# Beat Frequency Phenomenon - Interactive Visualization

Interactive simulation demonstrating beat frequency with real audio output using Web Audio API

> Canonical page: https://elysiatools.com/en/visualizations/beat-frequency

- **Category:** Physics

## Overview

Interactive beat frequency simulation with real audio output using Web Audio API demonstrating the acoustic beat phenomenon. Features wave equations: y₁ = A·sin(2πf₁t) (Tone 1), y₂ = A·sin(2πf₂t) (Tone 2), y = y₁ + y₂ (Superposition), beat frequency f_beat = |f₁ - f₂|. When two sound waves with slightly different frequencies interfere, they create amplitude modulation at the beat frequency - the characteristic 'wah-wah-wah' pulsating sound used in instrument tuning. Real-time waveform visualization: Tone 1 waveform (blue) showing individual sine wave at frequency f₁, Tone 2 waveform (red) showing individual sine wave at frequency f₂, Combined waveform (purple) displaying y₁ + y₂ superposition with audible beat pattern, Beat envelope visualization showing the amplitude modulation envelope (2A·cos(π·f_beat·t)) as dashed lines indicating maximum and minimum amplitude boundaries. Web Audio API implementation: Dual oscillator architecture with independent frequency control, master gain node for volume adjustment (0-100%), real-time frequency updates while audio is playing, analyser node for audio visualization. Adjustable parameters: Tone 1 frequency f₁ (200-800 Hz, default 440 Hz - A4), Tone 2 frequency f₂ (200-800 Hz, default 444 Hz), master volume control, wave type selection (sine, triangle, square). Automatic note name display with cent deviation (e.g., 'A4 + 4¢' for 444 Hz). Real-time beat statistics: beat frequency f_beat = |f₁ - f₂| in Hz, beat period T_beat = 1/f_beat in seconds, beats per second count. Quick presets: Slow Beat (2 Hz, f₁=440, f₂=442), Moderate (4 Hz, f₁=440, f₂=444), Fast Beat (8 Hz, f₁=440, f₂=448), Tuning Fork (440+442 Hz), Unison/No Beat (440+440 Hz). Educational content: mathematical foundation with all four formulas, real-world applications (instrument tuning by listening for beat disappearance, piano tuning with beat rate intervals, Doppler radar velocity measurement, music production synth detuning techniques, heterodyne radio detection), listening guide with tips for slow beats (2-4 Hz for clear wah-wah), envelope focus (amplitude modulation perception), and fast beat merging (>15 Hz creates difference tone perception). Playback controls with play/stop buttons and responsive canvas rendering for all screen sizes. Multi-language support (zh, en, es, fr, de, ru, pt, ja).

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