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Beats Demonstration

Current frequencies are 440Hz and Hz. Adjust second frequency with slider.

Manipulate axes with controls at lower right of graph.

To hear beats, choose a frequency close to 440Hz. If frequencies \(f_1\) and \(f_2\) are close, playing them simultaneously will sound like a note with frequency \(\frac{f_1 + f_2}{2}\) turning off and on with frequency \(\lvert f_1 - f_2\rvert\).

This is because adding sine waves with frequencies \(f_1\) and \(f_2\) gives: \[{\color{blue}{\sin(f_1 2\pi t) + \sin(f_2 2\pi t)}} = {\color{red}{2\cos\left(\frac{f_1 - f_2}{2}\cdot 2\pi t\right)}}\sin\left(\frac{f_1 + f_2}{2} \cdot 2\pi t\right)\]

Current frequencies are 440Hz and Hz. Adjust second frequency with slider.