Explore constructive and destructive interference from two wave sources. Adjust frequency, phase, and source separation to reveal interference patterns, nodal lines, and standing waves.
When two waves overlap, their amplitudes add: y_total = A·sin(kr₁ − ωt) + A·sin(kr₂ − ωt + φ). Constructive interference occurs when |r₁−r₂| = nλ (path difference is integer wavelengths). Destructive interference when |r₁−r₂| = (n+½)λ. The 2D ripple tank shows brighter regions where waves reinforce and darker regions where they cancel.
For two point sources, intensity I ∝ cos²(πd·sinθ/λ) at angle θ from the centre. Young's double-slit experiment (1801) demonstrated the wave nature of light using exactly this geometry. The fringe spacing Δy = λL/d (L=screen distance, d=slit separation). Same physics governs radio antennas, sonar arrays, and optical interferometers.
When two identical waves travel in opposite directions, they form a standing wave: y = 2A·cos(kx)·sin(ωt). Nodes (always zero amplitude) are fixed in space at kx = (n+½)π. Anti-nodes have maximum amplitude. Musical instruments, microwave cavities, and laser resonators all use standing waves. In this sim, set phase = π to see the standing wave pattern.
This simulation visualises the superposition of circular waves emitted by up to six coherent point sources. A GLSL fragment shader running on your GPU evaluates, for every pixel, the summed amplitude A = Σ cos(kd−ωt+φ)/√d, where k = 2π/λ is the wavenumber and d is the distance to each source. Bright crests and dark nodal lines emerge wherever the contributions reinforce or cancel, producing a live two-dimensional interference pattern.
You can set the number of sources, the wavelength λ, the wave speed, the two-source separation, and the relative phase φ applied to the second source. A toggle switches between signed amplitude (blue↔red) and intensity (|A|² glow), and you may click or drag the canvas to reposition the nearest source. The same physics underlies Young's double-slit experiment, phased radio antenna arrays, sonar beamforming, and optical interferometers.
What does this simulation show?
It shows the interference pattern formed when circular waves from several coherent point sources overlap. Where the waves arrive in step they add to give bright crests (constructive interference); where they arrive out of step they cancel to give dark nodal lines (destructive interference).
How does it calculate the pattern?
A fragment shader sums the contribution of each source at every pixel using A = Σ cos(kd − ωt + φ)/√d, where k = 2π/λ, d is the distance to a source, and ω is the angular frequency. The 1/√d factor models how a two-dimensional circular wave spreads its energy outward.
What is constructive and destructive interference?
Constructive interference occurs where the path difference between two sources is a whole number of wavelengths (nλ), so crests align and reinforce. Destructive interference occurs where the path difference is a half-integer number of wavelengths ((n+½)λ), so a crest meets a trough and they cancel.
Sources sets how many emitters are active (1 to 6). Wavelength λ sets the spacing of wavefronts and hence the fringe spacing. Wave speed controls how fast the pattern animates outward. Source separation positions the two-source pair, and Phase φ shifts source two between 0 and 2π. You can also drag a source directly on the canvas.
For two sources a distance d apart, the far-field fringe spacing is Δy ≈ λL/d, where L is the distance to the observing screen. Increasing the wavelength or reducing the separation widens the fringes; the readout panel reports this value live using a reference distance of L = 700 pixels.
Phase φ offsets the oscillation of the second source relative to the first, measured in multiples of π. At φ = 0 the sources are perfectly in step. Setting φ = π makes them exactly out of phase, which shifts every bright fringe to where a dark one was and reverses the pattern.
Amplitude mode shows the signed displacement of the wave field using a diverging blue-to-red colour map, so you can see crests, troughs, and the instantaneous wave shape. Intensity mode shows |A|², the time-related energy or brightness that a detector or screen actually records, highlighting the bright fringes only.
It captures the correct phase relationships, path-difference rules, and fringe geometry that govern real interference. It is a qualitative 2D model, however: distances are in pixels rather than physical units, the 1/√d falloff is an idealisation, and effects such as diffraction at finite slits are not separately modelled.
The two-source preset reproduces the geometry Thomas Young used in 1801. By splitting light into two coherent sources he produced alternating bright and dark fringes, proving light behaves as a wave. The intensity for two sources follows I ∝ cos²(πd·sinθ/λ) as a function of viewing angle θ.
Interference is central to many technologies: phased-array radar and radio antennas steer beams by adjusting relative phase, sonar and ultrasound arrays focus sound, optical interferometers measure tiny distances, and anti-reflection coatings, holograms, and noise-cancelling headphones all rely on controlled constructive or destructive interference.