Visualize superposition, constructive and destructive interference, and standing waves. Click anywhere on the 2D field to inspect path difference and resulting amplitude.
Wavelength (λ)
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Path difference (Δr)
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Resultant amplitude
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Interference type
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Interference field
Click anywhere on the field to inspect that point's path difference and amplitude.
When two or more waves overlap in the same medium, the resulting displacement at any point is simply the sum of the individual displacements. This is the principle of superposition — it's why interference patterns appear wherever waves cross paths, from ripples in water to sound waves to light.
Path difference and constructive interference
Δr = r₂ − r₁ = mλ (m = 0, ±1, ±2, …)
At any point, each source's wave has traveled a different distance — the path difference Δr. When Δr is a whole number of wavelengths, the two waves arrive in phase and add constructively, producing a bright spot (in light) or loud point (in sound) with amplitude up to 2A.
Destructive interference
Δr = (m + ½)λ (m = 0, ±1, ±2, …)
When the path difference is a half-integer number of wavelengths, the waves arrive exactly out of phase — crest meets trough — and cancel each other out. For equal-amplitude sources, this produces zero net displacement: a dark or silent point.
Phase difference from source phase
Δφ_total = (2π/λ)·Δr + Δφ_source
Interference depends on total phase difference, which combines the geometric path difference with any built-in phase offset between the sources themselves. This simulator lets you set both — adjust the source separation for geometric path difference, and the phase slider for source-level offset.
Standing waves
y(x,t) = 2A·sin(kx)·cos(ωt) λₙ = 2L/n
A standing wave forms when two identical waves travel in opposite directions — for example, a wave and its reflection on a fixed string. The result doesn't travel; it oscillates in place, with fixed points of zero motion (nodes) and maximum motion (antinodes). Only specific wavelengths "fit" the boundary conditions — these are the harmonics.
Nodes and antinodes are evenly spacedAdjacent nodes are always λ/2 apart, and antinodes sit exactly halfway between nodes. The harmonic number n tells you how many half-wavelengths fit in the string length.
Constructive interference doesn't create energyThe 2A amplitude at a bright spot is balanced by zero amplitude at a dark spot elsewhere — total energy is conserved, just redistributed in space.
Two-slit patterns depend on spacingWider source separation packs more bright/dark fringes into the same viewing angle — this is the basis of diffraction gratings and Young's double-slit experiment.
Phase offset shifts the whole patternChanging the source phase difference doesn't change the fringe spacing — it shifts the entire interference pattern sideways, since it adds a constant offset to Δφ everywhere.
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Start with Two-source modeWatch the circular wavefronts spreading from each colored source. Where the bright and dark "fringes" appear is where constructive and destructive interference happens.
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Click anywhere on the fieldA crosshair marks your chosen point. The stat cards and table update instantly with the path difference, resultant amplitude, and whether that point is constructive, destructive, or in between.
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Change wavelength and source separationShrinking the wavelength packs more fringes into the same space. Widening the source separation does the same — both increase how many bright/dark bands fit on screen.
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Add a phase differenceDrag the phase slider away from 0°. Notice the whole fringe pattern shifts sideways, but the spacing between fringes stays the same.
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Switch to 1D Cross-sectionThis isolates a horizontal line through both sources so you can see the individual waveforms and their sum stacked clearly — easier to see exactly how superposition adds the two curves point by point.
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Switch to Standing wave modeUse the harmonic slider or jump buttons to see the 1st through 6th harmonics. Count the nodes (always still) and antinodes (maximum swing) for each harmonic number.
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Copy or export resultsHover any stat card or table row for a copy icon. Use 📋 Copy CSV for the session log, or 🖨️ Report for a printable HTML report with a field snapshot.
Experiments to try
Find a dark fringe: In Two-source mode with default settings, click at different points along a line equidistant from both sources first (the bright central fringe), then move outward until you land exactly between two bright fringes — that's a destructive point with Δr = λ/2.
Anti-phase sources: Load the "Anti-phase sources" preset. Notice the center point (equal distance from both sources) is now dark instead of bright — the 180° source offset flips constructive and destructive everywhere.
Standing wave harmonics: In Standing wave mode, step through harmonics 1 through 6 using the jump buttons. Count nodes each time — the nth harmonic always has n+1 nodes including the two fixed ends.
Wavelength vs. fringe spacing: In Two-source mode, halve the wavelength using the slider. The fringe pattern should roughly double in density — finer bright/dark bands packed into the same space.
