If two speakers play loud sounds it obviously requires energy, but if they are set up so the sound waves cancel, where does the energy go?
Off the top of my head I’d say that the energy (at that point) goes into cancelling each other out. The energy is still there and the sound waves still travel past the interference point.
It’s not that they disappear, think of two waves (of the same size) on the surface of a pond travelling toward each other and meeting. At the point where the low point of one wave and the high point of the other meet and overlap they will interfere and balance each other out.
I would say heat, either in the speaker cone or the air (or medium)
The sound waves don’t cancel everywhere. There will be regions of constructive interference, where the energy is more than the sum of the energies, as well as the destructive interference you’re talking about. For two speakers about the same distance away and playing at the same volume, the constructive interference bands will have four times the sound intensity of a single speaker at that volume and distance (that is, twice the sum of the two individual speakers’ intensities).
Are you talking about sound-cancellation that’s distant from a loudspeaker? If so, then in that case a pattern of standing waves or “interference pattern” is set up, and the sound energy is just shuffled around. For every “cancelled” spot, there is another spot nearby where the sound is twice as loud.
A similar thing can happen with laser light. If two coherent beams overlap, a pattern of dark and bright stripes will appear. There is light energy missing from the dark stripes. But that’s OK, since the bright stripes are much brighter than the sum of the two laser beams. As with sound, energy in interference patterns gets shuffled around.
On the other hand, if the cancellation takes place within 1/4 wavelength of the loudspeaker, then various things can happen. The loudspeaker can behave like a barrier, or the speaker can actually absorb energy (it can act “black” or “silver” as far as the incoming beam of sound is concerned.)