H2 Physics Superposition Notes | A-Level 9478

Q: What does A-Level Physics: 11) Superposition Guide cover?
A: Standing waves, double-slit fringes and the Rayleigh criterion sound abstract until you melt chocolate in a microwave or tune a guitar string.

TL;DR
Superposition links every wave idea you meet from sound to quantum. Master:
1\) the add-and-subtract rule for overlapping waves,
2\) node-antinode spotting to read standing-wave diagrams at speed,
3\) three exam-grade formulae - $x_n = \dfrac{n \lambda D}{a}$ (fringe spacing $\Delta x = \dfrac{\lambda D}{a}$), $a \sin \theta = n \lambda$ and $b \sin \theta = \lambda$ - plus the Rayleigh test for “too blurry to separate”. These show up often in structured questions, so practising them pays off.

Concrete example: how to use this page

If two waves arrive together, add their displacements, not their speeds. Bright and dark fringe questions usually become easier after you translate path difference into phase difference.

Decision map - choose the superposition route first

Many superposition errors come from using a diffraction formula on an interference question, or treating a stationary pattern as if the whole pattern travels. Sort the phenomenon first.

Misconception check: stationary waves are made from travelling waves. The nodes stay fixed, but the incident and reflected waves still carry energy in opposite directions.

Keep the wave toolkit coherent by revisiting our free H2 Physics notes; it chains this topic with Wave Motion, Diffraction, and Quantum follow-ups so your superposition drills stay contextual.

1 Where this sits in the syllabus

The SEAB 2026 H2 Physics document parks Superposition under Section III “Waves” and lists 13 learning outcomes, from the principle itself to the Rayleigh criterion for resolution.
Parents: this topic is commonly examined in both conceptual MCQs and multi-mark structured questions, making it a high-leverage chapter to revise.

2 Principle of superposition

Rule: If two or more disturbances overlap in a linear medium, the resultant displacement is the algebraic sum of the individual displacements.

2.1 Quick check

Add two sine waves of the same frequency but a phase difference $\phi$:

$$\begin{aligned} y &= y_1 + y_2 \cr &= A \sin(\omega t) + A \sin(\omega t + \phi) \cr &= 2A \cos\Bigl(\dfrac{\phi}{2}\Bigr) \sin\Bigl(\omega t + \dfrac{\phi}{2}\Bigr) \end{aligned}$$

Amplitude modulation pops straight out of the maths - the entire idea behind noise-cancelling headphones.

2.2 Mini-drill

Sketch the resultant at $t = 0$ for $\phi = 0, \pi/2, \pi$. Label points of constructive and destructive interference.

3 Standing waves

A standing (stationary) wave forms when two identical waves travel in opposite directions and superpose. Nodes (zero displacement) and antinodes (max displacement) appear at fixed positions.

3.1 Graphical formation

Plot incident and reflected waves every $\dfrac{T}{4}$. Nodes stay put at multiples of $\dfrac{\lambda}{2}$; antinodes halfway between.

3.2 Measuring sound wavelength

For a pipe closed at one end, first resonance occurs at $L = \dfrac{\lambda}{4}$. Measure $L$ with a metre rule and compute $v = f \lambda$ to within 3%.

4 Two-source interference

Ripple tank, twin loudspeakers or Young's double-slit-same physics. Conditions:

• Coherence (constant phase difference),
• Similar amplitudes,
• Path-difference governed phase.

4.1 Double-slit formula

Derivation assumes $D \gg a$ and small angles:
$$x_n = \dfrac{n \lambda D}{a}, \quad \Delta x = \dfrac{\lambda D}{a}.$$

Path-difference checkpoint

Before substituting into the fringe formula, decide whether the question gives path difference, phase difference, or screen position. These are linked, but they are not the same quantity.

Misconception check: bright fringes do not require zero path difference. Zero path difference gives the central bright fringe; other bright fringes occur whenever the path difference is a whole-number multiple of the wavelength.

5 Diffraction grating

Large arrays of slits sharpen the interference:

$$a \sin \theta = n \lambda$$

For a typical $600 \space \pu{mm-1}$ grating, $a = 1.67 \times 10^{-6} \pu{m}$. First-order green $( \lambda = 550 \pu{nm} )$

Exam tip: higher orders may “walk off the screen”. Check $|\sin \theta| \le 1$.

Grating order checkpoint

Before solving for an angle, check whether the order can physically exist. The grating equation has a built-in limit because $\sin\theta$ cannot be greater than 1.

Worked check: for a $\pu{600 mm-1}$ grating, $a = 1/(600 \times 10^3) = \pu{1.67e-6 m}$

Misconception check: the order number must be an integer. A value like $3.03$ is a limit, not an actual bright order.

6 Single-slit diffraction

The first minima satisfy\

$$b \sin \theta = \lambda.$$

Narrowing the slit widens the central peak-exactly the opposite of photographic aperture behaviour.

7 Rayleigh criterion - resolving power

Two point sources are just resolved when the principal maximum of one falls on the first minimum of the other:\

$$\theta \approx 1.22 \dfrac{\lambda}{b}.$$

Smaller $\theta$ means sharper detail; astronomers fight atmospheric seeing to get below 1 arc-second. (The $1.22$ factor comes from the first minimum of the circular Airy disk.)

Aperture width checkpoint

Single-slit diffraction and Rayleigh resolution both compare a wave's wavelength with an opening size. The question changes the interpretation of $b$, but the direction of change is the same: a larger opening gives a smaller diffraction angle.

Worked check: if a telescope aperture is doubled while $\lambda$ stays the same, $\theta \approx 1.22\lambda/b$ halves. That means the telescope can separate sources with half the angular spacing, so the resolution improves.

Misconception check: "smaller angle" is good for resolving power here. It means the diffraction blur is narrower, not that the image has become harder to see.

8 Three common exam traps

1. Mixing displacement and pressure nodes - in sound pipes, they are half a loop out of phase.
2. Using $v = f \lambda$ for standing waves - remember the wave still travels even though the pattern is stationary.
3. Using degrees in calculator set to radians - spot when your $\sin \theta$ looks off.

9 WA timing rules (tested in our tuition drills)

1. Use syllabus pacing as a guide: Paper 2/3 average ~1.6 min/mark; Paper 4 ~3 min/mark.
2. Label nodes/antinodes first - avoids losing easy pictorial marks.
3. Match sig-figs to the data - avoid over-precision.

Need structured practice on Superposition? Our H2 Physics tuition programme covers this topic with weekly problem sets and Paper 4 practical drills.

Comprehensive revision pack

9478 Section III, Topic 11 Syllabus outcomes

Candidates should be able to:

• (a) explain and use the principle of superposition in simple applications.
• (b) show an understanding of experiments which demonstrate standing (stationary) waves using microwaves, stretched strings and air columns.
• (c) explain the formation of a standing (stationary) wave using a graphical method, and identify nodes and antinodes, differentiating between pressure and displacement nodes and antinodes for sound waves.
• (d) determine the wavelength of sound using standing (stationary) waves.
• (e) show an understanding of the terms diffraction, interference, coherence, phase difference and path difference.
• (f) show an understanding of phenomena which demonstrate two-source interference using water waves, sound waves, light and microwaves.
• (g) show an understanding of the conditions required for two-source interference fringes to be observed.
• (h) recall and use the equation $\dfrac{ax}{D} = \lambda$ to solve problems for double-slit interference, where $a$

Concept map (in words)

Start with linear superposition: add displacements to obtain resultant wave. Standing waves arise from counter-propagating waves with equal amplitude. Two-source interference depends on coherence and path difference. Diffraction creates envelope patterns that govern resolution; Rayleigh criterion links aperture size to angular detail.

Key relations

Term guide

Derivations & reasoning to master

1. Standing wave equation: superpose $\sin( kx - \omega t )$ and $\sin( kx + \omega t )$ to obtain $2A \sin( kx ) \cos( \omega t )$

Worked example 1 - string harmonics

A string fixed at both ends has length $0.90 \mathrm{m}$ and supports standing waves at frequencies $120 \mathrm{Hz}$ and $160 \mathrm{Hz}$ but not in between. Find the fundamental frequency and wave speed.

Solution: consecutive harmonics differ by $f_0 \Rightarrow f_0 = 40 \mathrm{Hz}$. Use $v = 2 L f_0 = 72 \pu{m.s-1}$

Worked example 2 - diffraction grating spectrum

Light of wavelengths $\pu{450 nm}$ and $\pu{650 nm}$ hits a grating with $\pu{500 mm-1}$. Determine the highest observable order for each colour.

Method: grating spacing a is approximately $\pu{2.00e-6 m}$. Because $\sin(\theta)$ must not exceed $1$, $n$ is limited by $a/\lambda$

Practical & data tasks

• Capture node/antinode positions with a stroboscope and string vibrator; measure $\lambda$.
• Use a laser and double slit to photograph the fringe pattern; calculate $\lambda$ from fringe spacing.
• Build a DIY spectrometer with CD grating; record angles versus colour for cross-checking grating equation.

Common misconceptions & exam traps

• Confusing phase difference with path difference (missing $\dfrac{2\pi}{\lambda}$ factor).
• Forgetting that intensity $\propto A^2$ when combining waves.
• Mixing displacement nodes with pressure nodes in wind instruments.
• Ignoring missing orders in grating problems when $n \lambda$

Quick self-check quiz

1. What condition defines constructive interference? - Path difference = $n \lambda$ ($n \in \mathbb{Z}$).
2. In a string with fixed ends, where are displacement nodes located? - At the fixed ends and at intervals of $\dfrac{\lambda}{2}$.
3. What happens to fringe spacing if slit separation doubles? - Fringe spacing halves.
4. Give one method to improve resolving power of a telescope. -

Revision workflow

1. Redo Young's double-slit derivation and practise with numeric problems weekly.
2. Tabulate formulas for open vs closed pipes and revise before practical sessions.
3. Reconstruct standing wave diagrams from scratch, labelling pressure/displacement features.
4. Attempt Rayleigh criterion questions from past years and interpret in terms of real instruments.

Practice Quiz

Test yourself on the key concepts from this guide.

10 Further reading

• SEAB H2 Physics 9478 syllabus (2026)
• Isaac Physics - Superposition
• Wikipedia - Superposition principle
• Khan Academy - Standing waves on strings

11 Call-to-action

Parents: book a 60-min Superposition clinic two weeks before WA 1-it typically lifts wave-topic marks by 15 %. Students: pin the three key formulae on your study wall and practise switching between degrees and radians on your calculator.

Last updated 14 Jul 2025. Next review when SEAB releases the 2027 draft syllabus.