Result

Interference condition

Δ r = mλ (constructive), Δ r = (m + \frac{1}{2}) λ (destructive)

Statement

Whether two coherent waves reinforce or cancel, based on their path-length difference Δr.

Symbols and units

$Δ r$	path-length difference	m
$λ$	wavelength	m
$m$	integer order	—

When it applies

Two coherent waves of the same wavelength reaching a common point.

Don't use when

Requires two coherent waves of the same wavelength — incoherent or differing-wavelength sources do not produce a stable interference pattern.

Derivation

A path-length difference converts directly to a phase difference between the two waves arriving at the point:

Δ ϕ = \frac{2 π}{λ} Δ r

In phase (Δφ a multiple of 2π) gives constructive interference; exactly out of phase gives destructive:

Δ r = mλ or (m + \frac{1}{2}) λ

Worked examples

A full wavelength of path difference

At Δr = λ exactly:

Δ ϕ = \frac{2 π}{λ} λ = 2 π

A full 2π phase difference is indistinguishable from being perfectly in phase — the waves interfere constructively, even though a whole wavelength appears to have been 'lost' along the way.

Watch out for

Δr must be measured as the actual difference in path length travelled by the two waves to the observation point — not the separation between the sources.