When sound is cancelled in an acoustic duct, where does the energy go? Let's say I have an acoustic duct setup and the contaminating sound is propagating through it. I can hear it at the end where I'm standing.
I now start the ANC algorithm and it cancels out the contaminant within the duct so that I can no longer hear it where I'm standing. Success!
But where did all that energy go? I read some posts here and it seems like there should be localized cancellation (near the error microphone), but not global. As far as I can tell, being at the output end of the duct, the cancellation on my side is complete. I can't audibly tell a difference on the other side (but I might not be a good measurement equipment). This is a metal pipe embedded in concrete.
Did the incoming sound reflect and travel back to its source? But then why didn't the cancelling wave reflect towards me? Or did it?
Or are they cancelling "globally" in my direction because they are travelling in the same direction and it is possible for them to always cancel from some point on? Seems like magic though :/
 A: Generally speaking, when you use one wave to get cancellation of another at some point, it means you get constructive interference somewhere else.  So the sound goes somewhere else.  There are ways to imagine idealized systems that produce cancellation everywhere, but I believe practical limitations (like the uncertainty principle, or non-infinite geometries) will always spoil efforts to get global cancellation, so you can only direct the sound elsewhere, like the nulls in an interference pattern.
A: Think of water waves in the sea. They are continuous dislocations of delta(m) in a sinusoidal dependence in space and time, with a specific kinetic  energy. Two water waves with the opposite direction and the correct phase will cancel each other. What happens to them, or more correctly to their energy? It turns into heat, i.e. vibrations of individual molecules incoherently. (Well, smaller waves that die out, actually, because it is hard to get exact cancellation on water waves. )Energy is always conserved
Sound is waves in air, it is sinusoidal dislocations in space and time of delta(m). The correct in phase opposing wave will flatten the sinusoidal dependence (sound) and turn it into  individual molecules kinetic energy, i.e. heat.
In your complicated apparatus, the answer is more complex, there will be reflections etc, but in the end, sound always degenerates to heat (molecular kinetic energy) , because of energy conservation, whether in the air itself , or in the solids it hits at the end.

Did the incoming sound reflect and travel back to its source? But then why didn't the cancelling wave reflect towards me? Or did it?

This is an interesting question related to laser light, which is a different story than a wave traveling in a medium.  have a look at this question where also I have answered , 
