Superposition

Learning objectives

• Apply the principle of superposition.
• Identify coherent sources.
• Distinguish nodes from antinodes.

The big idea

Where waves overlap, the resultant displacement is the sum of their individual displacements. Stable interference patterns require coherence: a constant phase relationship and the same frequency.

Two opposite-travelling waves of the same frequency can form a stationary wave. Nodes have zero oscillation amplitude and antinodes have maximum amplitude. A stationary pattern does not carry net energy along the medium in the same way as a single progressive wave.

1. Add displacement where waves overlap

Superposition says that resultant displacement at a point is the algebraic sum of individual displacements when the medium responds linearly. A +2 mm disturbance and a −2 mm disturbance can cancel locally, but the waves continue beyond their overlap. Cancellation of displacement does not mean their energy has been destroyed everywhere. For sustained interference, energy is redistributed between different regions of the pattern.

2. Coherence makes a pattern stable

A constant phase difference allows bright and dark or loud and quiet regions to remain in fixed positions. For initially in-phase sources, path difference nλ gives constructive interference and an odd half-wavelength gives destructive interference. Include any initial phase difference before applying those simple conditions. Equal amplitudes help produce complete cancellation but are not the definition of coherence. Diffraction can spread waves so that two paths overlap.

3. Young’s double slit is a geometric model

For narrow coherent slits, small angles and a distant screen, fringe spacing x is approximately λD/a. Increasing screen distance spreads the fringes; increasing slit separation makes them closer. If λ = 600 nm, D = 2 m and a = 0.30 mm, spacing is about 4 mm. Convert both nanometres and millimetres before substitution, and remember that this is an approximation rather than an exact equation for every geometry.

4. Stationary waves arise from opposition

Equal-frequency waves travelling in opposite directions can produce fixed nodes and antinodes. Adjacent nodes are λ/2 apart; a node and neighbouring antinode are λ/4 apart. At a node, displacement remains zero throughout the cycle. Points between the same adjacent nodes oscillate in phase in the ideal model, while neighbouring segments are in antiphase. Allowed modes depend on boundary conditions, such as fixed ends or open and closed air-column ends.

A method you can reuse

1. Identify coherence and any starting phase difference.
2. Calculate path difference before assigning interference type.
3. Check small-angle and distant-screen assumptions for double-slit calculations.
4. Mark nodes and antinodes before measuring a stationary-wave wavelength.
5. Explain the physical boundary condition behind the selected mode.

Equations and relationships

Investigate and explain

Use two coherent simulated sources and move a detector across regions of constructive and destructive interference.

Use two sources for interference and reflections for stationary waves. Keep source coherence and boundary settings fixed when testing a prediction; visual similarity does not make all patterns stationary waves.

Open the interactive investigation

Check your understanding