On the advice of a couple of friends I recently invested in some noise cancelling headphones, and I love them. They are amazing; engine noise on aeroplanes is almost completely silenced. They were so impressive that I wanted to learn more about how they work.

Turns out there are two types of noise reduction, active and passive.

Passive noise reduction comes from the headphones blocking sounds before they reach the listener’s ear, which is how earmuffs work. I’m not going to spend any more time on this – it’s just not as interesting.

Active noise reduction, or noise cancellation, is extremely cool, and is what the rest of this post is all about.

To understand how active noise reduction works, we first need to know a bit about noise. Noise, or sound, is actually a wave in the air. Waves can do all sorts of interesting things, especially when more than one is present. Two waves can either add together to make a stronger signal, that is a louder noise, or they can cancel each other making a quieter noise.

Waves have maxima, the peaks of a wave, and minima, the troughs of a wave. There are two cases of wave-wave interactions we need to explore.

If the peaks (and troughs) of both waves line up then the result will be a bigger wave, meaning the noise gets louder. The waves have interfered “constructively” with each other.

If the peak of one wave lines up with the trough of another wave that is about the same size, then the result will be nearly zero. The wave have interfered “destructively”. If the fact that two waves can be added together and the result is smaller than either of them seems impossible, think of it as adding a positive number and a negative number to get zero.

Active noise cancellation is all about producing sound waves that will interfere destructively with the background noise. The following video gives a crude example of active noise cancellation. There is an incoming noise and the active noise cancelling system produces a wave that destructively interferes with it. The bottom plot shows the amplitude of the incoming noise, the active noise cancelling signal and their combination at the listener’s ear. You can see that when the incoming noise is positive the noise cancelling signal is negative. This means that the resulting signal is much smaller than either the incoming noise or the noise cancelling signal.

If you’d like to explore this some more, here is a webpage that has a neat little app to add two waves together and see what the result is.

One thing that I didn’t mention above, is that sound waves are a particular type of wave called “longitudinal”. It makes no difference to the way they interact, but should be mentioned for completeness. The following video neatly illustrates the difference between longitudinal waves (e.g. sound) and transverse waves (e.g. waves on the surface of water).

The code I used to create the video for this post can be found here in a BitBucket repository.