Re this wiki on noise cancellation.
If I am in 10 x 10 room with standard timber construction and sheetrock walls and I shout something as loud as I can as a adult male, can active noise cancellation entirely make my shout disappear so no one could hear me outside the room?
My noise cancellation headphones turns sound which is really loud into sound that is kinda loud.
Based on that, I’d say no.
Outside the room, for a limited definition of “disappear”, with everything working right, quite possibly.
In your favour you have a few things.
A limited and well understood geometry of the room.
A single source of sound.
Knowing a-priori that we have a single source of sound (even if it moves) means a lot. We know that we can use a reasonably small (where small means say a dozen) microphones to sample the sound field and reconstruct the source of the sound in 4D. We need to do this to significant accuracy.
The range of frequencies we need to worry about is pretty broad. But at least it is no as broad as the range we can hear. We might reasonably limit ourselves to the range 100-10,000 Hz. That is arguably overly broad, but not ridiculous. We have a range of wavelengths here that is roughly from 3 metres long to 3cm long. Both ends are going to cause issues. At the low end we are seeing wavelengths that will energise room modes. Getting rid of that energy is not trivial. At the high frequency end, for successful cancellation we need to be measuring and cancelling energy to an accuracy of a fraction of the wavelength. That means millimetres.
Where it gets nasty is the moment the sound reaches the walls of the room, as it will now reflect. And the room will fill with a diffuse field of reflected sound, and any chance we have of predicting what that field is, in order to cancel it out, is gone.
So, we really need to somehow destroy the emitted sound before, or as, it hits the walls. This is actually just on the edge of doable. The trick is not to think so much about cancellation so much as swallowing the sound. We set the room up lined with wide-band transducers (ie small wide response speaker drivers) where each transducer is driven by its own amplifier and its own signal calculated by the noise cancellation system you could get the system to calculate exactly the sound pressure expected at each location on all the walls, and to move in such a manner that the wave is met by a transducer diaphragm that is moving exactly the same. Assuming the transducer has enough range of travel, it can match the motion of the air and effectively eat the pressure wave. In effect this is an active anechoic chamber.
But you can see that things are very carefully constructed. Breaking any one of the assumptions makes things go quickly wrong.
You are going to need a LOT of speakers and LOT of amplifiers and heck of a lot of compute.
Francis Vaughan explains things very well and thoroughly.
For a simpler answer that you can work out yourself, just remember what sound is: it is your perception of mechanical air pressure shifts. Cancelling a sound, consists of creating an exact opposite image of the air pressure fluctuations that are making the sound. Visualize the sound waves you are dealing with, and then imagine what it would take to balance them all away. Complicated and really hard to do? Yup, you got it.
I used to work for Bose and met Doctor Bose and his former graduate students that invented the concept of noise cancelling. The short answer is, no, you can’t at least with known technology. It takes processing time to figure out how to cancel out the noise signal and it isn’t instantaneous. The headphones or noise cancelling speakers would have to intercept a loud noise microseconds before the sound hit your ears, figure out how to cancel it out and then generate a reverse sound wave. That simply doesn’t work with known technology.
What you can do is process and cancel out ambient noise like an airplane engine or air conditioner. That works fairly well but you can hear the headphones adjust to the active cancelling over a few seconds as they learn to produce the cancellation waves.
However, what you describe is not really a problem. Most people want the ambient noise gone while still being able to hear people talking or sudden sounds like an alarm.
It is theoretically possible to cancel out almost any sound but it requires a carefully controlled environment and special equipment. In short, you would get near silence but you wouldn’t be able to move around to maintain the effect or hear anything if it was perfect. Most people do not want that and there is limited use for it.
Notice also that most noise cancelling is about counteracting the sound field as received at a particular point.
The OP is talking about counteracting the sound field at its origin so it’s imperceptible everywhere.
Those are very, very different things.
As a matter of basic physics, to achieve the OP’s goal of cancellation everywhere we’d have to have our anti-sound transmitter be in the same spot as the source and be close to the same size. To within a distance of a small fraction of a wavelength. At 10,000 Hz one wavelength is very roughly 1 inch. So we need size & location accuracy of 1/4" or maybe 1/10th inch to get good cancellation. Might be doable for noise from a point source loudspeaker. Not gonna happen for a human voice from a human mouth.
Let’s try a simpler goal: leave the shout heard inside the room but not outside. In principal that becomes sorta-doable with ANR-type tech.
Line the interior surface of the wall with microspcopic microphones and the exterior of the wall with microscopic speakers. Make the wall as thick as necessary to provide enough processing time to compute the necessary outputs for the millions of micro-speakers given the input of the millions of data points coming from the micro-mikes.
By the time you make the wall that thick, you’d do as well to fill it with ordinary sound-absorbing insulation. It’s certainly be cheaper than that much compute resources.
ANC (especially in a closed sound field) works best with low frequency (long wavelength) sound, and steady state low-level background noise. with higher frequency (shorter wavelength) sounds, you simply moving your head an inch or so can turn noise cancellation into noise enhancement.
It is impossible to destructively interfere with a wave everywhere: a wave has a certain amount of energy and you can’t make that energy just disappear in fact as the cancelling wave also has energy the total sound energy will be greater.
I’ll admit that I don’t know much about the specifics, but what you can do is use destructive interference so a sound wave is mostly cancelled over some region and then make sure outside of the region the sound energy is mostly converted into another form of energy (e.g. absorbed by some barrier and turned into heat energy).
Basically, sound is a series of pressure waves that originate from whatever makes the sound. It would be incredibly difficult to create a corresponding opposite wave to cancel out the sound everywhere, other than to have the cancelling wave originate from the same point. the alternative is simulate a wave front in all directions coming from a sphere surrounding the source - but if the source sphere is not actually a sphere centered on the source, you will hear something in spots.