Wave Superposition Beats Simulator
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About This MicroSim
When two waves arrive at the same place, their displacements simply add. That one sentence produces two very different-looking outcomes, and this sim puts them side by side.
The top two plots are the input waves alone. The bottom plot is their sum, on the same time axis, so you can line any moment up across all three.
Equal frequencies. The relationship between the two waves never changes, so the interference is steady. Phase alone decides the outcome: in phase and they reinforce to double amplitude; a half cycle apart and they cancel to nothing.
Unequal frequencies. Now the relationship drifts. The waves slide in and out of phase over and over, and the sum's amplitude pulses. That pulsing is a beat, and it repeats at exactly \(|f_1 - f_2|\) Hz.
The dashed purple envelope traces that pulsing. It comes straight out of the identity
where the slow \(\cos\) factor is the envelope and its zero crossings are the moments of complete cancellation.
How to Use
- Start with both frequencies at 300 Hz. Drag Wave 2 phase offset from 0 toward π and watch the sum shrink to nothing. Note that neither input wave changed size.
- Return phase to 0. Now set Wave 2 frequency to 320 Hz. The steady pattern is replaced by a pulsing envelope, and the beat readout appears.
- Count the pulses in the 100 ms window. At a 20 Hz beat you should see two.
- Widen the gap to 350 Hz. The beat gets faster; the readout confirms 50 Hz.
- Set the frequencies equal again and confirm the envelope and readout disappear — with no frequency difference there is no beat.
Reading the Shading
| Shading | Meaning |
|---|---|
| Green | The waves are near in phase — constructive |
| Red | The waves are near opposite phase — destructive |
| Dashed purple | Beat envelope (only drawn when frequencies differ) |
Lesson Plan
Grade Level
Undergraduate (college junior/senior)
Duration
12-15 minutes
Prerequisites
- A sine wave has frequency, amplitude, and phase
- Waves add point by point
Learning Objective
Students will be able to examine the summed waveform of two sine waves and compare regions of constructive interference, destructive interference, and the beat pattern produced when the frequencies are close but unequal.
Activities
- Phase only (5 min): At equal frequencies, students find the phase offsets for maximum and minimum sum amplitude and state them in radians.
- Introduce a difference (5 min): Students set a 20 Hz difference, count the beats in the window, and verify against the readout.
- Predict the beat rate (4 min): Given 440 Hz and 444 Hz, students predict the beat frequency and describe what a listener would hear.
Assessment
Ask: "Two guitar strings are played together and you hear the loudness pulse three times per second. What can you conclude about their frequencies, and what would you do to fix it?"
Related Resources
References
- Superposition principle — why displacements simply add.
- Beat (acoustics) — the beat frequency result and its use in tuning.
- Wave interference — constructive and destructive interference in general.