Interference patterns
Point sources emit circular waves that spread, overlap, and interfere. Where path differences equal whole wavelengths, crests reinforce. Where they differ by half a wavelength, they cancel. Drag the sources, adjust parameters, and watch the pattern reconfigure in real time.
Δφ = 2πΔr / λ constructive: Δr = nλ destructive: Δr = (n+½)λ
Wave superposition
When two or more waves overlap, the resulting displacement at any point is the sum of the individual displacements. This is the principle of superposition, and it holds for any linear wave system — water waves, sound, light, even quantum mechanical wavefunctions.
Path difference and interference
For two point sources of the same frequency, the interference at a point depends on the path difference Δr = |r₁ − r₂|, where r₁ and r₂ are distances from each source. When Δr is a whole number of wavelengths, crests arrive together: constructive interference. When Δr is a half-integer number of wavelengths, crest meets trough: destructive interference.
Nodal lines
The loci of destructive interference form curves called nodal lines (or antinodal lines for constructive interference). For two point sources, these are branches of hyperbolas with the sources as foci. The number of nodal lines between the sources equals the number of wavelengths in the source separation.
Multiple sources
Adding more sources creates richer interference patterns. Three or four sources arranged symmetrically produce patterns with rotational symmetry. Antenna arrays exploit this principle — phased arrays steer radio beams by adjusting the phase relationship between many sources.