Waves add, then interfere

Why this rule exists

Because the wave equation is linear, two waves crossing the same point simply add their displacements — superposition. Where crests align they reinforce; where crest meets trough they cancel. This adding-then-cancelling is interference, and it is the signature of waves, impossible for particles. The pattern encodes the waves' relative phase, which is why interference measures distances and wavelengths so precisely.

In practice

To combine waves, add their amplitudes with phase, not their intensities. Track the path difference: a difference of a whole wavelength gives constructive interference, a half wavelength destructive. Use this to predict bright and dark fringes, standing waves, and beats, and to design diffraction gratings, anti-reflection coatings, and noise cancellation.

Example

Waves from two slits meet on a screen.

The path difference d*sinθ sets the phase.

Constructive (bright) at whole wavelengths:
d*sinθ = n*λ,   n = 0, 1, 2 ...

Destructive (dark) at half-integers:
d*sinθ = (n+½)*λ

When it doesn't apply

Superposition holds only while the medium responds linearly. At large amplitude — shock waves, intense optics — the medium's response bends, waves interact, and they no longer pass through each other untouched.