Wave Superposition Simulator
Superpose two traveling waves and observe interference, beats, and standing waves. Adjust frequency, amplitude, and phase of each wave.
About Wave Superposition
The superposition principle states that when two or more waves overlap in the same medium, the resulting displacement at any point is the algebraic sum of the individual displacements. Waves pass through each other without permanently affecting each other.
Key Variables
| Symbol | Name | Unit | Description |
|---|---|---|---|
| A₁, A₂ | Amplitudes | m (normalized) | Peak displacement of each wave |
| f₁, f₂ | Frequencies | Hz | Number of complete cycles per second |
| λ₁, λ₂ | Wavelengths | m (normalized) | Spatial period; λ = v/f |
| φ₂ | Phase offset of Wave 2 | degrees | Phase shift of Wave 2 relative to Wave 1 |
| v | Wave speed | px/s (normalized) | Speed at which wave pattern travels |
| f_beat | Beat frequency | |f₁ − f₂| Hz | Rate of amplitude modulation when f₁ ≈ f₂ |
| y(x,t) | Resultant displacement | m | Sum of both waves at each point and time |
Types of Superposition
- Constructive interference: waves reinforce. Occurs when phase difference = 0°, 360°, … The resultant amplitude equals A₁ + A₂.
- Destructive interference: waves cancel. Occurs when phase difference = 180°. If A₁ = A₂ the result is zero.
- Beats: when f₁ ≈ f₂, the amplitude slowly oscillates at f_beat = |f₁ − f₂|. Audible as a pulsing volume.
- Standing waves: when identical waves travel in opposite directions, nodes and antinodes form fixed patterns.
Worked Example — Beats
Two tuning forks vibrate at f₁ = 440 Hz and f₂ = 444 Hz, both with A = 1:
You hear the amplitude pulse 4 times per second. Musicians use this effect to tune instruments — when the beat frequency reaches zero, the instruments are in tune.
Worked Example — Equal Frequencies, Phase Shift
Two waves with f₁ = f₂ = 2 Hz, A₁ = A₂ = 1, phase difference φ = 90°:
Key Formulas
Traveling Wave
Superposition (Resultant)
Wavelength–Frequency Relation
Beat Frequency
Resultant Amplitude (same frequency)
| Formula | Description | Notes |
|---|---|---|
| y = A sin(kx − ωt + φ) | Traveling wave (rightward) | k = 2π/λ, ω = 2πf |
| y_total = y₁ + y₂ | Superposition principle | Valid for all linear waves |
| f_beat = |f₁ − f₂| | Beat frequency | Heard as periodic loudness variation |
| λ = v/f | Wavelength from speed and frequency | v is wave speed in the medium |
| A_R = |A₁ + A₂| (φ=0°) | Maximum constructive amplitude | Waves fully reinforce |
| A_R = |A₁ − A₂| (φ=180°) | Minimum destructive amplitude | Waves partially or fully cancel |
Frequently Asked Questions
What is the superposition principle?
The superposition principle states that the net displacement at any point is the sum of the individual wave displacements at that point. Waves pass through each other undisturbed — after they separate, each wave continues unchanged.
What are beats and why do I hear them?
Beats occur when two slightly different frequencies interfere. The resultant amplitude oscillates at f_beat = |f₁ − f₂| Hz. Your ear perceives this amplitude modulation as a periodic 'wah-wah' sound. Musicians use beats to tune — when the beat disappears, the frequencies match.
What is the difference between interference and superposition?
Superposition is the general principle (displacements add). Interference is the result when two coherent waves superpose — we describe the pattern as constructive (crests align → larger amplitude) or destructive (crest meets trough → smaller amplitude).
How do standing waves form?
Standing waves form when two waves of equal frequency and amplitude travel in opposite directions. The superposition gives y = 2A cos(ωt) sin(kx). Nodes (always zero) and antinodes (maximum oscillation) are stationary in space — they do not travel.
Why does phase difference matter?
Phase difference Δφ determines whether the waves reinforce or cancel. Δφ = 0° gives maximum constructive interference (A_R = A₁ + A₂). Δφ = 180° gives maximum destructive interference (A_R = |A₁ − A₂|). Intermediate phases give intermediate amplitudes.