Wave Interference Formula
The wave interference formula calculates constructive and destructive interference patterns.
Learn path difference with examples.
The Formula
Destructive: path difference = (n + 1/2)λ
Wave interference occurs when two waves overlap and combine. Whether the result is louder (constructive) or quieter (destructive) depends on the path difference between the two waves.
Thomas Young first demonstrated light interference in 1801 with his famous double-slit experiment in England. This experiment provided strong evidence that light behaves as a wave. The same principles apply to sound waves, water waves, and all other wave phenomena.
Constructive interference happens when waves arrive in phase (crests align with crests). Destructive interference happens when waves arrive out of phase (crests align with troughs). The path difference determines which type occurs: whole wavelength multiples give constructive, half-wavelength offsets give destructive.
Variables
| Symbol | Meaning |
|---|---|
| d | Path difference between the two waves (meters) |
| λ | Wavelength of the waves (meters) |
| n | Order number (0, 1, 2, 3, ...) |
Example 1
Two speakers emit a 680 Hz tone (speed of sound = 340 m/s). A listener is 4.0 m from one speaker and 5.0 m from the other. Is the interference constructive or destructive?
Calculate wavelength: λ = v/f = 340/680 = 0.5 m
Path difference: d = 5.0 - 4.0 = 1.0 m
Check: d/λ = 1.0/0.5 = 2 (a whole number)
n = 2, so this is constructive interference (the sound will be louder)
Example 2
In a double-slit experiment, light of wavelength 600 nm passes through slits 0.1 mm apart. The screen is 2 m away. Where is the first dark fringe?
For the first dark fringe: the fringe position y = (n + 1/2) × λL/d
y = (0 + 0.5) × 600 × 10⁻⁹ × 2 / (0.1 × 10⁻³)
y = 0.5 × 1.2 × 10⁻⁵ / 10⁻⁴ = 0.006 m
y = 6 mm from the center of the pattern
When to Use It
Use wave interference formulas to predict how overlapping waves combine at any point in space.
- Designing acoustic spaces to avoid dead spots or echoes
- Understanding how noise-canceling headphones work
- Analyzing thin film interference (oil slicks, soap bubbles)
- Measuring wavelengths using interferometry