Destructive interference
The superposition of two or more coherent waves in anti-phase, producing an amplitude (and intensity) less than the sum. Total cancellation requires equal amplitudes and a path-length difference of (m+½)λ.
Definition
Destructive interference occurs when two or more coherent electromagnetic waves arrive at a point 180° out of phase — peaks of one coincide with troughs of the other — producing a resultant amplitude less than the sum of the individual amplitudes. When the two waves have equal amplitudes, the cancellation is total and the resultant intensity is zero, even though each wave by itself carried nonzero energy; the energy has been redistributed elsewhere by constructive interference, not destroyed.
The condition for total destructive interference is a path-length difference of (m + ½)λ for integer m, producing the dark fringes in Young's double-slit pattern (angles satisfying d sin θ = (m + ½)λ), the dark bands in thin-film interference at specific wavelengths, and the nulls in a radio-antenna array pattern. Anti-reflective coatings on camera lenses and eyeglasses exploit controlled destructive interference: a single quarter-wavelength layer of material with index n_c = √(n_air · n_glass) makes the reflection from the top surface cancel the reflection from the glass surface for one specific wavelength, giving the characteristic faint purple-green hue of coated optics. Active noise-cancelling headphones extend the same principle to acoustic waves, synthesising anti-phase signals in real time to cancel ambient noise in the wearer's ears.