Superposition and Coherence

So far we have treated waves singly. But the real world is full of situations where two or more waves overlap in space and time. When this happens, the principle of superposition tells us that the resultant disturbance at any point is found by adding the individual wave displacements. This simple rule has profound consequences: it produces interference, diffraction patterns, beats, stationary waves and many other of the beautiful phenomena of physics.

This lesson establishes the principle of superposition, defines the closely-related idea of coherence, and introduces the concepts of path difference, phase difference, constructive interference and destructive interference. These ideas are the foundation of Lessons 8 to 11.

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The Principle of Superposition

The principle of superposition states:

When two or more waves meet at a point, the resultant displacement is the vector sum of the individual displacements that each wave would produce at that point if the others were absent.

In mathematical form, for two waves with displacements y₁ and y₂ at the same point at the same instant:

y = y₁ + y₂

This applies to every type of wave — mechanical, sound, electromagnetic. It is an instantaneous rule: you add the displacements at each moment in time, not the amplitudes.

What It Does and Does Not Say

• It does say that waves pass through each other without destroying each other. After the overlap, each wave continues on its way unaltered.
• It does say that where two crests meet you get a bigger crest, and where a crest meets a trough you get partial (or total) cancellation.
• It does not say anything about energy being destroyed. In regions of destructive interference no energy arrives, but this is compensated by constructive interference elsewhere. Total energy is conserved.

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Constructive and Destructive Interference

When two waves of the same frequency and amplitude superpose, the result depends on their phase difference.

Constructive Interference

If the two waves arrive in phase (phase difference = 0, or a whole number of wavelengths of path difference), crests line up with crests and troughs with troughs. The result is a wave of double amplitude — constructive interference.

Constructive interference occurs when path difference = nλ, where n = 0, 1, 2, 3, … (a whole number of wavelengths).

Destructive Interference

If the two waves arrive in antiphase (phase difference = π rad = 180°, equivalent to a path difference of an odd number of half-wavelengths), crests line up with troughs. If the amplitudes are equal the result is zero — total destructive interference.

Destructive interference occurs when path difference = (n + ½)λ, where n = 0, 1, 2, 3, … (an odd number of half-wavelengths).

Partial Interference

For phase differences between 0 and π, the two waves partially reinforce. For phase differences between π and 2π, they partially cancel. The resultant amplitude varies continuously between 2A (fully constructive) and 0 (fully destructive).

flowchart TB
    subgraph Setup["Two coherent sources"]
        S1[Source 1]
        S2[Source 2]
    end
    S1 --> P[Point P receives both waves]
    S2 --> P
    P --> Q[Compute path difference, then phase difference]
    Q --> C[Constructive: delta = n lambda]
    Q --> D[Destructive: delta = n plus half lambda]
    Q --> M[In between: partial]

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Path Difference and Phase Difference

When two waves reach a point P from two different sources S₁ and S₂, the distances travelled are S₁P and S₂P. The path difference is the difference of these two distances:

Δ = |S₁P − S₂P|

The corresponding phase difference φ is given by:

φ = (2π/λ) × Δ (in radians)

In words: each wavelength of path difference corresponds to a full 2π of phase difference.

Path difference Δ	Phase difference	State
0	0	In phase — constructive
λ/2	π	Antiphase — destructive
λ	2π	In phase — constructive
3λ/2	3π	Antiphase — destructive
2λ	4π	In phase — constructive

You must be fluent in converting between path difference and phase difference.

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Worked Example 1 — Path and Phase Difference

Q. Two loudspeakers emit sound of wavelength 0.40 m. At a point P, the distances from the two speakers are 3.60 m and 4.20 m respectively. State whether P is a point of constructive or destructive interference.

A. Path difference:

Δ = 4.20 − 3.60 = 0.60 m

As a multiple of λ:

Δ / λ = 0.60 / 0.40 = 1.5

This is an odd half-wavelength multiple (1.5 = 3/2). The two waves arrive in antiphase and interfere destructively — P is a quiet point (in fact, if the two speakers are equal in output, a silent point).

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Worked Example 2 — Locating a Bright Fringe

Q. Two coherent radio transmitters emit waves of wavelength 2.0 m. A receiver is positioned so that the path difference between the two transmitters is exactly 6.0 m. Is the signal strong or weak? Explain.

A. 6.0 m / 2.0 m = 3, a whole number of wavelengths. The waves arrive in phase and interfere constructively. The signal is strong.

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Coherence

For a stable, observable interference pattern, the two sources must be coherent. This is an extremely important concept.

Definition of Coherence

Two sources are coherent if they have:

1. The same frequency (and hence the same wavelength in a given medium), and
2. A constant phase difference between them.

If either condition fails, the interference pattern changes so rapidly with time that it averages out and becomes invisible.

Why Coherence Is Necessary