Superposition, Standing Waves & Resonance — Print · Grade 12 Science
Chapter: Waves, Sound & OpticsSuperposition, Standing Waves & Resonance
Learning goals
- I can combine wave displacements and identify nodes and antinodes.
- I can calculate fixed-string harmonics and explain resonance limits.
Readings for this unit
Before this unit
Wave frequency, wavelength, speed, and signed displacement.
Use the detailed parent plan: two core 50–60-minute sessions per focused unit, with a third assigned for selected calculation/modeling topics. First study the core explanation and Parts 1–2 with their figures/checks; next study Parts 3–4, the whole-unit application, worked cases, varied practice and exit check. Attempt reasoning before opening guidance. Extend time when actual understanding requires it; investigations and chapter assessments have separate sessions.
Start with something to explain
Why can the same rope support only certain standing patterns?
Illustrative evidence case. Both ends are fixed and different driving frequencies produce nodes at specific positions.
Notice and wonder
Record two observations using the stated quantities or relationships. Keep observations separate from possible causes. Write one question that a new measurement could help answer.
Why can the same rope support only certain standing patterns?
Your first model
Label nodes, antinodes and the length boundary.
Build the language from the idea
- node
- A position of zero displacement in an ideal standing wave.
- antinode
- A position with maximal oscillation amplitude.
- resonance
- Large driven response near a system’s natural frequency.
The superposition principle states that overlapping small-amplitude linear waves add their displacements at each position and time. Two positive crests can reinforce, while a crest and matching trough can cancel momentarily. After passing, ideal waves continue; cancellation does not mean the wave energy has permanently vanished.
Constructive interference occurs where wave phases align; destructive interference where they oppose. Path difference mλ gives constructive interference for coherent in-phase sources, and (m+½)λ gives destructive interference in the simplest equal-source model. Source phase and geometry must be specified.
Oppositely traveling waves of the same frequency can form a standing pattern. Nodes have zero displacement and antinodes maximum oscillation in the ideal model. Adjacent nodes are λ/2 apart. The pattern does not travel down the string even though it arises from traveling-wave components.
For a string fixed at both ends, allowed standing wavelengths are λn=2L/n and frequencies fn=nv/(2L), n=1,2,3…. The fundamental is n=1. An air column’s allowed modes depend on whether each end is open or closed; do not apply the fixed-string formula to all instruments.
Resonance occurs when a drive near a system’s natural frequency transfers energy efficiently over cycles. Damping limits amplitude and broadens the response. Resonance can be useful in instruments and sensors or problematic in structures; the actual response depends on coupling, damping, and forcing amplitude.
Use a soft rope or audio examples at comfortable volume. A measured standing-wave pattern can test the wavelength relation, but the ideal linear model fails for large stretches or complicated boundaries. Identify mode number, length, and boundary conditions before calculating speed.
Build your understanding
Use the detailed parent plan: two core 50–60-minute sessions per focused unit, with a third assigned for selected calculation/modeling topics. First study the core explanation and Parts 1–2 with their figures/checks; next study Parts 3–4, the whole-unit application, worked cases, varied practice and exit check. Attempt reasoning before opening guidance. Extend time when actual understanding requires it; investigations and chapter assessments have separate sessions.
Part 1 · Apply superposition within a linear model
Superposition in a linear model means total displacement is the algebraic sum of contributions at the same location and instant. Two in-phase equal-amplitude waves can produce 2A locally; two opposite-phase contributions can cancel displacement at a point. Cancellation does not mean energy has been universally destroyed: the full wave and boundary account determines its redistribution. Separate diagrams on stacked rows are a comparison device, not three simultaneously separate ropes.
Read the visual model
Identify the labeled structures and units, then trace the arrows or compare values under the caption’s stated conditions. Separate the displayed evidence from a further inference. Write your reasoning before opening the guidance.
What total displacement results from+3 mm and−2 mm contributions?
Visual reasoning — after your attempt
+1 mm at that same point and instant under the linear model.
When compatible small disturbances overlap, a linear model adds signed displacements at each point. Constructive and destructive interference depend on phase. Destructive interference at one location does not mean energy is universally destroyed; the full pattern and energy account must be considered.
Two equal opposite displacements can sum to zero momentarily at a point while the waves continue to propagate elsewhere.
Pause and explain
Does a zero local displacement prove no waves are present?
Check your reasoning after answering
No. Opposing contributions can cancel locally; observations at other points or times can reveal the individual waves.
Part 2 · Connect path difference and phase
Phase specifies where an oscillation lies within its cycle. For coherent, initially in-phase sources in the same medium, a path difference of one wavelength adds a full cycle and restores phase agreement; half a wavelength produces opposition. Coherence means a stable phase relationship over the relevant interval. Source phase, reflections and unequal amplitudes can alter the simple bright/dark or loud/quiet pattern, so state those conditions before using path-difference rules.
Read the visual model
Identify the labeled structures and units, then trace the arrows or compare values under the caption’s stated conditions. Separate the displayed evidence from a further inference. Write your reasoning before opening the guidance.
Does half-wavelength path difference always guarantee complete cancellation?
Visual reasoning — after your attempt
No. It also needs the appropriate source phase, matched contributions and the linear coherent model.
For coherent waves of the same frequency with the same initial phase, integer-wavelength path differences give constructive interference; odd half-wavelength differences give destructive interference. Source phase differences can modify those conditions. State coherence and initial phase rather than using the path rule universally.
A half-wavelength longer path introduces a half-cycle phase difference in this same-phase-source model.
Pause and explain
Why is path difference alone sometimes insufficient?
Check your reasoning after answering
The sources may already have a phase difference or lack stable coherence, changing or washing out the predicted pattern.
Part 3 · Use standing-wave boundary conditions
A standing wave has nodes of zero displacement amplitude and antinodes of maximum amplitude, produced by counterpropagating coherent waves under appropriate conditions. A string fixed at both ends has nodes there and supports L=nλ/2. The displayed second harmonic has an additional middle node. Air columns with an open end have different displacement/pressure boundary conditions; do not transfer the fixed-string formula without identifying what is constrained.
Read the visual model
Identify the labeled structures and units, then trace the arrows or compare values under the caption’s stated conditions. Separate the displayed evidence from a further inference. Write your reasoning before opening the guidance.
For L=1.2 m in the shown second harmonic, what is λ?
Visual reasoning — after your attempt
L=2λ/2, so λ=1.2 m.
Standing waves arise from suitable overlapping waves traveling in opposite directions. Nodes have zero displacement amplitude; antinodes have maximum amplitude. Allowed patterns depend on boundaries. A string fixed at both ends has node endpoints and wavelengths λn=2 L/n in the ideal model.
A pipe open at both ends and a pipe closed at one end have different boundary conditions; do not copy the fixed-string pattern into every resonator.
Pause and explain
Why do boundary conditions determine allowed wavelengths?
Check your reasoning after answering
The pattern must fit the required nodes or antinodes at the ends, restricting which spatial arrangements persist.
Part 4 · Explain resonance and damping together
Resonance is an enhanced response when periodic driving couples effectively to a system’s natural oscillation. Damping removes mechanical energy and limits the steady response; the peak depends on the damping and on which response is measured. The plotted calculated displacement response is finite, rather than an unlimited growth claim. A real structure can have multiple modes and nonlinear behavior, so one simple resonance curve is a model to test.
Read the visual model
Identify the labeled structures and units, then trace the arrows or compare values under the caption’s stated conditions. Separate the displayed evidence from a further inference. Write your reasoning before opening the guidance.
Why does a resonance peak not justify saying the amplitude grows without limit?
Visual reasoning — after your attempt
Damping and other limits can balance input energy; the displayed model explicitly has a finite steady response.
Resonance involves a strong response near a system’s natural frequency under suitable driving. Damping and coupling determine response amplitude and width, so resonance is not unlimited growth in every real system. Compare response across driving frequencies while keeping other conditions controlled.
A broader response peak can reflect damping or other system behavior. One large oscillation alone does not identify the full resonance curve.
Pause and explain
What evidence supports a resonance-frequency claim?
Check your reasoning after answering
A controlled frequency sweep with measured response and repeated trials can locate a peak and characterize its width.
Apply the whole unit · Fit harmonics on a fixed string
Illustrative string length 0.800 m has wave speed 160 m/s and both ends fixed. Assume the ideal small-amplitude model.
Your task
Find the first three harmonic frequencies and wavelengths, describe nodes and antinodes, and explain why a differently constrained resonator needs a new pattern.
What your response should show
- Define the system, quantities, units, and relevant conditions before selecting a model.
- Show intermediate calculations or a labeled evidence account, including signs or denominators.
- Explain how the supplied evidence supports the conclusion rather than reporting only an answer.
- Identify a meaningful uncertainty, model limit, or measurement needed to test the conclusion.
Response guidance — after your attempt
Fundamental λ₁=2 L=1.60 m; f₁=160/1.60=100 Hz.
For n=2: λ₂=0.800 m and f₂=200 Hz.
For n=3: λ₃≈0.533 m and f₃=300 Hz.
The nth pattern has n half-wavelengths within the string, with nodes at both ends. The ideal frequency multiples depend on unchanged wave speed and fixed endpoints. Damping affects response, and a one-end-open acoustic resonator has different boundary conditions, so its allowed sequence cannot be inferred by copying these numbers.
Scientific references and source-model foundations
Parent / teacher teaching notes
Have students draw the first three spatial patterns and count half-wavelengths. Require a boundary statement rather than memorizing a frequency series. Ask the learner to explain each symbol, axis, unit and arrow before calculating. Calculated examples, supplied illustrative records and actual historical observations are labeled separately. Selected public OpenSciEd middle-school source figures are foundation examples, not complete high-school editions or an endorsement; all advanced explanations are supplied locally.
The key idea
Interference adds local displacement; boundary conditions select standing-wave modes.
Worked examples
Example 1
- f₁=v/(2L).
- 40/1.60.
Example 2 · Worked application
- The fundamental has half a wavelength over the string: λ₁=2L=2.4 m.
- f₁=v/λ₁=24/2.4=10 Hz.
- Both ends are displacement nodes.
Boundary conditions determine allowed wavelengths.
Example 3
- Add signed displacement at the same place/time.
- 3+(−2).
Example 4 · Find and fix the mistake
- Test the claim against the stated conditions: Boundary conditions determine allowed wavelengths.
- The pattern remains stationary while particles oscillate locally; nodes remain at zero displacement in the ideal mode.
Boundary conditions determine allowed wavelengths.
Example 5 · Reason from the visual model
- A standing wave has nodes of zero displacement amplitude and antinodes of maximum amplitude, produced by counterpropagating coherent waves under appropriate conditions. A string fixed at both ends has nodes there and supports L=nλ/2. The displayed second harmonic has an additional middle node. Air columns with an open end have different displacement/pressure boundary conditions; do not transfer the fixed-string formula without identifying what is constrained.
- L=2λ/2, so λ=1.2 m.
Original two opposite-phase snapshots of the same second-harmonic standing wave, not two traveling trajectories.
Avoid these mistakes
- A standing wave is a spatial pattern, not a wave moving with zero frequency.
Check your understanding
How could you estimate wavelength from a standing pattern?
Discuss the answer
Measure several adjacent node intervals, average, and double the node spacing.
Return to the phenomenon
Return to the initial case. Why can the same rope support only certain standing patterns? Revise your drawing or explanation using two named ideas and one stated limitation.
Compare explanations after your attempt
Counterpropagating contributions must satisfy endpoint constraints; damping and driving determine the response amplitude.
What should we investigate next?
Which comparison would test a remaining uncertainty? State what you would change, keep comparable and measure.
Investigate, calculate and revise
Keep a first model and an evidence-supported revision. Attempt each task before opening guidance; all required explanations and supplied-data routes are available locally.
Activity 1 · Return to the opening case
Use the opening phenomenon, unit definitions and explained Parts 1–4.
Your task
Return to the initial case. Why can the same rope support only certain standing patterns? Revise your drawing or explanation using two named ideas and one stated limitation.
Success criteria
- Use the stated observations and appropriate units.
- Connect a labeled model with the physical/environmental mechanism.
- State assumptions and an evidence limitation.
- Explain a substantive revision or next test.
Hint
Keep object/material identity, material transfer and energy account distinct.
Response guidance — discuss after your attempt
Counterpropagating contributions must satisfy endpoint constraints; damping and driving determine the response amplitude.
Activity 2 · Transfer to a changed case
Cover the worked answer and use the explained models, relationships and notebook.
Your task
A 1.5 m string fixed at both ends has wave speed 30 m/s. Find its fundamental and third-harmonic frequencies and the third-harmonic wavelength.
Success criteria
- Use the stated measurements or conditions, with correct units.
- Connect the result to the physical or environmental mechanism or evidence relationship.
- Identify the assumption, model limit or further evidence requested.
Hint
Name known quantities and conditions, show intermediate steps and check conservation.
Response guidance — discuss after your attempt
f1=v/(2L)=10 Hz; f3=30 Hz. λ3=2L/3=1.0 m, with fixed-end nodes in this ideal string model.
Words to know
- node
- A position of zero displacement in an ideal standing wave.
- antinode
- A position with maximal oscillation amplitude.
- resonance
- Large driven response near a system’s natural frequency.
In the real world
Use the chapter investigation to connect this unit to measured evidence and a practical decision.
Exit check
How could you estimate wavelength from a standing pattern?
Check your explanation after answering
Measure several adjacent node intervals, average, and double the node spacing.
Explain your reasoning without notes. If a key idea is missing, revisit the lesson and try again.
Problem 1
Hints
Write the relationship first; substitute with consistent units.
Answer and explanation
0.6 m
Node spacing=λ/2.
Problem 2
Answer and explanation
No; it describes local combined displacement. The waves can continue and energy distribution must be considered.
Response criteria
- State a scientifically accurate conclusion.
- Show the calculation or mechanism that supports it.
- Include units, conditions, or limits where relevant.
Problem 3
Hints
List the given quantities, select a relationship, and check units before substituting.
Answer and explanation
60 Hz
4×15=60.
Problem 4
Answer and explanation
Fixed, free, open, and closed boundaries allow different mode patterns.
Response criteria
- State a scientifically accurate conclusion.
- Show the calculation or mechanism that supports it.
- Include units, conditions, or limits where relevant.
Problem 5
Hints
Boundary conditions determine allowed wavelengths.
Answer and explanation
f₂=2f₁=20 Hz.
Problem 6
Hints
State what the supplied observations or conditions establish, and separate that from any assumption.
Answer and explanation
Their displacements cancel locally by superposition; this does not imply energy is destroyed everywhere.
Problem 7
Answer and explanation
L=2λ/2, so λ=1.2 m.
A standing wave has nodes of zero displacement amplitude and antinodes of maximum amplitude, produced by counterpropagating coherent waves under appropriate conditions. A string fixed at both ends has nodes there and supports L=nλ/2. The displayed second harmonic has an additional middle node. Air columns with an open end have different displacement/pressure boundary conditions; do not transfer the fixed-string formula without identifying what is constrained.
L=2λ/2, so λ=1.2 m.
Response criteria
- Interpret labels, structures, axes, units and directions correctly.
- Explain the physical or environmental relationship or mechanism.
- Respect stated assumptions and distinguish observation from inference.
Problem 8
Answer and explanation
f1=v/(2L)=10 Hz; f3=30 Hz. λ3=2L/3=1.0 m, with fixed-end nodes in this ideal string model.
f1=v/(2L)=10 Hz; f3=30 Hz. λ3=2L/3=1.0 m, with fixed-end nodes in this ideal string model.
Response criteria
- Use the stated measurements or conditions, with correct units.
- Connect the result to the physical or environmental mechanism or evidence relationship.
- Identify the assumption, model limit or further evidence requested.