# Beats Demonstration

• Press the start button. When two frequencies that are very close are played together, you hear a "wah-wah" sound.
• Move the slider below so the two frequencies are far apart. The "wah-wah" sound disappears.
• Read below for more information (or see the notes on Drew's page)

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.