Waveform Interference Lab (2D)

Superposition

Two coherent point sources at x = ±d/2 each emit s(k r − ωt + φ). At a point with distances r₁ and r₂ the disturbances simply add, with a mild decay 1/√(1 + 0.12 r). For sine waves the mean intensity is I ∝ 1 + cos(k(r1 − r2) − φ).

Antinodes, nodes and fringes

Constructive interference needs a path difference r1 − r2 = (m + φ/2π) λ; each integer m gives one hyperbola with the sources as foci (drawn in yellow). Nodes sit halfway between them. Far from the sources the order m leaves at d sin θ = (m + φ/2π) λ. Along the line between the sources the maxima are spaced λ/2, so wider separation or higher frequency packs in more lines.

Waveforms and Fourier series

A square, triangle or sawtooth wave is a sum of sine harmonics, for example (4/π)(sin x + sin 3x/3 + sin 5x/5 + …) for the square wave. Every harmonic has its own wavelength and so its own fringes; the sum blurs the bright and dark bands. Parseval's theorem ties the mean square of the wave to the harmonic amplitudes.

Honest simplifications

The waves are 2D circular ripples with a made-up gentle decay, constant speed 5 units per second (wavelength 5 at 1 Hz), no dispersion, no boundaries and no damping in time. The amplitude slider of the 3D page only scales the height of its mesh, so it has no counterpart here. The 3D page's antinode readout is a rounded estimate; this page counts the lines exactly.

• Start with the default and turn on Node hyperbolas: yellow and red dashed curves alternate with the bands.
• Pick Out of phase (180 degrees): the centre line becomes a node and the whole pattern shifts by half a fringe.
• Pick Wide sources, short wavelength, then watch the exact antinode count rise on the last graph.
• Pick Square wave and look at the harmonic spectrum: only odd harmonics 1, 3, 5 ... with amplitudes in the ratio 1 : 1/3 : 1/5.