Noise is noise, and if you cancel it out it's gone. Sound waves obey the superposition principle, which means that if I take 1 psi and superimpose -1 psi, I get 1-1=0 psi.

That's true, of course. But it's much harder to actively cancel higher frequencies. This is why noise cancelling works brilliantly on an airplane (relatively low frequency background noise) but it does almost nothing to filter out the sounds of conversations around you.

I always thought it was because noise cancelling can only block repeated waves (like the sound of an engine whirring) but conversation is constant changing sound so it can’t be blocked.

Much easier to match pace with a wave with long wavelength than one with short

Yes, but the question was essentially, "does the sound it cancels out and you don't hear still hurt you". If you still hear it, you're answering something else.

High frequencies are passively attenuated by the foam inside the earcups. It's the mid frequencies (speech, etc.) that survive both active and passive noise cancellation. Still, a high end pair of ANC headphones does a good job on speech/tv if you have some soft music playing through them.

In addition to canceling sound, the headphones actually block a lot of it to begin with, so they only have to cancel what's left. On the gun range, it's pretty popular to use quite the opposite of noise canceling headphones. They actually amplify sounds at normal human speech levels, then cut off anything above a certain amount. They make no attempt to cancel the gunshot sounds, they just don't reply them inside the headphones. And the blockage from just the physical headphones is enough to turn a deafening gunshot into something fairly comfortable to listen to for quite a long time.

I don't know if it truly protects them from damage overall. That would take science. See "complicated" below :)

But the basic answer to your question is, no, it doesn't double the sound energy.

To the extent the headphones are successful at creating an inverse sound to cancel incoming sound, that will reduce the sound energy reaching the inner ears. The sound and its "mirror" don't add up to increase energy, rather, the energy reaching the ears is reduced where waves cancel.

(The same thing happens in, say, optical interference patterns, and coastal waves. All the energy which seems to mysteriously disappear where waves cancel is, in fact, accounted for by increasing at the places the same waves interfere contructively. In the case of headphones, it's going to be rather more subtly accounted for either elsewhere in the spatial pattern around and on the speaker, and/or by the speaker acting as a net energy absorber.)

Active noise-cancellers also have a passive component. The big squishy cups on mine (Bose QC-35). These of course reduce incoming sound energy too.

But it's more complicated than that. Ears have a sort of physical volume control of their own (using tiny muscles), which reacts to the sound level. As far as I'm aware, damage is more likely deeper in the ear, past that stage.

So anything which alters the ear's coarse volume response might produce a more complicated damage response than simply measuring incoming levels would suggest.

I apply that theory: I sometimes listen to music on the headphones, in high volume environments. This seems to mask out the environmental noise better, and my theory is this isn't just psychoacoustic masking, it's causing my ears to block out more sound physically, helping to protect them physically. When doing this I listen to music which has a consistent sound level, rather than, say, a podcast or something subtle like classical.

I've read that noise-cancellers produce a hiss which annoys some people. I can't say I've noticed much volume to the hiss, and I used to assume it was residual signal from the signal processing. But it may be that's intended to interact with the ear muscles too.

won't that depend on how well synced the waves are? if you get the wrong phase, you're doubling the amplitude

If you get the wrong phase, you can hear it - it's louder.

So, if it's quieter, you're ok.

If the device makes the outside world louder, then yes, you might want to turn it off. (This actually happens on my headphones when taking a call.)

A good fraction of noise canceling for some over-ear headphones is the cups that go over your ear. Having that barrier can be a significant sound damper on its own.

Also, whatshisface is exactly correct.