AS Physics: Superposition — Revision Notes

Condensed for the final weeks. For the full explanation, use the Superposition study guide.

The principle of superposition

When two or more waves meet, the resultant displacement is the vector sum of the individual displacements.

Interference

Coherence — a constant phase difference, which requires the same frequency. Without it, the interference pattern shifts randomly and no stable fringes are seen. Coherence is the condition for a stable pattern, not for interference itself. Two independent light sources of the same colour will not produce a stable pattern — a single source split into two coherent paths, as in a double slit, is needed.

Condition	Path difference
Constructive	nλ
Destructive	(n + ½)λ

Young’s double slit:

lambda = a x / D        a = slit separation, x = fringe spacing, D = slit-to-screen distance

To increase fringe spacing: increase D, increase λ, or decrease a. The inverse relationship with slit separation is the one most often reversed.

Worked example. Slits 0.50 mm apart produce fringes spaced 1.2 mm apart on a screen 1.5 m away. Find the wavelength.

lambda = a x / D = (0.50x10^-3 x 1.2x10^-3) / 1.5 = 4.0x10^-7 m = 400 nm

Young’s experiment is the standard evidence that light behaves as a wave, because interference cannot be explained by a classical, ray-like particle model (the AS-level contrast is with light travelling as simple particles in straight lines, not with modern quantum theory, which does account for interference).

Diffraction

Waves spread out when passing through a gap or around an obstacle. Diffraction is greatest when the gap is comparable to the wavelength — which is why sound diffracts around doorways but light does not.

Diffraction grating:

d sin(theta) = n lambda        d = 1 / (lines per metre)

A grating gives sharper, narrower maxima than a double slit because many slits contribute, so the maxima are more precisely located — which is why gratings are used for accurate wavelength measurement. (Whether a grating maximum is also brighter depends on the illumination, slit width and how intensity is measured — sharpness, not brightness, is the guaranteed advantage.)

The maximum order is found by setting sin θ = 1, giving n = d/λ, then rounding down to a whole number.

Stationary waves

Formed when two waves of the same frequency and amplitude travelling in opposite directions superpose — usually a wave and its reflection.

	Progressive wave	Stationary wave
Energy	Transferred	Not transferred
Amplitude	Same for all points	Varies from zero at nodes to maximum at antinodes
Phase	Varies continuously	Points between adjacent nodes are in phase

Node — zero amplitude, from destructive interference. Antinode — maximum amplitude.

A stationary wave does not transfer energy along its length — energy is stored, oscillating between kinetic and potential forms within each section between adjacent nodes.

The distance between adjacent nodes is λ/2, not λ. That is the most common error in stationary-wave calculations.

Fundamental frequency for a string fixed at both ends: L = λ/2, so f = v/2L.

For a pipe closed at one end, there is a node at the closed end and an antinode at the open end, giving L = λ/4 and only odd harmonics.

Exam traps

• Saying coherence requires equal amplitude — it requires constant phase difference and equal frequency.
• Reversing the relationship between slit separation and fringe spacing.
• Using λ instead of λ/2 for the node separation.
• Rounding the maximum order up instead of down.
• Saying stationary waves transfer energy.
• Forgetting that a closed pipe supports only odd harmonics.
• Mixing up which quantity is a and which is x when rearranging λ = ax/D.
• Believing two independent lamps of the same colour will produce a stable interference pattern — they won’t, without a fixed phase relationship.
• Confusing the diffraction grating’s n (a whole-number order) with a refractive index or any other quantity.

Self-test

1. State the principle of superposition.
2. What does coherence require, and what does it achieve?
3. How does fringe spacing change if slit separation is halved?
4. Give three differences between progressive and stationary waves.
5. What is the distance between adjacent nodes?
6. Slits 0.50 mm apart produce fringes spaced 1.2 mm apart on a screen 1.5 m away. Find the wavelength.
7. Why does a diffraction grating give sharper, narrower maxima than a double slit?
8. Why can two independent lamps of the same colour not produce a stable interference pattern?
9. Explain why a stationary wave does not transfer energy, in terms of what happens between adjacent nodes.

Answers: 1. When two or more waves meet, the resultant displacement is the vector sum of the individual displacements. 2. A constant phase difference and the same frequency; it produces a stable, observable interference pattern. 3. It doubles, since fringe spacing is inversely proportional to slit separation. 4. Progressive waves transfer energy, stationary waves do not; for an ideal, undamped progressive wave in a uniform medium, amplitude is constant along its length, while a stationary wave’s amplitude varies from nodes to antinodes; in a stationary wave all points between adjacent nodes are in phase. 5. Half a wavelength. 6. λ = ax/D = (0.50×10⁻³ × 1.2×10⁻³) ÷ 1.5 = 4.0×10⁻⁷ m = 400 nm. 7. Many slits contribute to each maximum, so the maxima are narrower and more precisely located, giving sharper fringes (brightness depends separately on illumination and slit width). 8. They are not coherent — there is no constant phase relationship between them, so any interference pattern shifts randomly and no stable fringes are seen. 9. Energy is stored, oscillating between kinetic and potential forms within each section between adjacent nodes, rather than being carried along the wave.