Are noise cancelling headsets safe?
Naively, it seems that this would pump more audio power into the ear. Granted, it will be in opposition to the original noise, but it seems it would be very tricky to get the cancellation exactly in phase.


If the headphones succeed in reducing the noise you hear, then they will also have succeeded in reducing the power delivered to your eardrum (at the relevant frequencies) -- because those are one and the same.

It is conceptually and mathematically convenient to imagine that the original noise and the canceling noise both reach your eardrums and interfere there -- but if you actually work out how energy moves in the combined situation, what actually happens is that the cancellation speaker sets things up such that much of the sound energy will get reflected away from your ear canal instead of entering it.

The core mathematical point is that the air displacements and minute pressure differences that make up the sound do indeed (to a good approximation) behave linearly -- so you can take a description of the noise and a description of the cancelling noise separately and simply add the displacements at the location of your eardrum.  But power goes as the square as the displacement, so it does not add linearly. You can't take the power transport of one sound along and add it to the power of another sound to get the power of the combination (except averaged over time if the sounds are not coherent in frequency, which is specifically not the case for noise-cancelling headphones).