Beats

Let us asume that we have at the same point two harmonic oscillations (say two pure sounds):
x = A₁ cos(ω₁ t + δ₁) + A₂ cos(ω₂ t + δ₂) = 2 A₂ cos(ε t + μ) cos(ω t + ξ) + (A₁ - A₂) cos(ω₁ t + δ₁),
where
ε ≡ (ω₁ - ω₂)/2,
μ ≡ (δ₁ - δ₂)/2,
ω ≡ (ω₁ + ω₂)/2,
ξ ≡ (δ₁ + δ₂)/2.
Obviously an equivalent expression can be obtained by exchanging the indices 1 and 2 in the last expression.

In the animation the units of time and length and the time origin are chosen so that A₁ = 1 , ω₁ = 1 and δ₁ = 0.
You can select below A2/A1 = A₂, ω2/ω1 = ω₂ and δ = δ₂.
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