constructive and destructive interference

When two waves meet, sometimes they team up to make a bigger wave and sometimes they fight and shrink toward nothing. Teaming up is constructive interference, cancelling is destructive interference. An everyday image: push a child on a swing in time with their motion and the swings grow (constructive), push against the motion and you kill it (destructive). The question it answers is: when do overlapping waves add up, and when do they cancel?

Constructive interference happens when two waves arrive in phase, crest on crest and trough on trough, so their displacements add and the amplitude grows, for two equal waves the result is twice as tall. Destructive interference happens when they arrive out of phase, crest on trough, so they subtract, and two equal waves cancel completely to zero. The deciding factor is the path difference: constructive when the path difference equals m lambda (a whole number of wavelengths), destructive when it equals (m + 1/2) lambda, where m = 0, 1, 2, ...

This is the machinery behind noise-cancelling headphones (they add an anti-sound, a wave exactly out of phase, to cancel the noise) and behind the bright and dark bands in optics. An honest note: energy is not destroyed at a dark spot, it is redistributed. The energy missing from the cancelled regions reappears in the reinforced regions, so the total is conserved.