Since I went to the trouble to dig up this post I wrote previously on a different forum and since it probably won't be seen in the thread I applied it to, I'll put it here where people that are actually interested in horns will see it, and also expand on the original post a bit.
The common definition for acoustic horn is the simplistic version that follows -
An acoustic horn or waveguide is a tapered sound guide designed to provide an acoustic impedance match between a sound source and free air.
By that definition, the only "true" horns are full size ideal horns, for which the math is clearly defined and can be found all over the internet. Undersized horns don't meet this definition.
First, how do you think this "impedance match", not to mention the huge amounts of in band gain, is realized? It's resonances, all horns have resonances and without resonances you can't have a horn.
Next, let's look at the claimed advantages of horns and how much they apply to full size vs undersized horns, your so called "wonderful virtues of true horns". Hint - undersized horns don't have ANY of the wonderful virtues of true horns.
The claimed advantages are -
- pattern control (dispersion)
- acoustic impedance match between a sound source and free air
- high efficiency through the whole passband
- reactance annulling and lesser pressures on the cone
I'll go through these one by one and show that the undersized front loaded horns don't have any of these characteristics, only full size front loaded horns do.
And then at the end, I'll show how similar ported boxes are to front loaded horns. Ported boxes have a lot of resonances are operate very similarly to front loaded horns in this respect, although most of the ported box resonances are usually high above the passband and are usually ignored - but that doesn't mean they are not there.
To illustrate what's really going on let's examine a couple of different horns. They are both very large, but one is small compared to the low knee frequency and one is the ideal size (which means it's huge). Both these horns use the same driver but the horns are very different. The small one is still very large at almost 750 liters and with a mouth size of 6550 sq cm but it is still massively undersized compared to it's low knee frequency wavelength. The large one is ideally suited to it's low knee frequency, having a mouth size of over 50000 sq cm and a volume of almost 5000 liters.
Dispersion patterns are a function of size and frequency wave length. If there is a horn or waveguide involved the shape of the flare also has something to do with dispersion.
With tweeters the wavelengths are small because the frequencies are high so it doesn't take a large guide to control their dispersion pattern. With bass horns the wavelengths are large. Does that mean they can't be controlled? Absolutely not.
A small bass horn, the kind you are likely to see being built by a diy'er is much too small to control dispersion at subwoofer frequencies. But a large horn can have a great deal of pattern control.
Let's look at the dispersion patterns of the two example horns. The small one is on the left, the large horn is on the right. The top row is dispersion at 35 hz (near the bottom of the passband), the bottom row is dispersion at 100 hz (top of the passband).
So clearly the small horn is too small to have much effect on directivity even at the top of the passband. The large horn has a small amount of influence over directivity even at the bottom of the passband, and a huge impact at the top of the passband. What goes for tweeters goes for subwoofers too, the difference is frequency vs size, and undersized horns can't control dispersion.
The "impedance transformer" (cone to air) is one of those old sayings that gets passed on all the time but it's not really true at all for small horns. Let's look at the impedance curve of the two example horns. Small on left, large on right.
The small horn has an impedance curve that looks a lot like a transmission line, which we know is not considered an impedance transformer. The large horn's impedance curve is much different. The spikes are damped, the dips are not as deep and the average level in the passband is higher.
In the small horn there are narrow bands of very high efficiency corresponding to the huge impedance spikes and in between the spikes the efficiency is very low. The large horn has a higher level of efficiency across the entire passband due to it's sheer size and a much more calm impedance curve. This is what they mean by impedance matching and it doesn't apply to small horns. The impedance curve is where you see it but that's not really what they are talking about either. What they mean is that the impedance curve, the frequency response and everything else is smooth, not spiky so the cone is able to "see" a more constant radiation resistance in the air load at all frequencies in the passband.
Here's actual efficiency. Small horn on left, large on right, 120 liter ported box tuned to about 32 hz on bottom. The small horn does have a lot more efficiency than the much smaller ported box at most frequencies, that's to be expected based on size and having so many impedance peaks inside the passband. The large horn efficiency is much higher than both.
Clearly efficiency spikes at the same frequencies the impedance peaks are at. So if you line up a bunch of impedance peaks inside the passband you can have boosted efficiency for as much as a 3 octave passband. And the larger the enclosure, the more acoustic gain, the more efficiency.
Some people might not be aware of what the ported box is actually doing. The front wave is out of phase with the back across a lot of the passband so they are actually fighting each other. At some frequencies the direct radiator output wins and at others the box resonances win. It's a complex summation. This is the direct radiator output (left), the port output (right) and the summed response (bottom).
For people that only use WinISD or similar helmholtz type simulators it may not be clear that ported boxes have a series of resonances like a horn. But when you look at port output alone in a picture like that directly above it becomes clear that ported boxes have ALL KINDS of resonances. The people who are aware that there are ported box resonances usually just ignore them, thinking they will be too far above the passband to matter.
But when you look at the port output image it becomes clear that ported boxes and horns are not really all that different. The horn actively uses it's shape to line all those resonances up into a pretty row and give them the individual acoustic gain that they need to provide a propped up flat(ish) response over as much as 3 octaves. The ported box has the same type of resonances but they are at erratic frequencies where they can't help but can only cause trouble. Therefore the resonances need to be pushed up way out of the passband or stuffing must be applied to calm them down. MLTL and the various different types of TL are ported box variants that attempt to make use of ported box resonances instead of ignoring them or stuffing the life out of them. Some types of TL are remarkably similar to undersized horns.
The resonant enclosure types are not that different, they all lie on a continuum based on shape and layout. They all have a series of resonances. How they deal with these resonances might be different but they all have them.
And since that post hinted at but didn't include a picture of frequency response or excursion characteristics of a full size vs undersized horn, here's a random example showing the much flatter frequency response and excursion curve of a full size horn.
Full size on top, undersized on bottom. Both shown with equal input voltage.
The full size horn is about 4500 liters, the small horn is about 370 liters.
Both horns have approximately 2:1 compression ratio.
Let's go a step further and look at cone pressure and reactance annulling. These are the same horns shown in the last pic, shown with the same (equal) voltage applied.
Top row is the undersized horn, bottom row is the full size horn.
Left column is pressure on the horn side, right column is pressure on the sealed rear chamber side.
Notice two things in that image - first the smaller horn has much higher pressure on the horn side than the much larger full size horn all across the entire bandwidth. (The first big spike in pressure won't matter so much if the horn is high passed, but the higher pressure through the rest of the passband will still apply.)
Second, the pressures on both sides of the full size horn are fairly well balanced (peaking at similar values of pressure) whereas the pressures on either side of the undersized horn cone are at dramatically different pressure. We'll see what this means in the next image.
This is total pressure on the cone - the actual normalized pressure the cone experiences when considering pressure from both sides (not just either side separately as shown above).
Notice the same two things in this picture. First, the large full size horn experiences dramatically less cone pressure and stress across the entire bandwidth despite putting out dramatically higher spl.
Second, notice that the pressure peaks are fairly well balanced (peaking at the similar pressures) across the whole passband, and even below the passband. On the other hand, the undersized horn has very little pressure at frequencies below tuning and very high pressures at the peaks above tuning.
This is reactance annulling at work in the large horn and not so much in the small horn. When the pressure peaks below and above tuning are balanced you have good reactance annulling.
Note that the pressure peaks above and below tuning are not exactly the same in the full size horn but they are close. They could have been closer if this was an ideal horn for this driver, but an ideal horn for this driver would have an insanely high compression ratio so compromises must be made resulting in less than perfect reactance annulling.