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Conditions for constructive and destructive interference

Short answer

Constructive interference yields maximum intensity when path difference is an integer multiple of λ, Δ = nλ. Destructive interference gives minimum intensity when Δ = (2n+1)λ/2 (odd multiple of half wavelength).

What are constructive and destructive interference? Prove that the path difference for constructive interference is an integer multiple of λ and that for destructive interference is an odd integer multiple of λ/2.

Constructive interference: two coherent waves meet in phase, so their amplitudes add and the intensity is maximum (bright). Destructive interference: they meet out of phase, amplitudes subtract and the intensity is minimum (dark).

Proof

  1. Let two coherent waves of amplitudes A₁ and A₂ with phase difference δ superpose. The resultant intensity is
I = A₁² + A₂² + 2 A₁A₂ cos δ
  1. A path difference Δ corresponds to a phase difference
δ = (2π/λ) Δ

  1. Constructive: I is maximum when cos δ = +1, so δ = 2nπ (n = 0, 1, 2 ...). Then

    (2π/λ)Δ = 2nπ → Δ = nλ, an integral multiple of λ (equivalently an even multiple of λ/2). I_max = (A₁ + A₂)².

  2. Destructive: I is minimum when cos δ = −1, so δ = (2n+1)π. Then

    (2π/λ)Δ = (2n+1)π → Δ = (2n+1) λ/2, an odd integral multiple of λ/2. I_min = (A₁ − A₂)².

Phase differencePath differenceIntensity
Constructive0, 2π, 4π ...0, λ, 2λ ...(A₁+A₂)²
Destructiveπ, 3π, 5π ...λ/2, 3λ/2 ...(A₁−A₂)²

✓ Answer checked by Claude Opus 5.5 (high effort)

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