uncertainty, safe working, variable control, anomaly identification and maintenance of an individual contemporaneous practical record.

Risk assessment

The guide provides model risk-assessment information only. Complete the final centre assessment using the actual equipment ratings, approved string or wire, maximum total hanging mass, frequency and drive-amplitude limits, manufacturer instructions and local procedures. Assign an overall risk rating only after these factors have been reviewed.

Hazard / source Possible harm Control measures Emergency action
Falling mass hanger or slotted masses Foot or impact injury; damaged floor or apparatus. Secure pulley and stand; use a locking hanger; place a padded catcher just below the lowest expected position without supporting the hanger; keep feet clear; support the hanger while changing masses; switch off before adjustments. Switch off and isolate apparatus; keep others clear; provide first aid; report and inspect apparatus before reuse.
Tensioned string or wire snapping Recoil injury; eye injury; cuts from wire. Inspect before use; use the lowest effective tension; do not exceed ratings; keep faces away from the line of the string; use eye protection where required; replace damaged material immediately. Switch off; do not handle sharp wire with bare hands; provide first aid; dispose of broken wire in an approved sharps container.
Moving string and vibration generator Fingers, hair or clothing caught; minor impact. Keep hands clear while running; tie back long hair; secure loose clothing; switch off before touching the string or changing the setup. Switch off immediately; free the person without pulling against moving apparatus; seek first aid where required.
Unstable stands or pulley support Apparatus toppling; crush or pinch injury. Use heavy bases or bench clamps; keep the overhanging mass close to the bench; tighten bosses and clamps; keep equipment away from an unprotected edge. Keep clear of falling equipment; provide first aid; remove damaged apparatus from service.
Electrical equipment and leads Electric shock from damaged equipment; overheating. Use approved low-voltage equipment; inspect leads and plugs; keep equipment dry; make connections with power off; do not use damaged equipment. Isolate the supply; do not touch exposed conductors; report the fault and follow local electrical emergency procedures.
Trailing leads Trips and falls. Route leads away from walkways; shorten, secure or cover them; keep bags clear. Stop work, make the area safe and provide first aid.
Stroboscope, if used Headache, dizziness or photosensitive reaction. Use only after a separate centre assessment; warn users; do not expose anyone known to be susceptible; minimise duration and avoid unnecessary viewing. Stop the stroboscope immediately; assist the affected person and follow the centre’s medical response procedure.

Exam support and practical skills

Common exam questions

  1. What is a standing wave?
  2. What is the difference between a node and an antinode?
  3. Why does resonance occur?
  4. A standing wave has four loops. What harmonic is this?
  5. Why is the vibrating length measured between the end nodes?
  6. Explain why increasing tension increases wave speed.
  7. Why is wave speed approximately constant for different harmonics at the same tension?
  8. Why should measurements be taken for several harmonics?

Answer points

  1. A stationary interference pattern produced by superposition of two coherent waves of the same frequency travelling in opposite directions.
  2. A node has zero amplitude; an antinode has maximum amplitude.
  3. The driving frequency is close to a natural frequency, so energy supplied on successive cycles builds a large-amplitude mode limited by damping.
  4. The fourth harmonic, because the number of loops equals n.
  5. λn = 2L/n applies to the effective distance between boundary nodes; including string outside them overestimates L.
  6. At constant μ, v = √(T/μ), so increasing T increases v.
  7. For an approximately non-dispersive string, v depends mainly on T and μ; different modes have different f and λ but fλ is approximately constant.
  8. Several modes provide multiple graph points; the gradient gives a more reliable wave speed, while scatter and intercept reveal limitations.

Examiner advice summary

  • Measure the effective vibrating length between the end nodes.
  • Count loops before assigning the harmonic number.
  • Include the mass hanger in mtotal.
  • Convert mass to tension using T ≈ mtotalg.
  • Use amplitude, not instantaneous displacement, when defining nodes and antinodes.
  • Show units at every stage.
  • Use a best-fit gradient rather than relying only on a mean of individual wave speeds.
  • Do not force a graph through the origin without justification.
  • State instrument resolution and estimated measurement uncertainty.
  • Distinguish random uncertainty from systematic effects.
  • Qualify conclusions: wave speed is approximately constant for fixed T and μ.
  • Link every conclusion to the observed trend, gradient and uncertainty.

Thinking like a physicist (A/A*)

  • Look for patterns: resonance occurs at specific frequencies because standing waves obey mathematical relationships.
  • Connect theory to observation: use the wave model to explain visible nodes and antinodes.
  • Question measurements: consider how accurately node positions and length can be identified.
  • Control variables carefully: when investigating L, T or μ, change only one quantity and keep n and the other physical quantities constant.
  • Use evidence to support conclusions about wave speed and resonance.
  • Recognise ideal versus real systems: damping and imperfect end conditions affect results.
  • Seek underlying relationships such as v = fλ and v = √(T/μ).
  • Evaluate uncertainty and consider how it affects conclusions.
  • Consider phase: points within one loop oscillate in phase; adjacent loops oscillate in antiphase.
  • Interpret energy carefully: an ideal standing wave has no net time-averaged energy transfer along the string.

Frequently asked questions

Why should the first resonance observed not automatically be called the fundamental?

The first resonance found during a sweep may not be n = 1. Count the loops in the pattern: the number of loops equals the harmonic number.

Why is L measured between the effective end nodes?

The standing-wave relationship L = nλ/2 applies to the vibrating section between boundary nodes. Measuring to physical attachment points can introduce a systematic error.

Why approach a resonance from below and above?

A resonance has a finite width. Recording values from upward and downward sweeps and taking their mean reduces uncertainty in judging the peak frequency.

Why can the tension differ from mtotalg?

T ≈ mtotalg assumes the hanging mass is stationary and pulley friction is negligible. Pulley friction can make the horizontal string tension differ from the nominal value.

Why use several harmonics?

Several modes provide multiple points for a graph of fn against n. The gradient gives a more reliable wave speed and the intercept and scatter help reveal experimental limitations.

Why should the best-fit line not automatically be forced through the origin?

A non-zero intercept can contain useful evidence of end effects, pulley friction, a frequency offset or incorrectly assigned harmonics. Forcing the line through zero can hide these effects.

Why use a long sample when measuring μ?

Weighing too short a string sample produces a large percentage uncertainty. A longer sample reduces the relative effect of balance and length-scale resolution.

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About this guide

This guide was written and reviewed by Better Equipped's technical team and further reviewed by former A Level Science Teachers. Our technical team draw on experience supplying practical science equipment to schools, colleges, laboratories and science departments throughout the UK. They include ex-school laboratory technicians and are here to support schools, colleges and laboratories. If you have feedback on this guide, we'd love to here it so please contact us. Don't forget we will be regularly updating our guides and resources on our Better-Resources Hub. If you would like us to cover a particular subject matter in these guides or have some top tips you'd like to share then again we'd love to hear from you.

Last reviewed and updated: August 2026

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