Beats

The two sounds (pure sine waves) are at the bottom of the screen in black and blue. The resulting wave, showing beats, is in heavier black above that.

How does changing one of the frequencies (actually the 2πf, the ω values) affect the pattern of the combined wave?

How does changing one of the amplitudes (the a values) affect the beat pattern?

How does changing the relative phase of the two waves ( the φ values) affect the combined wave?

Lenore Horner, Created with GeoGebra