Superposition: Stationary Waves, Diffraction and Interference

This guide covers Topic 8, Superposition, in full — subtopics 8.1 Stationary waves, 8.2 Diffraction, 8.3 Interference and 8.4 The diffraction grating — from Cambridge International AS & A Level Physics 9702, 2025–2027 series. This is AS Level content.

Before studying this

This resource assumes the wave terms (displacement, amplitude, wavelength, v = fλ) from Waves: Progressive Waves, the Doppler Effect and Polarisation.

Syllabus coverage

CAMBRIDGE INTERNATIONAL AS & A LEVEL PHYSICS 9702 — AS Level, Topic 8

8.1 Stationary waves — explaining and using the principle of superposition; understanding demonstrations of stationary waves using microwaves, stretched strings and air columns (end corrections neglected); explaining stationary wave formation graphically, identifying nodes and antinodes; determining wavelength from node/antinode positions.

8.2 Diffraction — the meaning of diffraction; demonstrations of diffraction, including the qualitative effect of gap width relative to wavelength (e.g. water waves in a ripple tank).

8.3 Interference — the terms interference and coherence; demonstrations of two-source interference with water waves, sound, light and microwaves; the conditions needed to observe two-source interference fringes; recalling and using λ = ax/D for double-slit interference with light.

8.4 The diffraction grating — recalling and using d sin θ = nλ; describing the use of a diffraction grating to determine wavelength (the spectrometer’s structure and use are not included).

The principle of superposition

The principle of superposition states that, when two or more waves meet at a point, the resultant displacement equals the vector sum of the individual displacements. This single idea underlies every phenomenon in this topic.

Stationary waves

A stationary wave forms when two progressive waves of the same frequency and amplitude, travelling in opposite directions, superpose — for example, a wave reflected back along a stretched string, or in an air column. Unlike a progressive wave, a stationary wave does not transfer energy along its length.

• A node is a point of permanently zero displacement.
• An antinode is a point of maximum displacement amplitude.

Adjacent nodes (or adjacent antinodes) are separated by half a wavelength, so measuring the distance between them lets you determine wavelength directly.

How a stationary wave forms — the graphical method. Draw the two progressive waves of equal frequency and amplitude travelling in opposite directions at a series of time snapshots (e.g. every T/8), then add their displacements point by point at each snapshot. At points where the two waves are always exactly out of phase, the displacements cancel at every snapshot — these are the nodes. At points where the two waves are always exactly in phase, the displacements reinforce, swinging between a maximum positive and maximum negative value as time passes — these are the antinodes. Sketching even three or four snapshots (rather than just the two extreme “envelope” shapes) is what convinces an examiner you understand superposition is happening continuously, not just that you have memorised the final node/ antinode pattern.

Demonstrating stationary waves. Three standard apparatus set-ups appear on this syllabus:

• Stretched strings — a vibration generator shakes one end of a string fixed at the other end (or over a pulley with a weight); at certain driving frequencies, the reflected wave and the outgoing wave superpose into a clear stationary pattern of loops (antinodes) separated by stationary points (nodes), visible directly on the string.
• Air columns — a loudspeaker or tuning fork drives sound into a tube (open or closed at the far end, with end corrections neglected); at resonance, a stationary wave forms in the air column, with a displacement node always at a closed end and a displacement antinode always at an open end.
• Microwaves — a microwave transmitter is aimed at a metal reflecting plate; the wave reflects back and superposes with the outgoing wave, and a detector/probe moved along the line between transmitter and plate registers a series of strong and (near-)zero signal positions, mapping out the antinodes and nodes directly.

Harmonics. For a string fixed at both ends, the fundamental (lowest-frequency) mode has a node at each end and one antinode in the middle, so the string length L = λ/2, giving fundamental frequency f = v/2L. For a pipe closed at one end, there must be a node at the closed end (where air cannot move) and an antinode at the open end (where it moves freely), giving L = λ/4 for the fundamental — and, unlike the open string, a closed pipe supports only odd harmonics. Forgetting this restriction to odd harmonics for a closed pipe is a common error.

Diffraction

Diffraction is the spreading of a wave as it passes through a gap or around an obstacle. The effect is most noticeable when the gap width is comparable to the wavelength — a very wide gap relative to wavelength produces little noticeable spreading, while a narrow gap produces pronounced spreading, as demonstrated with water waves in a ripple tank.

Demonstrating two-source interference. The same interference pattern — alternating regions of reinforcement and cancellation from two coherent sources — can be shown with several types of wave:

• Water — two dippers vibrating in phase, driven from the same motor, on the surface of a ripple tank produce two overlapping sets of circular wavefronts; the resulting stationary pattern of enhanced and calm regions is visible directly on the water surface (or its shadow projected below).
• Sound — two loudspeakers connected to the same signal generator (so they are coherent) produce a pattern of loud and quiet regions in the space in front of them, detectable by walking along a line parallel to the speakers with a microphone, or simply by ear.
• Light — Young’s double-slit experiment (below) uses a single monochromatic source shining through two narrow, closely-spaced slits to create two coherent secondary sources, producing bright and dark fringes on a screen.
• Microwaves — a single microwave transmitter aimed at a pair of narrow gaps in a metal barrier creates two coherent secondary sources, and a detector probe moved across the far side registers alternating strong and weak signal regions, in direct analogy with the double-slit light pattern.

Interference

Interference is the superposition of two coherent waves, producing regions of reinforcement (constructive interference) and cancellation (destructive interference). Coherence means the two sources have a constant phase difference (in practice, usually the same frequency and a fixed phase relationship).

For observable interference fringes, the two sources must be coherent, and (for light) of similar amplitude and roughly monochromatic.

The condition for a bright or dark fringe can also be stated directly in terms of path difference — the difference in distance travelled by the two waves to reach a point:

Fringe	Path difference
Constructive (bright)	nλ
Destructive (dark)	(n + ½)λ

where n is a whole number (0, 1, 2, …). This path-difference form and the λ = ax/D formula describe the same pattern from two different angles: one gives the condition for a fringe, the other gives its position.

Double-slit interference with light uses:

λ = ax/D

where a is the slit separation, x is the fringe spacing, and D is the distance from the slits to the screen.

Worked example. In a double-slit experiment, slits 0.50 mm apart produce fringes spaced 1.2 mm apart on a screen 1.5 m away.

λ = ax/D = (0.50 × 10⁻³ × 1.2 × 10⁻³) / 1.5 = 4.0 × 10⁻⁷ m = 400 nm

The diffraction grating

A diffraction grating has many closely spaced slits, producing sharp, well-separated interference maxima described by:

d sin θ = nλ

where d is the slit spacing, θ is the angle to the maximum, and n is the order of the maximum (0, 1, 2, …). This is used experimentally to determine the wavelength of light by measuring the angle to a known-order maximum.

Common mistakes

• Believing a stationary wave transfers energy along its length — it does not; energy is stored, oscillating between kinetic and potential forms within each section between nodes.
• Confusing the node-to-node (or antinode-to-antinode) spacing with a full wavelength — it is half a wavelength.
• Forgetting the coherence condition for observable interference — two independent light sources of the same colour will not produce a stable interference pattern; a single source split into two coherent paths (as in a double slit) is needed.
• Mixing up the diffraction grating equation’s n (integer order) with a refractive index or another quantity — n here is always the whole-number order of the maximum.

Quick revision checklist

• The principle of superposition
• Node and antinode definitions, and the half-wavelength spacing between like points
• The gap-width-to-wavelength relationship in diffraction
• Interference and coherence, and λ = ax/D for double-slit interference
• d sin θ = nλ for a diffraction grating

Related resources

• Waves: Progressive Waves, the Doppler Effect and Polarisation — the previous AS topic, covering the wave terms used here
• Electricity: Current, Potential Difference and Resistance — the next AS topic
• Cambridge AS & A Level Physics hub

Written against Cambridge International AS & A Level Physics 9702, 2025–2027 series. Always check the current syllabus for your examination year.