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This is related to this question on Aviation.SE about noise cancelling headsets.

We know that soundwaves can be superimposed on each other. Therefore, by taking an existing sound wave and producing a second wave which is exactly its inverse, the two will sum up to zero. This is how active noise cancelling headset works - by "listening" to the exterior sound with a microphone, then reverting it in its speaker.

The problem is, where does the energy go? The noise sound is an energy source. Generating an inverted sound on the drums of the speaker is also an energy source (from the electricity which powers the speaker). So the total energy output cannot be zero. But we hear silence! Where does that energy go?

EDIT: I'm more interested in a theoretical answer, assuming a perfect world with no energy loss in converters / out-of-sync waves. Although an answer giving real world converter efficiencies is also useful.

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kevin
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The speaker mimics a black body radiator/absorber. If the speaker were 100% efficient and the impedance between the diaphragm and air perfectly matched, then no power need be sent to the speaker and all the sound power would return to the circuit driving the speakers, where its up to the design of the circuit what to do with it (for example, some class D drivers would return the power to the supply, while typically linear circuits would waste the power in a resistor). However, not only are speakers intrinsically inefficient, the diaphragm is horribly matched to the air. So to make the speaker behave as if its a perfect absorber, power must be sent to the speaker, where most is wasted as heat in the coils and magnet. The vast majority of original air power that was to be cancelled is also wasted as heat in the speaker, but a tiny fraction makes its way back to the driving circuit, and then its up to the circuit. But because the efficiency is far below 50%, far more power is spent than regained.

Digiproc
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  • I think you can neglect the speaker. assume you have a wave coming to you, and you have a wave to completely align it to the incoming noise. then them sum of them become zero. the energies will become inaccessible. Where the energy will go? – mohammadsdtmnd Jan 26 '23 at 21:50
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The sum will be zero (in the ideal case) on the side of the listener only. What happens in every other directions is a different story.

Sound waves will be reflected by the casing of the headphones in many directions. The energy is ultimately dissipated as heat as the wave travels through the viscous air.

Some is lost in friction amongst the mechanical components and dissipation in the electronics, although the majority of that comes from whatever source powers the headphones.

Cape Code
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Sound waves are by their nature longitudinal so they can be visualised as invisible springs in the gaps of each individual air molecule that carries the energy signature along its direction of travel and ultimately at the end of its journey through the passage of air is acoustically converted into an electric signal by the inner ear that is spontaneously experienced as a sonic sensation by somebody in the way. If you collided two opposing waves in one of those large slinky springs you get they would merge together and ultimately annihilate each other, but wave energy wouldn't just disappear but it would be rendered into micro vibrations that would dissipate as sound waves and heat - as the energy "had to go somewhere" as it can't just "vanish" and likewise the sound wave that for example is cancelled when you put on background cancellation headphones ultimately gets released as a tiny increase in heat energy into the environment.

Abdullah Aslam
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