Wave Interference / Two-Source

When two waves meet, their amplitudes add point-by-point — this is the superposition principle. For path difference Δr = mλ the crests align and intensity quadruples: I = 4I₀. For Δr = (m+½)λ a crest meets a trough and intensity drops to zero. The general result is Aₛᵤᵞ = 2A₀ cos(δ/2), so intensity I ∝ cos²(δ/2). This is exploited in noise-canceling headphones (inject δ = π), optical anti-reflection coatings (thin-film thickness λ/4), and radio phased-array antennas.

Two sources send out overlapping ripples. Pick any point on the screen: usually one wave has traveled slightly farther than the other, and that extra distance — the path difference — decides everything. When it equals a whole number of wavelengths, crests line up and you get a bright fringe; when it's off by half a wavelength, a crest meets a trough and the spot goes dark. Widen the separation d and the tool packs more fringes in, following sin θ = nλ/d.

You might assume interference needs a whole crowd of waves crossing at once, but it doesn't. Fire electrons through a double slit one at a time — so lonely that only one is ever in the apparatus — and each lands as a single dot, seemingly at random. Let thousands accumulate and the same striped pattern emerges. Each particle somehow samples both paths and interferes with itself, a result readers voted "the most beautiful experiment in physics."