One way to test aspects of horn design which are hard to simulate, is to make a scale model and measure it under similar conditions. A friend with a small CNC machine cut some plywood sheets based on exported on-the-flat profiles of a simplified version of the horn, having only the mouth bend. The scale is 1:5.
The mouth bend is the most important in this case, as this sets the footprint of the horn, and therefore enables us to position the horn in the same way as it will be positioned in the room.
To drive the horn, a scaled down driver is also required. I used a Celestion AN2775 driver, as I had some at hand.
The horn was then placed in a corner, and a few experiments were tried.
Below are frequency response curves for the horn alone in the corner, and with a cardboard baffle flush with the horn mouth. There is a 5dB loss of sensitivity, similar to the simulated loss, so we may assume we will get a similar loss for the big horn in real life.
I also tried making a Helmholtz resonator under the horn, but this only reduced the loss by 1dB.
The requirements for the bass horn was set up in the overview article, and some additional considerations are outlined in My Approach to Bass Horn Design. This article will outline the basic design of the Big BenD horn.
Initial Idea
I have wanted a bass horn ever since I moved from Øyfjell. In Trondheim I had to make due with an Altec 816A enclosure, which has a horn, but that's more of an upper bass/lower midrange horn that doesn't add much to the actual bass. Moving to England I didn't even have the Altecs, and the first house didn't have enough space to set up much more than the midrange horns with open baffles. The open baffles, using either Altec 515-8G or various drivers I built, had decent performance, which is probably one reason why I kept them for so long. Another reason is I didn't have time to work on a big horn while writing the Book. But I was of course thinking about various options.
I could easily fit a horn mouth of 70 by 100cm, this would take up similar space to my baffles. Using a J-folded horn with the driver near the ceiling, I could easily get about a 250cm horn length, and with a throat area of 300cm2 the horn cutoff would be about 35Hz. With an Altec 515-8G the response is quite smooth, if the horn is placed in the corner:
The throat impedance is also quite smooth:
This is for the horn placed in a corner, i.e. eighth space. In practice, however, the horn will be out on the floor, since I live in a rented house and can't make horn-sized holes in the walls. We'll see the disasterous effects of this shortly.
Horn Bends
There are many ways to fold and bend horns, as outlined in the Horn Book, chapter 26. Sharp folds cause powerful reflections, and typically the throat resistance falls to zero save for at the resonances, above a few hundred hertz. Gentle bends with a bend severity parameter (ratio of horn radius or half width to the bend radius) smaller than unity, on the other hand, have much less reflections, and less issues with loss of radiation resistance and peaks and dips in the response.
In order to make it easier to play around with the geometry, I rewrote the horn curving tool in my simulation software, so that bends could be specified in several ways. For variable radius bends, this included some math in order to find the length around the bend. I also wanted to be able to specify two out of three of the parameters bend length, bend radius (relative or absolute) and bend angle. For the one-dimensional simulations I had also implemented the 1D theory outlined in the Horn Theory chapter in the book. While an approximation, it would at least give an indication of the change in acoustic length at low frequencies.
New Design
As mentioned, I was also considering building a midbass horn, but with the effect of the rear wall causing a loss of radiation resistance right above the corner frequency of the horn looked like it could be troublesome. So after some thought, I decided to build a bass horn based on the initial ideas.
In the meantime I had also looked into designing a dedicated bass horn driver. Many of the classic bass horn drivers, like Altec 515 and JBL 2220 are 15" drivers, and building something similar was of course an option. But there are some factors that suggest this is not optimum for domestic use. A domestic bass horn does not need the power capacity of cinema and PA speakers. Even a couple acoustic watts is plenty, easily producing 120dB SPL in a small room. With the high efficiency of bass horns (25-50%) there is very little demand on the driver. I am all for having plenty of headroom everywhere in the system, so that amplifiers and speakers are only coasting along at even the highest peaks. But even with my open baffles, with C-weighted average SPL of 112dB in the listening position, the peak instantaneous power to the woofers was in the order of 30W. The bass horns should be at least 10dB more efficient.
Therefore a 12" driver would be a better choice. Cone breakup would also be pushed higher, and considering I wanted this horn to behave nice way above the intended 300Hz crossover frequency, that sounded like a good idea. Diaphragm displacement could still be kept low.
A 12" driver has a piston area of about 550cm2. I would like to keep the compression ratio low, so I selected a throat area of 250cm2. With an RE of 6.5ohms, a Bl of about 18Tm would be suitable. Diaphragm mass should be fairly light, and the suspension soft, in order to use a relatively small rear chamber. While I didn't have such a driver at hand, I planned to build one.
Geometry
The geometry went through several iterations. With the smaller throat area I had to make the horn longer. After some number crunching I also found that I could get a bit more deep bass out of it if I made it slightly longer, placing the cutoff at 30Hz. This became the base design. The mouth was kept at 7000cm2, 1m wide and 70cm tall.
The first part of the bending geometry to be worked out was the mouth bend. Making a bend often involves laminating thin plywood, or using bendable types. Or both. I therefore looked into having a constant width bend to simplify things: this way simple flat plywood side walls could be used. But this geometry has quite sharp discontinuities, and it's also not visually satisfying.
I therefore went for a simply curved bend. This means that the side walls also twist around the bend, necessiating a somewhat complex jig for laminating. But there is no longer any sharp discontinuities. I also decided to use a constant instead of variable curve radius, together with a vertically asymmetric geometry. The inner curve is therefore quite close to a constant radius too.
The mouth segment was designed to be fairly flat on the top, to provide a "shelf" for the midrange horn. This also set the depth of from the horn mouth to the front of the upright part, so to make it possible to fit the length of the midrange horn there.
The middle segment, which I call the spire since it goes straight up, is asymmetric. This is to push the horn into the corner, so it doesn't cover too much of the window.
The throat bend was designed last, and here I chose to use a constant width for most of the length. Initially I considered a 90 degree bend here too, but since this would place quite a strain on the joints from the lever action and the weight of the driver, I made it a 180 degree bend in the end.
At this point I basically had the geometry laid out. See 3D rendring below.
There is space below the horn, the reason for that will become clear soon.
Simulations
The horn was first simulated using a 1D model, with the horn mouth flush mounted in a corner. This is the infinite baffle + two reflecting surfaces boundary condition. It is easy to calculate the radiation impedance and SPL under these conditions. Here is the SPL response at 1m on-axis:
In the listening postion the response is not as extended, both due to directivity and interference from the floor and side wall reflections.
The throat impedance also looks pretty good:
Under these conditions, I also made an FEA simulation of the horn, to look at the wave fronts inside. Using gmsh for meshing, and Pafec for simulations, the geometry was exported, meshed ans simulated. Symmetry planes were used to create a corner boundary condition.
The geometry was simulated before I had settled on the 180 degree throat bend, but the rest of the geometry matches the final design. The wave fronts heave well inside the frequency range I intend to use the horn (up to 300Hz). At 460Hz we start to see mode formation in the mouth bend, and the wave fronts are somewhat rough even at 355Hz near the mouth. This is to be expected. Adding a lumped parameter driver model to the horn, we get the on- and off-axis results shown below (2m distance). There are no great peaks or dips in the working range.
Rear Wall
But the horn will not be flush mounted in a wall. The mouth will be about 1.25m in front of the rear wall. The SPL in this case is shown below. There is about 5dB loss in bhe low frequencies due to the rear wall reflection.
The reason can be seen in the throat impedance, which is much more peaky compared to the flush mounted condition:
And the reason for this is the effect of the rear wall reflection on the radition impedance.
There isn't really any practical way around this. Possible solutions would be
Knock a hole in the wall and flush mount the horn
Make the horn mouth a lot bigger
Move the bend nearer the mouth
Build a baffle in the room, effectively making a partition
None of these are very practical for me. In addition comes the complication of room modes, which I will return to later. However, there is an option that could be worth investigating: making the reflection weaker, for instance by using a Helmholtz resonator near the rear wall, under the horn (this was the reason for the space underneath the horn). I will investigate this using a scale model. Possibly an active solution could help too. But regardless of this, an impedance matched driver will give the smoothest possible response in any situation, so this must be implemented.
In July this year (2019), it is 100 years since Arthur Gordon Webster published his seminal paper "Acoustical Impedance and the Theory of Horns and of the Phonograph". Who was Webster, and what significance had his paper?
Arthur Gordon Webster was born in 1863. In 1885, he graduated from Harward College, and in the years from 1886 to 1890 he studied under Herman von Helmholtz for his Ph.D. From 1892 he was head of the Physical Laboratories at Clark University, and was appointed full professor in 1900. In 1899 he helped founding the Americal Physical Society, and became its president in 1903.
Webster was a proficient mathematician and a competent experimentalist, and spoke several languages fluently. His research was centred around acoustics and mechanics, and he invented the phonometer, an instrument to measure absolute sound intensity.
Webster committed suicide in 1923 following rumours that his physics department would be closed by the new president at Clark University.
His paper on horn theory was published in 1919, but it was actually originally read at a meeting of the American Physical Society at Philadelphia in December, 1914. It is not the first paper dealing with the horn equation, which was treated by Euler, Bernoulli and Lagrange in the 18th century, as described by Eisner (Edward Eisner, "Complete Solutions of the Webster Horn Equation", J. Acoust. Soc. Am. 41, 4 (2) (1966), pp. 1126–1146.). So why has Webster’s name been associated with it?
There are several reasons for this. First of all, it was a relatively accessible analysis of horns, presented at a time when it was needed. The old papers analysing sound in expanding tubes were not readily accessible, and Lord Rayleigh’s “Theory of Sound” was out of print. Second, Webster introduced the concept of acoustic impedance (although with a slightly different definition that what was later adopted), which tied in nicely with the impedance concept used by electrical engineers to analyse complex circuits. Thirdly, he showed how the equations for various types of finite horns could be developed, and in this way, the effect of the conditions at the mouth could be taken into account.
Why was it important? Up to this point, gramophone recording and reproduction had developed mainly through empirical methods, and radio broadcasting was in its infancy. With the vacuum tubes available at the time, output power was very low. To free the listener from using headphones an efficient loudspeaker (using a horn) was needed. The engineers needed a way to estimate and optimise the performance of the horn. As stated by professor V. Karapetoff in 1924, “This problem of horns is a 'house-on-fire' problem, in the sense that loud speakers are now being manufactured by the thousand, and while they are being manufactured and sold, we are trying to find out their fundamental theory.” Webster’s analysis made possible to find guidelines to use in the early stages of design, although the math was complex enough that for most cases rules of thumb based on equations that could be calculated by hand had to be employed.
The paper does not appear to have been widely recognized at first. The first citation seems to be one by Kennelly in 1923. But it was no doubt studied by engineers involved in loudspeaker design. And by the publication of the paper “The function and Design of Horns for Loud Speakers” by Hanna and Slepian in 1924, a much wider community was introduced to Webster and the horn equation. It is perhaps less widely known that at the same time, Paul B. Flanders and Donald A. Quarles at Western Electric Engineering Department, later Bell Labs, also studied horns and derived equations for predicting their performance. Even the performance of an entire system of driver, throat chamber and horn. But this work was only published in internal memos, and not publicly.
But the significance of Webster’s paper at the time was great. It was now possible to calculate the performance of horn loudspeakers, and one early user of this method (outside the Bell Labs) was Harry F. Olson of RCA, who published his findings in several papers. He and others studied the effects of horn flare, horn length and mouth size.
At this time, about 1925-1940, it appears to have been widely known that the equation was an approximation. As W. M. Hall puts it 1932, “The use of such approximate methods in the theory is justified only by the approximate nature of the results it is desired to obtain.” Experimenters often commented that the discrepancies between theory and experimental results were due to the approximate nature of the horn equation. This knowledge seems to have been more or less lost after the publication of Olson’s “Elements of acoustical engineering”, and Beranek’s “Acoustics”. Neither of these books mention the approximations in deriving the horn equation, and judging from published literature, it took several decades before it again became common knowledge among audio engineers.
Today, the main function of horns in pro audio is to provide directivity control. Compression drivers are crossed over at much higher frequencies than was common in the decades after Webster’s paper was published, and the higher power handling and cheap electrical power makes the acoustic load less important than it once was. And the only thing the horn equation really can do, is to predict the acoustic load, and the “transformer properties” of the horn. It is one-dimensional, and cannot predict directivity, which is a 3-dimensional quantity. For this we need more advanced methods, like FEM, BEM or MMM.
But for bass horns, it is still very useful. When the mouth is small enough to not have significant directivity, the predictions can be quite close to reality. The free software Hornresp has been successfully used by many to design low frequency horns. Since the throat impedance can be predicted with reasonably good accuracy, the power response of a speaker can be calculated. This applies to any frequency range.
The horn equation is also useful for instrument modelling, as the wave propagation in wind instruments is essentially one-dimensional, and the important property is the location of resonance frequencies.
In summary, Webster’s horn equation was a great help to the early electroacoustic engineers in the “roaring twenties”, when loudspeaker design progressed from a pure art to something that may be called science. But even today, it is not completely outdated. It still serves horn designers, and serves them well, if they ask the right questions. It cannot do everything, and it never claimed that either.
Does Webster deserve to have his name attached to it? I definitely think so. Webster was not the first to derive this equation, but he introduced it to the community of electrical engineers when it was needed, in a language they could understand. And to this day, nearly every paper dealing with the theory of horns, will cite his paper. It has become his equation.
Details
Written by Bjørn Kolbrek
Over the last nearly 15 years Thomas Dunker and I have worked on a rather big project: writing a book about horn loudspeakers. The idea came after we had collected a fairly large number of references through our own research, and we thought the best way to organize the growing archive was to write a book.
Also, there has not been a proper book about horn loudspeakers before, the information has been scattered around in hundreds of papers and book chapters. It is quite a job, even with a list of references, to actually get hold of all of them, and then you have to digest the information and apply it to your problem.
The project grew over the years (and was also the reason why I did an MSc and a PhD on horn simulation), and we ended up with this 1070-page volume containing the history, theory and design of horn loudspeakers. The full story will be told at our dedicated information website, hornspeakersystems.info, where we will also post updates from the production of the book, purchasing information and information about signing events.
Details
Written by Bjørn Kolbrek
New interesting compression driver
Celestion recently came out with a new compression driver, the Axi2050. For people like me who are interested in large midrange horns covering most of the vocal range (and I'm not talking about the 300Hz-3kHz telephone range, but more like 100Hz to 5kHz), this driver is very interesting. Especially since the high end response is very good for such a large unit. It's more of a mid-high than a midrange driver. Take a look at the frequency response curves, taken from the Celestion data sheet:
The PWT response is only shown down to 100Hz, but it is basically flat down to that frequency. In addition, the HF response looks good up to about 10kHz. That's two decades! And the HF sounds pretty good too; I got a change to hear one such driver on a straight circular Hypex horn at the Celestion factory, and it actually had a nice, smooth high end. Moreover, the driver is rated for 150W down to 300Hz, and the diaphragm is large, so for domestic use, using it down to 100Hz or even lower would not be a problem.
I would love to try a pair of these drivers on my midrange horns, but unfortunately they are listed as OEM only. Also, I would have to design a new throat and middle segment for the horn, as the driver has 2" exit, and the horn is currently less than 2" in height for about half its length.
I first became aware of this driver at the 2015 AES convention in NYC, where Jack Oclee-Brown and Mark Dodd presented two papers on the design of this driver:
Wideband Compression Driver Design, Part 1: A Theoretical Approach to Designing Compression Drivers with Non-Rigid Diaphragms (preprint 9386) and
Wideband Compression Driver Design. Part 2, Application to a High Power Compression Driver with a Novel Diaphragm Geometry (preprint 9391).
At the 2016 AES Convention in Paris, I got the chance to look at the driver itself. It's fairly big, but not very heavy for its size (about 7kg).
And just for the record: when writing this, I'm still at the university. I'm not trying to sell the driver or to advertise for Celestion, I just want a pair!
