sounds waves go positive and negative. think of a graph of a sine wave (google it to see a visual). half the time, it's above the line (positive), and half the time it's below the line (negative). that represents the waves moving through the air. that's all sound is. like when you push and pull really quickly on a slinky, you can see the waves ripple down the slinky. sound is like that.

sound waves have what's called a phase. remember the positive and negative thing above? if you reversed the positives and negatives, did you know you still get the same sound? the sound will just be out of phase, but to the person listening, there's no difference. a really really good way to actually hear noise cancelling for yourself is to find a living room setup that has big floor standing speakers. listen to a song with a lot of bass. then on ONE speaker, reverse the wires. meaning connect the + to -, and the - to +. listen to the same song. you'll notice all the bass is nearly gone. that's because waves that have the opposite phase cancel each other out.

think of the times the waves is above the line in the graph as +1. at the point it's crossing the line, it's 0. when it's below the line, it's -1. so one single period, meaning one full section of the wave before it starts repeating over and over, would look like this:

..--.._

or by the numbers we mentioned, like this:

0, +1, 0, -1, 0

now the wave with the opposite phase:

.._..--..

and the numbers for it:

0, -1, 0, +1, 0

now what happens if we add the two waves together?

(0+0), (+1-1), (0+0), (-1+1), (0+0)

0, 0, 0, 0, 0

if we graph that, we just get a flat line. the two waves completely cancel each other out.

in electrical terms, the positive and negative is just voltage, and it's very easy to reverse that with electronics. noise cancelling headphones just have microphones at the edges of your headphones, reverse the wave, and add it to the sound you hear. so if you're listening to music, it'll add the reversed phase wave to your music. you don't hear anything because its values exactly cancel out the original sound. also, this only works because electricity travels nearly at the speed of light, and that's way way way way faster than the speed of sound, so even through the microphone and your ear are only an inch or two apart, it has plenty of time to record, reverse, and add the reversed signal to your audio output. it's really neat stuff