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We know that active noise cancelling headphones work by playing a signal in your ear which is 180º out of phase with the ambient noise. If we compare the two waves, the peak in one wave is completely inverted as a valley in the other. Together, they cancel each other out and we don't hear anything.

However, when the inverse wave is superimposed on the normal wave, we see that they are mirrors of each other. If one has a peak at $2$ Pa, the other has a valley at $-2$ Pa. But can we say that this is exactly the same as a flat line with no vibrations? Because unlike the flat line which is one wave, noise cancellation works by having two waves. Of course, the consolidation of those two waves yields a flat line, but isn't it different in reality because they are two separate waves? In other words, one wave doesn't change the other, only the sum is zero.

Maybe it's something like when you put on noise cancellation headphones, you feel pressure inside your ears despite hearing nothing. The waves or vibrations can still be felt even if the consolidated sound is nothing because the waves are still there.

Can you help me with understanding this please? Thanks!

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  • $\begingroup$ True, one wave doesn't change the other. But the ear responds to the sum. And in this case the sum is zero. You don't feel additional pressure in your ears; the displacement of the molecules is zero. This can happen only approximately, and only over a limited spatial extent. Out side of that spatial region the waves will regain their distinguishability. $\endgroup$
    – garyp
    May 5, 2021 at 15:18

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As is usually the case, there is a model notion of destructive interference and that is that you simply superimpose waves and if the calculation results in zero, then there is nothing left of the wave at the calculated position.

This is of course a simplification. Both waves have an energy content and the energy cannot simply disappear. For sound waves it is the case that the colliding air molecules are deflected sideways or in rare cases collide head-on. In the first case, the energy is dispersed and deflected sideways and in both cases the internal energy of the molecular subatomic particles increases.

The waves or vibrations can still be felt even if the consolidated sound is nothing because the waves are still there.

This is not the case. Both waves, the one from the environment and the one from the headphones, disperse in chaotic movements of the air molecules.

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