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.
Put the mouse pointer over an element to get the corresponding tooltip.

Activities

Check the trigonometry in the two expressions for the superposition of both oscillations.
Check that the angle between both phasors is φ = (ω₁ t + δ₁) -(ω₂ t + δ₂) = 2 ω t + 2 μ .
First, let us assume both frequencies are equal: ω₁ = ω₂. Can you write the full oscillation in the form of a single sinusoid? What will happen with the relative phase φ? Use the simulation to check your answers.
Now, let us assume both frequencies are different but very similar: ω₁ ≅ ω₂, so that ε is small. What will happen with the amplitude of the first term |2 A₂ cos(ε t + μ) |? What about the angle φ? Use the simulation to check your answers.
Can you now explain why beats are heard when two very similar frequencies are excited?
Which is the beat frequency, |ε| or |ω₁ - ω₂|? Why?

This is an English translation of the Basque original for a course on mechanics, oscillations and waves.
It requires Java 1.5 or newer and was created by Juan M. Aguirregabiria with Easy Java Simulations (Ejs) by Francisco Esquembre.
One may also use Dynamics I thank Wolfgang Christian and Francisco Esquembre for their help. Solver to explore this problem: see this simulation.