Yes it's possible that the design could be less effiicient than one with rigid back, because when one is pushing in, the opposite side is pushing against it. But as long as it's more than 50% as efficient as a rigid back, there should be a net benefit from having twice the area resonating.
The BBC does not agree with you!

They have an extensive library of acoustic research, both theoretical and practical, and many of their papers highlight issues that they ran into when building tuned devices and the box wasn't rigid enough. The devices did not perform as designed, and had to be modified with thicker, stronger, more rigid boxes. And we are not talking about replacing a rubber membrane with plywood! Rather, things like having to replace 12mm plywood with 16mm plywood, because the 12mm plywood back was flexing too much, and de-tuning the box. I'm not sure if you have built many small boxes with 12mm plywood, but it's not something that one would think of as "flexing too much".... yet it did. Enough to mess up the tuning. When they replaced that with the thicker stuff, the problems went away.
Check out their library for all the details.
But as long as it's more than 50% as efficient as a rigid back,
If the rigid-back version has a coefficient of absorption of, for example, 0.86 at the resonant frequency, and yours has only 0.43 then it won't be a lot of use.
Which of these two do you think would make a good membrane trap, designed to resolve a modal issue at 80 Hz?
Limp-trap-examples.jpg
I know which one I'd prefer!
Notice the Q in each case....
there should be a net benefit from having twice the area resonating.
.... nope! You'd have twice the area, yes, but that area is now NOT all sharply tuned to the same frequency. Only a part of the area is actually tuned to the frequency you want, since the Q is low.
And even if you do get 0.4 instead of 0.8 with twice the area, all that you have accomplished is getting the exact same effect with a device that takes up a lot more space inside your room.
My understanding is that a membrane absorber is already low Q by design
Really? So the green curve in the above diagram is low Q? Ummmm..... I guess I must not be understanding "Q" very well....
As I understand it, the depth doesn't determine the centre of the frequency, but instead the LOW CUTOFF, and that it goes up from there.
Then your understanding is wrong. Not mine! The equation defines the resonant frequency, which is usually a bell-shaped curve with the center of the bell at the resonant frequency.
There is no such thing as a "low cutoff" with a resonant trap. That term is normally applied to numeric-based diffusers (Schroeder, skylines, QRD, PRD, BAD, etc), where there really is a fairly sharp fall off at a certain point, but even then the effect does not "go up from there". It only goes up to the high cut-off, where the same issue happens again. Diffusion is also usually a bell-shaped curve, albeit broader, tilted, and not so well defined. The ONLY device that has a low frequency cut-off of sorts, and then "goes up from there" is a purely porous absorber. Tuned devices have bell-shaped curves, with one peak absorption frequency, and the absorption falling off on each side of that (higher and lower). some devices might have more than one peak, but each peak behaves roughly the same.
So a membrane absorber might do from 60Hz up to a few hundred Hz -
I'm not aware of any such device. Maybe you could show the design for one, and the equations that describe it's operation? It certainly wold not be a membrane trap. Normal membrane traps are tightly tuned, high Q.
higher if there is 703 in front of it but not touching.
Huh? What "703 in front of it"? I have no idea what you are trying to describe, but it is not like any membrane trap that is commonly used! Membrane traps have insulation inside, not in front of them, and 703 might not be such a god choice, depending on what the trap is tuned to.
If I have that part correct, then I care not if, as it's vibrating, the plywood is moving my cutoff from 60Hz to 59 or 61.
There is no such cut-off, so your point is moot.
Actually, I would suspect it would be fractions of a cycle, but either way, who cares...if it's working?
Do the math, and see. Or check the BBC papers, since they already did the math, then built them, the re-built them when they did not work. IT will save you the trouble if you read up on what they did, and the results they got, and how they fixed that.
Irrelevant to this specific discussion because these are realities whether I'm building a hole panel Helmholtz, a slot resonator, a rigid backed diaphragmatic, or this double walled diaphragmatic absorber.
Sorry, but it is ENTIRELY relevant! If you don't see why, then I'm not sure that I can help you much.
None of them can be practically adjusted after they are cut, built, and glued
Yes they can.... Sorry to disagree with you so much, but you are just plan wrong. All of them can be adjusted.
with the exception of the Helmholtz, in which one can drill more holes.
Nope! That would not necessarily work. It might, but it might not. For example, what if drilling more holes would increase the percentage open area such that it no longer acts as series of individually tuned devices, and instead starts acting as a low-Q broad-band device? And you are talking about the specific case of a perforated panel device, not the general case of Helmholtz resonators. Many Helmholtz resonators can be tuned rather easily, by changing the neck length, for example.
The whole reason we are attacking this problem is because we have waves that are 15-20' long. Really, even if we turned one side away, 12" difference on a 20' wave is going to be close enough to the same part of the cycle that each will be compressing or rarefacting at the same time.
What makes you think you could build a membrane trap tuned to 56 Hz that would be only 12" deep? How would you go about that? What materials would you use, and what would their physical properties be? How big would it be, and how far out into the room would it protrude?
This would suggest that some 703 or similar in front would be of benefit - not touching the membrane of course.
Once again, membrane traps do not have 703 in front of them. And even if they did, it would not change the way the trap itself operates. It would merely "filter" the sound in some fashion, so the trap would not "see" the full sound field in the room.
I have room in my studio for 2' panels protruding from corners and ceilings
That's great! You have a large room! Which means that modal issues are likely not such a big problem anyway, and can easily be dealt with. How big is the room, and what is the Schroeder frequency for it at present?
Also, if you have 2 feet to spare, I would go for hanger traps, for sure. Tom Hidley style (or John Sayers style...). Proven to work down to very low frequencies, with good efficiency. With 2 feet to spare, you have enough space to do that, for sure. I have done several rooms with hangers in a bit less than that (around 20"), with good success.
I'm trying to determine if, theoretically, there is a net benefit, or a net detraction to the concept of a double walled absorber to tackle low end problems.
It's an interesting experiment, certainly, and I'd love to see the results, but I'm very unconvinced about the practicality, even if you can figure out how to keep the Q high.
I suppose the way to do this would be to build 1/2 dozen panels, do some REW tests, then screw a rigid back on one side and test again to see which works better.
If you have the time, and the money, and the inclination to do that, that would be fantastic! I'd really love to see the results.
- Stuart -