John Carlos Baez on Nostr: Without any electronic equipment, piano tuners can tell if two strings are vibrating ...

Without any electronic equipment, piano tuners can tell if two strings are vibrating at almost but not quite the same frequency. They do it by listening for 'beats' - pulsations in loudness.

How does this work? If you add two sine waves of slightly different frequencies, say sin(ωt) and sin(ω't), they will add up and be loud for a while, but then drift out of synch and cancel out for a while. Then they'll drift back into synch and get loud again, etc.

There's even a formula for this:

sin(ωt) + sin(ω't) = 2 sin((ω+ω')t/2) cos((ω-ω')t/2)

We get a sine wave whose frequency is the average (ω+ω')/2, slowly pulsing because it's multiplied by a cosine wave with the low frequency (ω-ω')/2.

I got the animated gif from here:

https://ophysics.com/waves10.html

If you go there and choose ω = 500 hertz and ω = 510 hertz, you can hear nice pulsations at 10 hertz: that is, ten times a second.