# Theory

 I don't want to go into extensive covering of basic horn theory, Witold Waldman's Site has a very good overview of what is available in terms of general loudspeaker theory and documentation thereof, so please go there to find the way to more specific stuff. Briefly however all acoustic plane wave horns share a common formula, the various horn shapes are obtained by tweaking or omitting variables. The Duelund Horn is simply a chosen version of that formula that comes with the advantage of the most linear loading of the unit, and with the slight disadvantage compared to an exponential horn that it will theoretically pay for that linearity in its lowest octave with a slighty lower maximum SPL if distortion parameters are to be applied. This because it has a somewhat slower initial expansion rate. In horn "family name terms" it is a hyperbolic horn, exactly the intermediate between a catenoidal horn and an exponential horn, i. e. it is exponential-catenoidal. Here is the formula describing the linear area expansion of all planar wave horns, tractrix horns (a.k.a. known as "Kurvenwellentrichter") have their own formula with a few extra hieroglyphs added. ``` [1] Sx = St(cosh mx + Tsinh mx)2'ed ``` Where S is the cross-sectional area, Sx the area at the distance x, St is the area at the throat, x is the distance from the start of the horn, and m is a constant derived from the intended cut-off frequency FcutoffHz determined by: ``` mc [2] FcutoffHz = --- 2 pi ``` i. e. (if I didn't get it wrong) ``` 2 pi FcutoffHz [3] m = -------------- c ``` Where c is the speed of sound. The Duelund Horn is characterized by the paramater T in equation [1] being chosen as 0.6. Solving Equation [1] with a start area of zero shows why it is such a wonderful advantage to use a large beginning area, i. e. that the way to make a horn shorter for the use in any given band is to use a larger initial area, which implies a larger loudspeaker unit. There are some additional concerns in horn design, it is beyond this basic description of the formula chosen for The Duelund Horn to go into them. But if you want to construct a different implementation of The Duelund Horn, you may have to so do in case you want to minimize distortion and maximize efficiency. There is a simple and basic difference between rear loaded horns and front loaded horns that you must understand. A rear loaded horn has a chamber between the unit and the start of the horn. This chamber functions as a cross-over, sound below the Fs (resonance frequency) of the loudspeaker in it gets passed on via the horn throat into the horn. A front loaded horn should have the smallest possible volume of air between the loudspeaker membrane and the horn so as to have the widest possible frequency range. To also have maximum efficiency and minimum distortion it is required that the unit has a rear chamber, a conventional closed cabinet, with a size that results in a compliance that is equal to the compliance of the horn mouth, otherwise the units membrane will move in an asymmetric way and thus create second order distortion. This usually means that the rear chamber should be perhaps surprisingly small, and usually filled up with suitable damping material to avoid midrange resonances - it can with advantage also be irregular. You may also want to look for litterature on matching the loudspeaker unit compliance to the horn compliance, they should match for maximum efficiency, the Qt may a good parameter to look for and at. Loudspeaker units that are designed for horn use usually have very low Qt's, i. e. in the range below Qt = 0.3. If the maximum linear bass reflex cabinet gets ridiculously small in a bass reflex modelling, then the unit is more likely to give good results when horn loaded. Compression is usually the name of the game, i. e. that the area of the horn throat must he [the compression factor] smaller than area of the unit. Compression factors usually end up being in the range between 2 and 3, so a good "dumb guess" probably is a compression factor of 2.5. However, while compression improves coupling to the horn, and thus improves efficiency, then it comes with a penalty: air is not linear, and will exhibit significant harmonic and intermodulation distortion at levels in the 150 dB SPL range and beyond. It follows from this that a horns maximum SPL at the mouth (!) when distortion parameters are considered gets smaller with increased throat compression.