A key thing about sound waves is how they combine. If you take two identical sounds and play them together, the peaks of the waves align and amplify to make a very loud sound. This is called constructive interference.
But you can also do a sort of reverse. If you have a wave with peaks and troughs at certain points, you can invert the wave by taking every peak into a trough and every trough into a peak. If you play these two sounds together, the peaks of the original sound line up with the troughs of the inverted wave. These cancel out giving you destructive interference.
The neat thing is that you can take a combined signal of multiple sound waves, and further combine it with just the inverted wave of the original sound coming from the microphone and that will perfectly cancel out the microhppne sound without cancelling any of the normal audio you do want to keep.
This is also how noise cancelling headphones work. They sample the noise coming from outside the headset, invert it and add it to the headset sound. This played sound combines with the outside sounds and causes the same destructive interference, cancelling out the outside sounds so you then only hear the sound from the headset. It’s not an *exact* match to what waves actually make it through into your ear so it doesn’t completely cancel all background sound, but it’s close enough to significantly dampen the external sounds. Combined with regular sound blocking and sound proofing you get a high quality noise reduction.
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