Wave Superposition, Interference, and Diffraction

The principle of superposition states that when two or more waves overlap, the resultant displacement at each point is the sum of the individual displacements; interference is the resulting pattern of reinforcement (constructive) and cancellation (destructive), and diffraction is the spreading of a wave as it passes through a gap or around an obstacle. In brief: two waves in phase add up to a larger amplitude, while two waves in antiphase (half a wavelength out of step) cancel. Constructive interference occurs for a path difference of nλ, and destructive interference for (n + ½)λ. Diffraction is most noticeable when the gap is about the same size as the wavelength.

Noise-canceling headphones, the colors on a soap bubble and the ability to hear around a corner all depend on these effects. This guide explains superposition, interference of two sources and diffraction, compares the cases in a table, works through path difference calculations and corrects common mistakes.

What is superposition?

Waves pass through each other without changing, and at each point the total displacement is the algebraic sum of the displacements of each wave. If two pulses of amplitude 2 cm meet, the displacement at the moment of overlap can be 4 cm (both up), or 0 (one up and one down). After they have passed, each pulse continues unchanged. The principle holds for waves of small amplitude in a linear medium, for sound, light and water waves.

What is interference?

Interference is the superposition of waves from two or more coherent sources, sources with the same frequency and a constant phase difference. Two loudspeakers connected to the same signal generator or two slits illuminated by the same laser are coherent.

• Constructive interference: the waves arrive in phase (crest meets crest), and the resultant amplitude is the sum of the amplitudes. This occurs when the path difference is a whole number of wavelengths: Δ = nλ (n = 0, 1, 2, …).
• Destructive interference: the waves arrive in antiphase (crest meets trough), and the amplitude is the difference. This occurs when the path difference is a half-odd number of wavelengths: Δ = (n + ½)λ. For equal amplitudes the result is zero.

The pattern of loud and quiet spots between two speakers, or of bright and dark fringes in Young’s double-slit experiment, is the result. Applications include noise-canceling headphones (a microphone detects noise and the headphone produces the opposite wave), the anti-reflective coating of lenses (reflections from two surfaces cancel), and the colors of oil films and soap bubbles.

What is diffraction?

When a wave passes through a gap or around the edge of an obstacle, it spreads out into the region behind. Diffraction is greatest when the gap width is comparable to the wavelength; if the gap is much wider, there is little spreading and a shadow forms. Sound with wavelengths of meters (low pitched sounds from 100 Hz to 1 kHz have λ from 3.4 m to 0.34 m) diffracts easily around doorways and corners, which is why we hear people in the next room but not see them, since light has a wavelength of about 500 nm. Ocean waves bend around a headland and spread into a harbor through a narrow entrance. A single slit produces a central bright band and fainter bands to the sides, with the width of the central maximum inversely proportional to the slit width.

Constructive and destructive interference compared

Worked example: path difference and diffraction

1. Two loudspeakers. Two speakers emit the same 686 Hz tone in phase. The speed of sound is 343 m/s, so λ = 343 ÷ 686 = 0.50 m. A listener is 4.00 m from one speaker and 4.75 m from the other. Path difference = 0.75 m = 1.5λ, which is a half-odd number of wavelengths, so there is destructive interference and the sound is quiet.
2. Moving the listener. If the distances are 4.00 m and 4.50 m, the path difference is 0.50 m = 1λ, so there is constructive interference and the sound is loud.
3. Superposition of pulses. Two pulses of heights +3 cm and +2 cm overlap: the resultant at the overlap point is +5 cm. Pulses of +3 cm and −3 cm cancel to 0 at that moment.
4. Diffraction of sound and light. A doorway 1.0 m wide: sound at 340 Hz has λ = 1.0 m, which equals the gap, so it spreads strongly. Light of wavelength 5.0 × 10⁻⁷ m has a ratio of gap to wavelength of 1.0 ÷ 5.0 × 10⁻⁷ = 2 × 10⁶, so the diffraction is negligible and the light casts a sharp shadow.

How were these effects discovered?

Thomas Young demonstrated the interference of light with his double-slit experiment in 1801, which supported the wave theory. Augustin Fresnel developed the theory of diffraction in 1818; Siméon Poisson deduced from it that there would be a bright spot in the center of a circular shadow, a consequence which seemed absurd until François Arago observed it. Francesco Grimaldi had already described the diffraction of light in the 1660s.

Which mistakes are common?

Mistake 1: Thinking that destructive interference destroys energy

The energy is redistributed to the places with constructive interference. The total is conserved.

Mistake 2: Using the distance to the source instead of the path difference

Interference depends on the difference between the two path lengths.

Mistake 3: Believing that diffraction occurs only with narrow slits

It occurs at all gaps and edges, but it is noticeable when the size is similar to the wavelength.

Mistake 4: Forgetting that interference needs coherent sources

Two independent light bulbs do not produce a stable pattern, because their phase difference changes randomly.