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Horn Size and ShapeHorn loudspeakers can be roughly divided up into bass, midrange, and tweeter horns. Tweeter horns are very small and front-facing with a straight horn axis. Midrange horns can vary in size (depending on your classification of midrange) but might be anywhere from 20 cm diameter (round horn) to perhaps 80 cm in diameter. See for example the Avantgarde horns. They are usually front-facing with a straight axis, though the larger ones can have a bend in the axis. Bass horns have to be long to work and require large mouths. Bass horns are usually folded to allow a "normal" size, since they can often be 2.5 m long, or longer. Some bass drivers have short horns on the front (see altec bass cabinets) which help the upper bass region. The true benefits from horns of acoustic impedance matching do not occur until you reach at least a quarter wavelength of your lowest frequency, and more likely a half wavelength. That results in horn lengths of 1.7 m horn (1/4 wavelength) for 50 Hz, or 3.4 m for 50 Hz at 1/2 wavelength. Pretty long! Horn MouthAny loudspeaker can radiate into free space (hanging 100 feet in the air over a cornfield!), 1/2 space (sitting on the ground/floor somewhere, away from walls), 1/4 space (on floor against a wall) or 1/8 space (in a corner). As you reduce those spatial dimensions, the loudspeaker itself looks more and more like it is sitting in a horn. A bass horn, which reproduces long wavelengths, benefits the most from room reinforcement. Midrange and tweeter horns are designed as if they are radiating into free space (because the wavelength is short compared to the distance to the room walls/floors). For a horn loudspeaker to reproduce with high quality the lower frequencies, it needs a certain size mouth. For a free space horn with a circular mouth, the mouth size should be equal to one wavelength at the flare frequency. Thus, the free mouth area will be:
For a bass horn with a flare frequency Fo of 50 Hz, the free space mouth will be 3.8 m2 or 38000 cm2. This is very large. Building a midrange front horn, the size factor (see formula below) is 1 (equivalent to free space placement). However, if you are building a bass horn, it is effectively constrained in space and you can reduce that mouth size by a factor of 2 (middle of floor), 4 (wall placement), or 8 (corner placement). So a Klipsch corner horn can have a mouth size 8 times smaller than the flare frequency would indicate. As you reduce the mouth size, of course you also shorten the horn. The formula then becomes:
If you have too small a mouth for the flare and placement (i.e., overly shorten the horn), then you will get impedance peaks (and hence bass peaks and troughs) below the frequency dictated by the mouth size. This is apparently the case for almost all rear-loaded Lowther horns. Horn Expansion ContourEvery horn follows some type of expansion contour. This is the type of curve between the throat and the mouth and include:
Below is a comparison of several expansion contours for a bass horn (sorry if they are hard to see). The shortest curve is a Tractrix expansion. The middle is exponential, and the longest is an exponential / hyperbolic contour with m = 0.6. Note the flare at the end of the Tractrix contour. Expansion Contour & FrequencyFor 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. Some people are using the Tractrix expansion in bass horns for Lowther drivers, for example the Big Fun horn and the commercial Carfrae. The rules of best expansion contour might break down when horns are shortened so much as to be sub-optimal. A shortened Lowther horn might possibly sound good with a shortened Tractrix expansion. Nonetheless, mid range and high frequency horns do not have the same size restrictions and can therefore use the most appropriate expansion type. Calculating the ContourThe formula for calculating an exponential contour is:
The formula for an exponential / hyperbolic contour is:
This is more easily calculated on a hand calculator or programmed for a computer using the following: Note that when M is equal to one, the formula above reduces to the exponential formula at the top. Therefore, you can use the formula above to calculate both exponential and exponential / hyperbolic horns. The Tractrix formulation is:
This formula is presented differently to the other formulas with x being the calculated variable. To use this formula, choose your rm, select a value for rx of between zero and rm, then calculate the distance from the mouth x (at this radius). These formulas are used in my Contour Calculator Applets:
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