Destructive Interference Formula: 5 Essential Wave Equations

The destructive interference formula tells you exactly where and when two waves cancel each other out. The constructive interference formula does the same for waves that reinforce. Together, these equations are the mathematical backbone of everything from double-slit experiments to noise-cancelling headphones.

This guide compiles all five essential interference formulas — path difference, phase difference, the conversion between them, fringe spacing, and intensity — with two worked examples that show how to apply each one.

Think of two stones dropped into a still pond. Where the outward ripples meet, crest-on-crest, the water jumps higher — that is constructive interference. Where crest meets trough, the surface flattens — that is destructive interference. The formulas below are just a precise way of describing that same meeting of waves, whether the waves are light, sound, or water.

The five essential interference formulas at a glance

Each formula is a different lens on the same physics. Choose the one that matches what you know about the system.

Formula 1: path difference (the most common)

The path difference Δr is the difference in distance two waves travel from their sources to the meeting point. This is usually the easiest quantity to measure in a lab setup.

• Constructive interference formula (path difference): Δr = nλ, where n = 0, 1, 2, 3, …
• Destructive interference formula (path difference): Δr = (n + ½)λ, where n = 0, 1, 2, 3, …

In plain terms: if the extra distance one wave travels is a whole number of wavelengths, the waves arrive in step and reinforce. If it is a half-integer number of wavelengths, they arrive out of step and cancel.

Here is a quick reference for the first few orders:

For a more detailed explanation of what these conditions mean physically, see our guide on constructive vs destructive interference.

Formula 2: phase difference

Phase difference Δφ describes the same relationship but in angular terms — how far through its cycle each wave is when they meet.

• Constructive: Δφ = 2πn (0, 2π, 4π, … radians) or 0°, 360°, 720°, …
• Destructive: Δφ = (2n + 1)π (π, 3π, 5π, … radians) or 180°, 540°, 900°, …

The phase difference method is more natural when you are dealing with electronic signals, oscillators, or coatings that introduce a fixed phase shift.

Formula 3: converting between phase and path difference

This is the bridge formula — it lets you move from one description to the other.

Δφ = (2π / λ) × Δr

The relationship is linear. Every full wavelength of path difference corresponds to 2π radians (360°) of phase difference. Every half-wavelength of path difference corresponds to π radians (180°).

Example conversion: If two light waves (λ = 550 nm) have a path difference of 825 nm, the phase difference is:

Δφ = (2π / 550 nm) × 825 nm = 3π radians (540°)

That is an odd multiple of π, so the interference is destructive.

This conversion formula is useful whenever you have path length data but need to plug into a phase-based intensity equation, or vice versa.

Formula 4: fringe spacing in a double-slit experiment

For interference patterns on a distant screen, the fringe spacing formula predicts how far apart the bright bands will be.

Δy = λD / d

Where:

• Δy = distance between adjacent bright (or dark) fringes
• λ = wavelength of the light
• D = distance from the slits to the screen
• d = separation between the two slits

The formula says: wider slits squeeze the fringes closer together. Moving the screen further back spreads them out. And longer wavelengths — red light rather than blue — give wider spacing.

For a full walkthrough of Young's experiment and how to set it up, see our explanation of the double slit experiment.

Formula 5: resultant intensity for unequal amplitudes

The simple destructive interference formula Δr = (n + ½)λ assumes the two waves have equal amplitude. In the real world, amplitudes often differ — one slit might be slightly wider, or one speaker slightly louder. The intensity at any point is then:

I = I₁ + I₂ + 2√(I₁I₂)cos(Δφ)

Where I₁ and I₂ are the intensities of the individual waves and Δφ is the phase difference.

• When Δφ = 0 (constructive): I = I₁ + I₂ + 2√(I₁I₂) = (√I₁ + √I₂)²
• When Δφ = π (destructive): I = I₁ + I₂ − 2√(I₁I₂) = (√I₁ − √I₂)²

If I₁ = I₂, destructive interference gives I = 0 — perfect cancellation. If I₁ ≠ I₂, the minimum intensity is greater than zero. This matters in practical applications like noise-cancelling headphones, where the anti-noise wave must match the ambient noise amplitude for effective cancellation.

For more on how patterns form from multiple sources, see our guide on interference patterns.

Worked example 1: light interference (Young's double slit)

A red laser (λ = 650 nm = 6.5 × 10⁻⁷ m) shines through two slits separated by d = 0.25 mm = 2.5 × 10⁻⁴ m. The screen is D = 2.0 metres away.

Step 1 — find the fringe spacing.

Δy = λD / d = (6.5 × 10⁻⁷ × 2.0) / (2.5 × 10⁻⁴) = 5.2 × 10⁻³ m = 5.2 mm

The bright red fringes will be spaced 5.2 millimetres apart on the screen.

Step 2 — find the position of the third bright fringe.

The third bright fringe (n = 3) is located at a path difference of Δr = nλ = 3 × 650 nm = 1950 nm.

On the screen: y₃ = nλD / d = 3 × 5.2 mm = 15.6 mm from the centre.

Step 3 — find the position of the second dark fringe.

The second dark fringe (n = 1 for destructive, as n starts at 0) has Δr = (n + ½)λ = 1.5 × 650 nm = 975 nm.

Position: y = (n + ½)λD / d = 1.5 × 5.2 mm = 7.8 mm from the centre.

Worked example 2: sound interference (two speakers)

Two speakers are 4.0 metres apart, facing the same direction. They both emit a 440 Hz tone. The speed of sound is 343 m/s. You stand 5.0 metres in front of the midpoint and walk to the side.

Step 1 — find the wavelength.

λ = v / f = 343 / 440 = 0.780 m (78 cm)

Step 2 — decide which formula to use.

You are not on a screen but in open space. The relevant quantity is the path difference between the two speakers to your current position. Use the path difference formulas.

Step 3 — calculate the path difference at a given position.

You stand at a point where the distance to speaker A is 5.4 m and the distance to speaker B is 6.3 m.

Δr = |6.3 − 5.4| = 0.9 m

Compare to λ: 0.9 / 0.78 ≈ 1.15

This is not a whole number or a half-integer — it is somewhere in between. The interference is partial (neither fully constructive nor fully destructive).

Step 4 — find a destructive interference position.

For destructive interference: Δr = (n + ½)λ = (0 + ½) × 0.78 = 0.39 m.

You move along the line until the path difference is 0.39 m. At that point, the tone drops to near silence (provided the speakers are equally loud).

For constructive interference: Δr = nλ = 1 × 0.78 = 0.78 m.

Walk to the position where the path difference is 0.78 m, and the tone is loudest.

Common misconception: interference formulas tell you everything

The formulas above give the positions of maxima and minima, but the real pattern is a continuous variation of intensity between these extremes. At a point mid-way between a maximum and a minimum, the interference is partial — the waves neither fully reinforce nor fully cancel.

The intensity formula I = I₁ + I₂ + 2√(I₁I₂)cos(Δφ) captures this continuous behaviour. The path difference and phase difference formulas are just the special cases where cos(Δφ) = ±1.

For more on how the complete pattern relates to diffraction effects, see our comparison of diffraction vs interference.

Key takeaways

• The path difference formulas (Δr = nλ for constructive, Δr = (n + ½)λ for destructive) are the most widely used interference formulas.
• The phase difference formulas (Δφ = 2πn / (2n + 1)π) are equivalent — use them when you know the phase relationship between sources.
• The conversion formula Δφ = (2π/λ) × Δr links the two descriptions.
• Fringe spacing Δy = λD / d applies specifically to double-slit setups with a distant screen.
• The intensity formula I = I₁ + I₂ + 2√(I₁I₂)cos(Δφ) accounts for unequal amplitudes and gives the full continuous intensity pattern.
• Partial cancellation occurs when the path difference is neither a whole nor a half-integer multiple of the wavelength.

External resources

• The Physics Classroom: Two-Point Source Interference — clear diagrams and step-by-step derivation of interference conditions for light waves
• HyperPhysics: Interference — authoritative reference with interactive formula tables and worked examples from Georgia State University
• Encyclopaedia Britannica: Destructive Interference — mathematical treatment with historical context and real-world applications