When you design a horn, you have several criteria:
- What frequency response you desire
- What driver you will use (and its parameters)
- How much compromise you are willing to accept (size, shape, materials, etc.)
The driver and frequency response determine the type and size of the ideal horn, and for a bass horn the maximum size will affect the output and probably the design criteria.
Note that many existing horn designs (and this is especially true for bass horns) often have massive compromises or deviations from what might be an ideal horn. Many Lowther rear-loaded bass horns have been designed by large amounts of experimentation to achieve the best sound given the compromises (often size) that are imposed. Beware when designing your own horn that you understand why things were designed a certain way. This is especially true if you are using or building on current designs. If you start from scratch using the best available theory (such as Leach, below), you might get quite different designs than appear commonly today.
The W. Marshall Leach Horn Model
One of the most important papers on horns is by W. Marshall Leach, Jr., On the Specification of Moving-Coil Drivers for Low-Frequency Horn-Loaded Loudspeakers. Available at the AES website as Preprint number 1405. This paper contains a model of horn loading based on electrical circuit parameters. Thomas Danley, a horn and sound pressure expert from Servodrive says this model is the most accurate for horn design that he has come across.
I would encourage anyone who is considering building a horn to purchase this paper. There are quite a number of calculations which you go through in order to fit a horn to driver parameters, or find a driver given your ideal horn. I recommend you work through by hand (look at both examples) and then set them up in a spreadsheet. This allows you to change input values easily and see the outcomes.
In essence, you design a horn for a particular frequency range, which gives relatively flat output over that range, and has shoulders where low and high frequencies roll off. These shoulders (Fl and Fh) are used in the calculations, along with the driver parameters, to determine the size of front and rear compression chambers, and the overall efficiency of the horn. I believe the answers are for an "ideal" horn, one that has a mouth area properly terminated for the space it is radiating into (see Mouth Area, below).
Note that most of the information below discusses horn design using a different model to the Marshall Leach model.
The first thing to calculate is the throat area. This is determined by the driver parameters. There might be existing well-know throat sizes for certain drivers. Lowther drivers are often used in rear-loaded horns with throats ranging from about 50 cm2 to 150 cm2.
So here is one formula for calculating throat area:
- At is the area of the throat (m2)
- FS is the nominal air resonance of the driver
- QTS is the total Q of the driver (note, some formulas use QES, the electrical Q here - Bruce Edgar for one)
- VAS is the equivalent volume suspension (m3)
- c is the speed of sound (344 m/s)
The Leach model calculates throat area with a different formula.
Now that we have the throat (and therefore the beginning of the expansion), we need to know when to stop. We need to work out the mouth size. The formula for this is:
- Am is the mouth size (m2)
- SF is the size factor (1, 2, 4, or 8)
- c is the speed of sound (344 m/s)
- Fo is the flare frequency (or cutoff frequency) in Hz
The size factor is 1 (mid or high frequency front horn), or bass horns: 2 (middle of floor), 4 (wall placement), or 8 (corner placement). Once you have calculated the mouth area, you can determine the profile of the horn. NOTE, if you are using a calculating program, such as my horn contour calculators, you might not need to supply the mouth size. Instead, you supply only the size factor, along with the other horn parameters, and the program stops at the correct mouth size.
The Leach model probably uses this mouth area as the ideal termination.
This is tied to mouth area. If the mouth area is too large, or the horn length is too long, you might want to shorten the horn. This is a compromise. To reduce the size of a bass horn, you can either shorten the horn or increase the flare cutoff frequency. If you increase the flare frequency, you can still build a horn with "ideal" properties, but it will only play down to the flare cutoff frequency, then drop like a stone at lower frequencies. If you keep the flare frequency but shorten the horn (also reducing mouth size), you keep the lower frequencies somewhat, but you get impedance peaks below the ideal mouth size. These peaks create a bumpy frequency response below the mouth cutoff. See for example the simulations (with David McBean's Horn Response Analysis program) in this diagram.
The size of the compression chambers depends on the drivers and the frequency range you want from the horn. For a rear-loaded bass driver such as in a Lowther horn, the front compression chamber volume tends to vary from about 2.5 liters to 4 or more liters. Of course, a Lowther horn does not have a rear compression chamber (because of the radiation from the front of the driver).
A number of different formulas exist for compression chambers, and any one of them may not give the best size for your application. These formulas should be considered a starting point. One formula for a compression chamber volume:
- VB is the back volume (volume of compression chamber next to horn throat, the front chamber in the diagram on the previous page)
- Vas, Fs, and Qts are standard driver parameters
- Fc is the lower cutoff frequency of the horn flare (I think)
Another formula comes from the paper by D.B. Keele, Jr., Low-Frequency Horn Design Using Thiele/Small Driver Parameters from the AES (see References) which is a simple form first derived by Klipsch in 1941:
- VB is the back volume (volume of compression chamber)
- ST is the throat area of the horn
- FC is the horn cutoff frequency
- c is the velocity of sound
Front and rear compression chambers are calculated differently in the Leach model.
David McBean's program (see below) can show the effects of different size front and rear chambers on the reactance and SPL generated by a horn.
First you need to choose the appropriate contour. For bass horns (less than about 300 Hz or so), the exponential or exponential / hyperbolic contours are proving to be the best, according to Dr. Bruce Edgar. The Tractrix contour is very well suited to mid range and high frequencies.
Calculate the contours using one of the formulas on the previous page, or use one of the applets below:
Front loaded horns are are usually only designed for mid range, unless you have a horn dedicated to bass. For examples of square front loaded horns, see the Front Horn Page.
Front horns can be built with bending plywood, veneers, cardboard, and other materials. Richard Vance has built a nice 80 Hz horn out of curved Masonite (fiberboard). This is not as strong as 1/4 inch plywood, but dampens a little better and can be roughened so fiberglass both impregnates and sticks to it. In the US, Masonite costs almost nothing, making this the cheapest big project imaginable. Read the 80 Hz horn article.
Rear-Loaded Bass Horns
For information on rear-loaded horns, check out the Rear-loaded Horn pages.
Horn Shape and Materials
Horns are normally built from wood, a very useful and adaptable material. The ideal horn has a circular cross section but this is hard to construct in practice. Short horns such as front horns can have a square cross section, such as Bert Doppenberg's front horn. A square cross section is harder for a large bass horn, but probably could be constructed with a lot of woodwork.
The rectangular cross section is the most common for bass horns with small drivers. Larger drivers like the Altec or Tannoy Churchill often play out of a front horn that is approximately square. For rear-loaded Lowther horns, a rectangular cross section is normal. See the cabinet plans at Lowther's website and at Marc Wauters site.
Simulating Horn Performance
Some software programs can calculate horn impedance and resistance curves and also horn sound pressure levels (SPLs) given driver parameters. One excellent program I would recommend is David McBean's Horn Response Analysis program, which is free.
Other horn simulations can be modeled in AJ-Horn, Speak 32 and AkAbak.
However nothing beats actually building the horn and measuring it. There are plenty of good speaker measurement software packages available today. Look around. I'll let you know of anything that I come across.
Also check out the Software page on this website.