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Wave Interference Simulator

Watch two waves combine in real time. Adjust frequency, amplitude, and phase of each wave to see constructive and destructive interference. Switch to the 2D ripple tank to see how two point sources create an interference pattern.

Wave Controls

Wave 1

Frequency f₁

Amplitude A₁ (0–1)

Phase φ₁ (°)

Wave 2

Frequency f₂

Amplitude A₂ (0–1)

Phase φ₂ (°)

Scenarios

Result

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Δφ (φ₂ − φ₁)

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Δf |f₁ − f₂|

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Beat period

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Sum peak amplitude

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Superposition principle

y_sum(x, t) = y₁(x, t) + y₂(x, t)

y₁ = A₁ · sin(k₁x − ω₁t + φ₁) ; y₂ = A₂ · sin(k₂x − ω₂t + φ₂)

When f₁ = f₂: peak sum amplitude = A₁ + A₂ (constructive) or |A₁ − A₂| (destructive)

When f₁ ≠ f₂: beats appear at frequency Δf = |f₁ − f₂|

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Live visualization

Animating · 1× · t = 0 s

Speed:

Wave 1 Wave 2 Sum (interference)

About Wave Interference

When two waves occupy the same space at the same time, their displacements add — this is the superposition principle, a cornerstone of linear wave physics. The result is a new wave whose shape depends on the frequencies, amplitudes, and phase difference of the components. This wave interference simulator lets you adjust each wave's parameters and watch the two wave sum evolve in real time. To explore how individual harmonics stack up to form complex tones, try the harmonic series calculator.

Constructive interference (Δφ = 0)

When two same-frequency waves are in phase, their peaks align and their troughs align. The sum amplitude is the literal sum of the individual amplitudes: A_sum = A₁ + A₂. Two identical waves combining constructively produce a wave twice as tall — but with the same energy density per cycle, so the sound (or light, or water wave) is 6 dB louder. This principle underlies acoustic resonance in instrument bodies and speaker enclosures, where carefully timed reflections reinforce the output.

Destructive interference (Δφ = 180°)

When two same-frequency waves are opposite in phase, peaks of one align with troughs of the other and they cancel: A_sum = |A₁ − A₂|. Equal-amplitude waves cancel completely, leaving silence. This is exactly how active noise cancellation in headphones works — a microphone samples the incoming noise and a speaker emits an anti-phase copy that subtracts it before it reaches your ear. The same destructive-interference principle is used in acoustic panels and anechoic chamber design to eliminate unwanted reflections.

Beats (f₁ ≠ f₂)

If the two frequencies differ slightly, the relative phase drifts over time. The sum amplitude swells and fades at the beat frequency f_beat = |f₁ − f₂|. Musicians use this to tune instruments — when two strings are almost in tune, you hear the beat pulse slow down as they approach perfect pitch. At exactly in tune, beats vanish. You can hear and measure this directly with the beat frequency calculator, which plays both tones and computes the resulting beat period. The same effect explains why slightly detuned chorus effects in synthesizers sound fuller: beating between two slightly different oscillators creates the characteristic shimmer.

2D ripple tank

Two point sources emitting circular waves create an interference pattern in the plane. Along directions where the path difference to both sources equals an integer number of wavelengths (Δd = nλ), the waves arrive in phase → bright fringes (constructive). Where the path difference is a half-integer multiple (Δd = (n + ½)λ), they cancel → dark fringes (destructive). This is the foundation of Young's double-slit experiment, and the math is the same one used for radio antenna arrays, phased-array radar, and acoustic holography. Changing the source separation slider rescales the fringe spacing — closer sources, wider fringes; farther apart, tighter fringes — exactly matching the formula fringe spacing ≈ λ·D/d.

Frequently Asked Questions

What is "phase" intuitively?

Phase is where in its cycle a wave is at a given moment. 0° means the wave is just starting its positive half. 90° means it's at its positive peak. 180° means it's at the zero crossing going negative. 360° (= 0°) brings it back to start. Two waves with the same phase rise and fall together; with opposite phase, one rises as the other falls.

Why does the sum amplitude depend on phase, not just amplitudes?

Adding waves is vector addition in their cycle plane, not just adding their peak values. Two waves of amplitude 1 with the same phase sum to amplitude 2. The same two waves with 180° phase difference sum to amplitude 0 — they perfectly cancel. With 90° phase difference (quadrature), the sum is √(A₁² + A₂²) ≈ 1.414 for two unit amplitudes — the diagonal of a unit square in phase space.

What's the difference between beats and interference?

They're the same phenomenon viewed differently. Interference usually refers to spatial patterns (where in space waves cancel/add). Beats refer to temporal modulation (waves cancel/add over time because their frequencies differ). If f₁ = f₂, the interference pattern is static in time. If f₁ ≠ f₂, the pattern shifts over time at the beat frequency.

Why do the 2D fringes shift when I change phase?

The fringes are surfaces where the path difference Δd between the two sources equals an integer (n·λ) for bright, or half-integer ((n+0.5)·λ) for dark. Changing one source's initial phase by Δφ shifts that condition by Δφ·λ/(2π) — the whole pattern translates without changing the fringe spacing. This is exactly how phased-array antennas steer beams.

Is this a real wave simulation?

Yes — for two ideal sinusoidal sources in a non-dispersive medium with no boundaries, the math is exact. We compute A·sin(k·d − ω·t + φ) for each source at each point and sum them, which is what the wave equation produces for linear superposition. Real ripple tanks add complications (boundary reflections, viscosity damping, finite source size, nonlinear effects at high amplitude) that this simulator doesn't model.

What does "ω" mean and how does it relate to f?

ω (omega) is the angular frequency in radians per second: ω = 2π·f. It comes from preferring to write sinusoidal functions as sin(ω·t) rather than sin(2π·f·t) — cleaner math, especially in calculus. Similarly k (wavenumber) = 2π/λ, so the wave equation sin(k·x − ω·t) describes a wave traveling at speed ω/k = f·λ.

How does wave interference apply to musical intervals and tuning?

When two notes form a simple frequency ratio — an octave (2:1), a perfect fifth (3:2), a perfect fourth (4:3) — their harmonics align and the interference pattern is stable, which our ears perceive as consonance. Ratios farther from simple integers produce many near-misses among harmonics, creating rapid beating and a sense of tension or dissonance. This is why equal-temperament tuning is a compromise: every interval except the octave is slightly mistuned, generating very slow beats that most listeners find acceptable but that trained ears can detect.

Can this simulator model more than two waves at once?

No — this tool is designed specifically for two-wave superposition, which covers the core physics clearly. Real instruments, room acoustics, and antenna arrays combine many waves simultaneously; for that, you need a Fourier-synthesis approach. To hear what stacking many harmonics sounds like in practice, the harmonic series calculator lets you build a tone from multiple partials and see how they sum.

What is a standing wave, and how does it differ from the waves shown here?

A standing wave forms when two waves of identical frequency and amplitude travel in opposite directions along the same path. Instead of the pattern moving forward, nodes (points of permanent cancellation) and antinodes (points of maximum amplitude) stay fixed in space. This simulator shows waves traveling in the same direction and summing; for the counter-propagating special case, the standing wave calculator shows the resulting node and antinode positions for strings, tubes, and rods.

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