BEATS
y₁+y₂ = 2A·cos(πΔf·t)·cos(2πf̄t) — the throb piano tuners count.
Use the simulation above to change the variables and play through the guided stages. The explanation below describes the default starting values; the simulation updates its explanation as you experiment.
Setup
Two tuning forks, almost — but not quite — in agreement. Struck together they make a sound neither makes alone: a tone that swells and dies, swells and dies. The disagreement is audible, and countable.
Superpose
Just add them. The two waves start in step and drift apart at the difference frequency — constructive, destructive, constructive again. Neither parent wave ever changes; only their agreement does, and the sum breathes with it.
Solve
One trig identity turns the sum into a product: a fast carrier at the average frequency, multiplied by a slow envelope at half the difference. The ear hears loudness peaks twice per envelope cycle — so beats arrive at exactly |f₁−f₂|.
Listen
Slow motion, honestly labeled. The dashed envelope is not itself a wave — it is the moving boundary of where the two waves agree. Watch the violet sum fill it and abandon it, on schedule, forever.
Audit
The audit: sum form and product form agree at every sample to machine precision — the identity is exact, and the identity is the physics. Count the swells and you have measured a frequency difference by ear; piano tuners, radio mixers, and droning twin engines all run on this page.