The 'Circles of Doom'.....Open baffleless full range speakers.

U or H frame actually making transmission line long as frame is. First resonance will show at frequency which 1/4 of lambda (from here quarter wave line name came) is equal to length of frame (from opening to speaker baffle). Because of that H frame is much more convenient as for same separation has half length and resonance will occur at double frequency than U frame.
First resonance cannot be tamed effectively by stuffing because at same time output is taming as well so we on same level. Light stuffing can help with higher modes resonances.
Best option is to use H frame and cross it bellow first peak (what we actually always want, especially for bass) which is always bellow first resonance so we are on safe side
 
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Could a shallow (say 100mm) U or H frame get close to the same boost but allow crossover at a higher frequency - maybe 500hz?

No.

The issue with U- and H-frames that limits their upper range has to do with what is called a "tunnel resonance",

I beg to differ. An H frame for a 15" driver will be 400mm or so across. If that H frame were 100mm deep, each side is only 50mm deep minus timber thickness, so around 38mm deep on each side. You won't measure a tunnel or cavity resonance in an H baffle of those dimensions.

As I showed above, this would behave in a similar manner to a clamshell with around a 200mm spacing.
 
You call that an H-frame? Hardly! 10cm deep is essentially nothing. My H-frame subwoofer is about 1 meter deep (0.5m per side), and 0.6m wide, loaded with an 18" driver.

But you are correct in that, with your dimensions where width >> depth there will be little to no tunnel resonance and it will be at a very high frequency anyway.

I just answered the question as posted without adding any irrelevant criteria. A 10cm H frame behaves about the same as the clamshell arrangement that is the topic of discussion. I've made several 200mm to 300mm deep H frames and they work well without resonances.
 
Since you've gone into H frames, and not to derail this very interesting thread, could you please point me to good sources (threads, other) to learn about design of H, M (Linkwitz?) config for OB subs?

Through this great thread I learnt clamshell has the advantage of a compact design, easy to mechanically decouple the vibrations, reaches 6dB gain and could be xo somewhere up to 150Hz (I'm guessing, BM xo at 100Hz). But there is a limit to how large the phantom baffle can be made which is a function of driver diameter.

I'm exploring GR Research 12" servo subs designed for dipole (two drivers per sub), and clamshell is an option yet I'm thinking the smaller size of these drivers might run out of steam in the 20 to 50Hz or so region. Would like to explore other OB sub configurations to compare. Small form factor is always nice in the living room but within certain limits I could extend to a larger size if I can gain additional SPL without distortion and acceptable looks. GR Research shows theirs in an H frame that seems relatively shallow, but show no dimensions. Maybe Linkwitz arrangement in the LX521 is better than H? My subs would be separate from the main speaker. Off the bat it seems U baffle, being inherently asymmetrical for driver load, shouldn't be the best approach, but maybe I'm missing something. Looking forward to learn.

Thanks in advance!
 
Since you've gone into H frames, and not to derail this very interesting thread, could you please point me to good sources (threads, other) to learn about design of H, M (Linkwitz?) config for OB subs?

Through this great thread I learnt clamshell has the advantage of a compact design, easy to mechanically decouple the vibrations, reaches 6dB gain and could be xo somewhere up to 150Hz (I'm guessing, BM xo at 100Hz). But there is a limit to how large the phantom baffle can be made which is a function of driver diameter.

I'm exploring GR Research 12" servo subs designed for dipole (two drivers per sub), and clamshell is an option yet I'm thinking the smaller size of these drivers might run out of steam in the 20 to 50Hz or so region. Would like to explore other OB sub configurations to compare. Small form factor is always nice in the living room but within certain limits I could extend to a larger size if I can gain additional SPL without distortion and acceptable looks. GR Research shows theirs in an H frame that seems relatively shallow, but show no dimensions. Maybe Linkwitz arrangement in the LX521 is better than H? My subs would be separate from the main speaker. Off the bat it seems U baffle, being inherently asymmetrical for driver load, shouldn't be the best approach, but maybe I'm missing something. Looking forward to learn.

Thanks in advance!

I don't know of a "design formula for H-, U-frames" page that would be like a cook book. There are some pages on the theory that you can review:
Electro-acoustic models
Frontiers
DIY-dipole-1
http://www.quarter-wave.com/OBs/U_and_H_Frames.pdf

For both U- and H-frames, there are three contributions to the output:
1. The driver output
2. The front-to-back pathlength
3. Tunnel resonance

(1) The driver output can be modeled as a first approximation to be the free-air closed box response. In reality, when you enclose the air on one or both sides of the driver this increases the air load a bit and that will cause Fs to decrease, Qts to increase, and efficiency to decrease, all slightly. There is a table in the pdf linked above showing these effects.

(2) The front to back pathlength is what determines how much low frequency cancellation will occur, and also will determine where the dipole peak will occur. This is what tends to limit both the upper and lower ends of the usable frequency range of a U- or H-frame system. Smaller front-to-back pathlengths allow for a higher upper end, but at the cost of increased low frequency losses, and vice versa. One way to think about this is that the dipole response "shape" will stay the same, but will shift to higher or lower frequency depending on the pathlength involved.

(3) The the 'frame encloses space in a tunnel, and this space can undergo a quarter-wave resonance. The longer the tunnel is compared to its width the more prominent (higher Q) the resonance will be. The frequency of the resonance is determined by the depth of the tunnel, plus some fraction of the width (like a port end correction). As the tunnel becomes shorter and shorter for a given width the resonance frequency moves up and the Q decreases until for very short or no tunnel you no longer see it. I like to try to keep the width equal the depth to keep the Q to a reasonable level so that it is relatively easy to work with. This means that if you want a long pathlength to minimize dipole losses, the frame also needs to be rather wide and you might as well put the largest driver you can inside it, since the Sd*Xmax volume displacement is a limit on SPL and larger drivers can offer more of both of those.

Keeping the points above in mind, you can start to design the frame. It's not so easy to model it a priori (with the exception of perhaps HornResp), so building a flimsy mock up and making some measurements can help to guide you. I use measurement-based modeling to figure out what my mockup will do before building the real thing. You CAN make far-field measurements outdoors with everything (mic included) on the ground but you need a wide open space and a very calm day to do it.
 
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Speaking of the "clamshell" (compound dipole), let's compare that to an H-frame using SL's analysis.

SL's model (valid at low frequencies) finds that the output (SPL) for the compound dipole is proportional to two times the effective width of each driver that makes up the compound dipole (usually they are identical). Let's say we use two 15" drivers and they are mounted in 20"x20" baffles. SL's analysis says the output from the compound dipole will be like an H-frame that is 2*20" or 40" long! That is pretty large. But the advantage of the compound dipole is that it will (probably) not have the resonance of an H-frame (because there are no fully enclosed front and rear "tunnels"). This might make it usable to a higher frequency, and based on my quick measurements from a few days ago I would say that is indeed the case.

It would be interesting to compare an equivalent H-frame and compound dipole where D=2*d1 (following SL) via measurements to see the similarities and differences in LF SPL and higher frequency resonances.
 
design formula is always same, doesn't matter OB, U or H frame. For U and H should be calculated quarter wave resonance and output will be little bit different, as Charlie said, due to air mass load.

I found Kreskowsky papers most thorough analyses of OB. On his pages could be downloaded "A,B,C dipole" spreadsheet.
A, B, C, Dipole
On MJK pages you can find (for small fee) Mathcad spreadsheets for modeling several different box kinds and I found it quite useful for modeling OB's and TL's
 
It's not so easy to model it a priori (with the exception of perhaps HornResp)

The Hornresp Loudspeaker Wizard can simulate H and U frame loudspeaker designs in real time using theoretical models developed by 'bolserst'. The model predictions have been validated against measured results.
 

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I have had a few requests to have a listen to my speakers - people often have never heard open baffle speakers, and can only guess how they sound.

One of the best on-line listening experiences I have had was from this youtube video.
It is recorded binaurally with an $8500 special binaural mic.

When listening through a decent pair of headphones I really felt it gave a good flavour of the differences I have found in my own listening experiences. Something you can't glean from measurements alone. That 'open' feel.

This is a particularly nice video as it has some very highly rated traditional speakers from Revel (excellent measurements) and some horn based speakers, along with a pair of open baffle speakers.

Whatever speakers you prefer - doesn't matter really as this is very personal! - this excellent recording I feel highlights some differences to be heard in person.

Spatial Sapphires, X5s, Concept 500, Klipsch Heresy and Revel's! Binaural Shootout! - YouTube

***NOTE YOU HAVE TO LISTEN THROUGH GOOD HEADPHONES DUE TO THE NATURE OF THIS RECORDING!***
 
I used to have a nice lovely set of open backed HD580 Sennys but alas they are no more.
My Brainwavz HM5 are a very solid closed back and I can hear a difference clearly, most obviously between the revel and the spatial, 7:00-7:20 vs 32:10-32:30.

There's a transition that appear quite different between the models.
 
I have had a few requests to have a listen to my speakers - people often have never heard open baffle speakers, and can only guess how they sound.

One of the best on-line listening experiences I have had was from this youtube video.
It is recorded binaurally with an $8500 special binaural mic.

When listening through a decent pair of headphones I really felt it gave a good flavour of the differences I have found in my own listening experiences. Something you can't glean from measurements alone. That 'open' feel.

This is a particularly nice video as it has some very highly rated traditional speakers from Revel (excellent measurements) and some horn based speakers, along with a pair of open baffle speakers.

Whatever speakers you prefer - doesn't matter really as this is very personal! - this excellent recording I feel highlights some differences to be heard in person.

Spatial Sapphires, X5s, Concept 500, Klipsch Heresy and Revel's! Binaural Shootout! - YouTube

***NOTE YOU HAVE TO LISTEN THROUGH GOOD HEADPHONES DUE TO THE NATURE OF THIS RECORDING!***

Can watch my video if did't see. Not a comparison, but maybe some day.. Binaural recording of loudspeakers in room - YouTube