(A) "This" works for a wavelength < L.
> and above a certain frequency all you have is passive muffling
(B) "This" does not work for a frequency > F. <=> (B') "This" works for a frequency < F.
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Don't A and B' (or B) contradict each other? Because an upper limit for the wavelength equates to a lower limit for the frequency - not an upper limit.
What is your background that justifies your very confident demeanour?
My background is audio processing is one of my hobbies, I have torn down and repaired noise canceling headphones and understand how they work (and have previously talked about this on Reddit and been corroborated by an actual engineer at a company designing such headphones), I have implemented laser galvanometer closed loop feedback systems and worked with them (which is both a similar control system problem, and a different part of OP's research paper on the light path), I have attempted room and speaker response cancelation (and wrote my own DSP firmware for implementing biquad filters, which I use for my DIY living room rig), making an open source speaker calibration system is in my future project list and I've recently done a lot of thinking about this... and, most importantly, since this is neither my professional nor academic field I have no reason to overstate my claims or mislead about the importance of my work, unlike way too many academic researchers (see also: that guy who I recently called out for being a paper mill about side channel leakage methods after he successfully spun a false media story about generating Wi-Fi signals from RAM using a misleading paper title; in that case infosec is in fact one of my professional fields).
I don't like to brag or anything, I'm just a curious guy who likes tinkering with (lots of) things... but if you insist on asking "what are your credentials", well, there you go.
I'm a silence junkie living in world getting noisier with every day it seems - so this subject is of high personal importance to me. That's why I'm interested in getting an idea about where to place your assessment.
But room-scale or free-space noise cancellation only works when either 1) your noise source is extremely predictable, or 2) your noise source is low-dimensional. It's just never going to work for the general case unless you can somehow sense the entire soundfield. The entire sum of what waves are traveling in which directions. Not just a few microphone inputs. It's going from 1D data to 3D data, the difference between a light sensor and a CAT scanner. And it needs to work at 20kHz response.
I guess hypothetically you could do something like wrap a room in microphones and compute the sound coming in in all directions, then with a very good adaptive model be able to cancel noise from outside within it. But if you're going to draw a boundary and wrap it in a massive array of microphones... Aren't you better off just adding insulation material? :-)
Edit: just as a baseline, I very crudely tested my Bose QC20 earbuds (which aren't exactly bleeding edge tech) and I think they get about 10dB active cancellation at 1kHz, which goes down to nothing at 2kHz. Human voice fundamental frequencies are typically 85-180 Hz for adult cis men and 165 - 255 Hz for adult cis women, so this level of ANC does get rid of the fundamental and a few harmonics. I might test a friend's AirPods Pro later and see if they're better. 1kHz has a wavelength of ~34cm, so I think there is room for improvement here in the earbuds case. I'm not experienced enough in this field to have a good feel for the numbers beyond order of magnitude estimates, but I think going up to 4kHz might be doable, given that for earbuds we're talking about cavities in the ~5cm range. Might need more mics/drivers and more miniaturization to pull it off, not sure.
For those wondering, the test methodology was to play a tone, turn on cancellation (which has a delay), then turn off cancellation (which is instant) simultaneously with decreasing the amplitude of the tone, and trying to match the perceived loudness between both cases.
At my current place I've pretty much concluded that anything but deep bass just doesn't transfer. Even at 5 in the morning, I can listen to stuff at a reasonable listening volume without disturbing my neighbor, as long as I'm not pumping out bass below 80Hz or so. And the neighbor recently got a dog and I haven't heard her bark even once when she's home :)
Well, let's do the math here. I can't find the data sheet for the weight of an electret microphone cartridge, but let's assume 4 grams. A thousand microphones -- assuming we want 100 the length of the airplane and 10 around -- is around 4 kilos. Add another 2 kilos for computation, and a few more for wiring, and you've added the weight of a piece of luggage.
Now, let's say you want to add 2 inches of mineral wool sound insulation. I'm assuming 100 meters x 10 meter circumference (I have no idea, but the numbers scale the same). You've added (quite literally) half a metric ton to your airplane. And made it 2" thicker, either reducing cabin space or increasing drag.
Plus, sound insulation does very little for low frequencies, which is the majority of airplane noise.
This is all assuming this whole idea works, which it probably won't, because it's not just about the microphones on the fuselage but also how sound is transmitted inside the plane and other noise sources.
1) Assuming system-level model
10,000 microphones * 8 kilosamples per second = 800 megasamples per second.
NVidia Tesla does 100 teraflops. I get about 125,000 FLOPS per sample. That feels adequate to me!
2) One filter per microphone per passenger. We need to divide by 250.
500 FLOPS per sample per passenger. That's more than enough for a very fancy IIR.
Of the two, I think #1 is more likely to work than #2, precisely because you want a coherent model. If you want to adjust your model for someone walking down the aisle (or any kind of system ID), that's a lot easier with 125,000 FLOPS per sample than 500 FLOPS per sample.
Again assuming this all works, which is a massive if.
Look, this isn't practical.
The answer to that might be right now (we couldn't do it before, and we probably can today) or it might be in a decade. But electronics will keep falling in price. 10,000 microphones * $5 per microphone+electronics = $50k.
The advantages of cancelling an entire wavefront go well beyond passenger comfort too. Industrial noise cancelling systems are more about equipment life than about employee comfort. I had laptop screws unscrew on airplanes before, due to vibration. If planes need less maintenance as a result of active vibration reduction throughout the airplane, you'll make up that $50k virtually overnight.
As a footnote, the crazy part here isn't the 10,000 microphones, but the speaker-at-every-seat part. You'd almost certainly want both the microphones and speakers in the skin of the airplane. But that's a story for another day.
A long wavelength implies that the phase of the reference input and feedback input are similar, which is what allows you to just subtract them.
Your processing delay, the distance between the speaker and the microphone, and the speed of sound are all fixed, so as the wavelength gets shorter, the relative phase difference between the two signals you're trying to subtract increases. At a certain point, you run out of phase margin and can no longer just subtract them anymore.
If you were designing an ANR system, you'd probably want to put a low pass filter somewhere around this point to prevent your feedback system from oscillating.