Waves and superposition: models, interference and diffraction
Key idea: Use signed displacement and phase consistently across time and space, then apply superposition to standing waves, interference and diffraction without confusing their distinct conditions.
Before you start: Oscillations objective chainQuantities & Measurement objective chain
By the end, you can
- Model mechanical and electromagnetic progressive waves, interpret time and position graphs, and use wave quantities, phase and energy transfer.
- Apply intensity–amplitude and inverse-square relationships, and use polarisation and Malus' law for electromagnetic waves.
- Apply superposition and explain standing-wave experiments, boundary conditions, nodes, antinodes and sound-wavelength measurement.
- Analyse coherent two-source interference and Young double-slit fringes using phase and path difference.
- Use diffraction gratings, single-slit first minima and the Rayleigh criterion with their distinct aperture spacings and assumptions.
Starting-point self-check
1. Check your starting point
Attempt all seven groups without notes and mark the first graph axis, phase conversion, boundary condition or aperture spacing you could not justify. Use the recorded topic diagnostic above when you want scoring and a personalised repair plan.
Superposition and standing-wave experiments 11(a)–(d)
Question 1
A string of length 1.20 m is fixed at both ends and has wave speed 240 m s⁻¹. Find its fundamental frequency, explain how the pattern forms, and identify one way microwaves or an air column can demonstrate fixed nodes and antinodes.
Check the model response
The ends are displacement nodes, so λ₁ = 2L = 2.40 m and f₁ = v/λ₁ = 100 Hz. Equal-frequency, equal-amplitude progressive waves travelling oppositely superpose to form fixed nodes and antinodes with no net energy transfer along the pattern. A movable microwave probe maps minima and maxima, or a resonance tube locates successive air-column resonances.
repair
2. Repair the eleven common breaks
Use only the repair matching an error, then redraw the axes, signed displacement, boundary conditions, rays or aperture before retrying.
Superposition and standing-wave experiments 11(a)–(d)
Check this idea
Misconception: An open end is both a displacement node and a pressure node.
Repair: An open end is approximately a displacement antinode and pressure node. A closed end is a displacement node and pressure antinode.
Check this idea
Misconception: Superposition means adding wave intensities point by point.
Repair: Add signed instantaneous displacements first. Intensity depends on the resulting amplitude.
worked example
3. Follow seven worked models
Follow how each solution fixes the graph type, energy-spreading model, transmission axis, boundary condition, path difference or aperture before calculating.
Superposition and standing-wave experiments 11(a)–(d)
Model 1
A pipe closed at one end has length 0.850 m and sound speed 340 m s⁻¹. Find its fundamental and next allowed frequency, identify end conditions, and state how resonance can measure wavelength.
Check the model response
Closed end: displacement node/pressure antinode; open end: displacement antinode/pressure node. For the fundamental L = λ/4, so λ = 3.40 m and f = 100 Hz. Only odd harmonics occur, so the next is 300 Hz. Successive resonant lengths differ by λ/2, allowing λ and then v = fλ to be found.
guided practice
4. Guided practice
Use each hint only to select the graph interval, spreading surface, polarisation reference, mode shape, interference condition or aperture equation.
Superposition and standing-wave experiments 11(a)–(d)
Question 1
Successive resonant lengths of an air column differ by 0.170 m at 1000 Hz. Find sound wavelength and speed, and state the pressure/displacement conditions at an open end.
Hint: Successive resonances are separated by λ/2.
Check the model response
λ/2 = 0.170 m, so λ = 0.340 m and v = fλ = 340 m s⁻¹. An open end is approximately a pressure node and displacement antinode.
independent practice
5. Independent practice
Solve without repair notes and state the graph, source, boundary, coherence and aperture assumptions used.
Superposition and standing-wave experiments 11(a)–(d)
Question 1
Compare how a microwave probe, a vibrating string and a resonance tube reveal standing waves, including what is measured and the relevant node or antinode.
Check the model response
A microwave probe moved through the field records alternating intensity minima and maxima. A driven stretched string shows stationary displacement nodes and antinodes at resonant frequencies. A resonance tube shows sound maxima at allowed air-column lengths; the closed end is a displacement node/pressure antinode and the open end the reverse. Adjacent node or resonance spacing determines λ/2.
Practice exit check
6. Practice assessment
Use this as extra closed-book practice, then complete the separate recorded assessment in your plan.
Superposition and standing-wave experiments 11(a)–(d)
Question 1
Explain standing-wave formation by superposition, distinguish pressure and displacement end conditions in an air column, and describe how one of microwaves, strings or air columns can determine wavelength.
Check the model response
Equal-frequency, equal-amplitude waves travelling oppositely add signed displacements to form fixed nodes and antinodes with no net energy transfer. At an open air end, displacement is an antinode and pressure a node; at a closed end the reverse applies. Map adjacent microwave minima, string nodes or successive air-column resonances: their separation is λ/2.
Re-test practice
7. Delayed re-test practice
Return after at least three days and solve these fresh contexts without reopening earlier responses. The recorded plan enforces the delay and uses a separate re-test family for selected-response skill-group evidence.
Superposition and standing-wave experiments 11(a)–(d)
Question 1
A string fixed at both ends has length 0.90 m and third-harmonic frequency 300 Hz. Find wavelength, wave speed and node spacing, then state why the pattern transfers no net energy along the string.
Check the model response
For the third harmonic, L = 3λ/2, so λ = 2L/3 = 0.60 m. v = fλ = 180 m s⁻¹ and adjacent nodes are λ/2 = 0.30 m apart. Equal opposite progressive waves carry equal energy fluxes in opposite directions, so the stationary pattern has no net transfer.