Resonance And Entrainment Phenomena In Binaural Beats: A Mathematical Acoustics Perspective
DOI:
https://doi.org/10.29284/29tcge61Keywords:
Binaural Beats, Wave Superposition, Beat Frequency Resonance, Forced Oscillations, Coupled Oscillators.Abstract
This study investigates resonance and entrainment phenomena in oscillatory systems subjected to beat-frequency modulation using a mathematical acoustics framework. When two sinusoidal signals of slightly different frequencies are superimposed, a beat frequency fb=∣f1-f2∣is generated, producing a low-frequency amplitude modulation that acts as an external periodic driving force on the system.
A controlled simulation-based methodology was employed in which an idealized damped harmonic oscillator system was driven by beat-modulated signals at 6 Hz, 10 Hz, 20 Hz, and 40 Hz. System response was analyzed using spectral, time–frequency, and synchronization metrics.
The results demonstrate clear frequency-dependent resonance behavior. Maximum amplitude amplification was observed at 10 Hz, where spectral power increased from 6.1 to 10.4 units (≈70% increase). Similarly, 6 Hz stimulation showed a 48% increase in power (from 5.2 to 7.8 units). In contrast, higher frequencies exhibited weaker responses, with 20 Hz and 40 Hz showing only 25% and 31% increases, respectively.
Synchronization analysis using the Resonance Strength Index (RSI) revealed strong entrainment at low frequencies, with values of 0.70 (10 Hz) and 0.50 (6 Hz), compared to weaker entrainment at higher frequencies (0.25 at 20 Hz, 0.31 at 40 Hz). Coupling metrics further confirmed enhanced system coordination, with global efficiency increasing from 0.51 (baseline) to 0.66 (10 Hz).
These findings confirm that maximum energy transfer and synchronization occur when the driving frequency approaches the system’s natural frequency, consistent with classical resonance theory. The study introduces a unified framework combining wave interference, forced oscillation, and coupled oscillator dynamics, along with a quantitative Resonance Strength Index (RSI) for measuring entrainment strength.
This work provides a purely physics-based interpretation of beat-induced synchronization and offers potential applications in signal processing, resonant systems, and nonlinear dynamical analysis.
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