Beats
Author: Lenore Horner

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?

sound 1

sound 2

total sound

a_1 = 1 (slider from -5 to 5, step 0.1)
ω_1 = 50 (slider from -5 to 50, step 0.1)
φ_1 = 0 (slider from -5 to 5, step 0.1)
f(x) = a_1 * sin(ω_1 * x + φ_1)
a_2 = 1 (slider from -5 to 5, step 0.1)
ω_2 = 50 (slider from -5 to 50, step 0.1)
φ_2 = 0 (slider from -5 to 5, step 0.1)
g(x) = a_2 * sin(ω_2 * x + φ_2)
h(x) = f(x) + g(x) + 1.5 * (a_1 + a_2)