I believe that should read "has the same frequency but is out of phase by HALF A wave length".

> active noise cancelling has the same effect as putting earplugs in due to wave mechanics.

Simply from experience, I would disagree with this. Every time I have ever enabled active noise cancelling, I have experienced an increase in eardrum-pressure, volume, or both (both with and without music playing).

I understand that wave mechanics would theoretically work as you state, but in practice, does not.