Play the same tone from two speakers and the room fills with fixed LOUD and DEAD spots — where the two waves arrive in step they add (loud), where they arrive half a wavelength off they cancel to silence. Walk across the room and the volume pulses even though nothing changed. It’s the whole trick behind noise-cancelling headphones. Slide your listening position through the pattern.
Two coherent sources emit the same wavelength λ; at a listening point the two paths differ by Δ = |r₁−r₂|. When Δ is a whole number of wavelengths (nλ) the waves arrive in phase and add — CONSTRUCTIVE, double amplitude; when Δ is a half-integer ((n+½)λ) they arrive opposite and cancel — DESTRUCTIVE, silence. The resulting loudness is |cos(πΔ/λ)|, a fixed field of maxima and nulls. A fail-loud self-check throws unless Δ=0 is loud and Δ=λ/2 is silent. Active noise cancellation manufactures exactly the λ/2 null at your ear. ◆ real wave acoustics, node-verified.
Two idealised point sources, equal amplitude, one wavelength shown (the exact two-source pattern); a real room adds reflections and finite source size that blur the deepest nulls — the constructive/destructive path-difference rule is exact.