Superposition and Interference

When two or more waves meet at a point, they combine according to the principle of superposition. This leads to the phenomena of constructive and destructive interference, which provide some of the most striking evidence for the wave nature of light and sound.

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The Principle of Superposition

The Principle of Superposition: When two or more waves meet at a point, the resultant displacement at that point is the vector sum of the individual displacements due to each wave.

This principle applies to all types of wave. The key word is vector sum — displacements have a sign (positive or negative relative to the equilibrium position), and these must be added algebraically.

• If two crests arrive simultaneously, the resultant displacement is the sum of both amplitudes (a large positive displacement).
• If a crest and a trough of equal magnitude arrive simultaneously, the resultant displacement is zero.
• In general, the resultant is the algebraic sum of the individual displacements at each instant.

After passing through each other, the waves continue unchanged. Superposition does not permanently alter the waves.

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Constructive and Destructive Interference

Constructive Interference

Constructive interference occurs when two waves arrive in phase — their crests (and troughs) coincide. The resultant amplitude is the sum of the individual amplitudes.

For two waves of equal amplitude A arriving in phase:

• Resultant amplitude = A + A = 2A
• Resultant intensity ∝ (2A)² = 4A² — four times the intensity of one wave alone

Destructive Interference

Destructive interference occurs when two waves arrive in antiphase (180° or π radians out of phase) — the crest of one coincides with the trough of the other.

For two waves of equal amplitude A arriving in antiphase:

• Resultant amplitude = A − A = 0
• Resultant intensity = 0

Common Misconception: Energy is not destroyed in destructive interference. It is redistributed — regions of destructive interference have zero intensity, but regions of constructive interference have enhanced intensity. The total energy is conserved.

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Conditions for Observable Interference

For a stable, observable interference pattern, the sources must be coherent:

Coherence

Two sources are coherent if they have:

1. The same frequency (and therefore the same wavelength)
2. A constant phase difference between them (which may be zero)

If the phase difference between two sources varies randomly over time, the interference pattern shifts so rapidly that no stable pattern is observed — the waves are said to be incoherent.

Two separate light bulbs are incoherent because each atom emits light independently in short, random bursts. A laser produces highly coherent light. To produce coherent light sources without a laser, a single source is split into two (e.g., using a double slit).

Exam Tip: "Coherent" does NOT mean "in phase." Two coherent sources have the same frequency and a constant phase difference. If their phase difference is zero, they are coherent AND in phase.

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Path Difference and Interference Conditions

The path difference is the difference in the distances travelled by two waves from their respective sources to the point where they meet.

For two coherent sources in phase, the interference conditions are:

Condition	Path Difference	Result
Constructive interference	nλ (n = 0, 1, 2, 3, ...)	Maximum amplitude
Destructive interference	(n + ½)λ (n = 0, 1, 2, ...)	Zero amplitude

If the two sources have a constant phase difference of π (antiphase), the conditions are reversed: