Horns fundamentals

In the early days of sound reproduction, amplifiers supplied extremly modest powers and loudspeakers were not very efficient at all. But genial, yet simple, idea had accoured, like cupping your hands to your mouth to shout, a horn increases sound pressure of not very efficient driver. In 1928, the wider bandwidth of the Rice-Kellogg direct radiator loudspeaker and availability of higher-power amplifiers all but removed the horn loudspeaker from home audio systems. After 1928, horn loudspeakers were only found in theater PA systems, until the introduction in the late '40s of the Klipschorn, which spawned a revival of horn loudspeakers in the '50s. The introduction of high-power solid-state amplifiers and small bookshelf speakers in the '60s removed horns from home audio systems. In the early '90s, the hipe of SET tube amplifiers spurred a new interest in horn designs.
A horn can be seen as an acoustic transformer that couples the air at the surface of the diaphragm with the air in the listening room, thus matching high pressure/low volume to low pressure/high volume.The name acoustic (impedance) transformer derives from this model of description.

Another explanatory model of horn behaviour exists and, by my opinion, describes functioning of a horn, equaly well. Horn can be seen as guideing path to the driver's output radiation, preventing it from going to other areas. This ability to control (reducing radiation from full space or half space to horn mouth area) loudspeaker's dispersion pattern is a reason for high efficiency.



Even though modern amplifiers can power cone and compression drivers to very high output levels, increasing a driver's efficiency through horn-loading reduces strain and minimizes distortion. At the high output levels demanded by today's loudspeaker standards, harmonic distortion remains a problem for poorly designed drivers and systems.

A horn will be effective in any given frequency range depending only on the size of its mouth and the rate of its flare. Bass reproduction from a horn is NOT bassless by default. Usually this happens when the designer compromises to much, and foreshortens the horn in order to get the physical dimensions down. Because wavelengths in the lower octaves are so large (allmost 7m at 50 Hz), the construction of practical and "mobile" horn is not very possible.

A horn system is a lot more efficient than any other type of enclosure. At the same SPL driver in horn design has smaller excursion than in other type of boxes or in open air, thus horn driver will operate linearly, producing less distortion. The horn permits the driver displacement to be small without sacrificing acoustic output, this means lower distortion and/or higher power handling.

very dynamic
extremely low THD, IM, and FM distortion
excellent impuls (transient) behaviour
high efficiency and ease of drive
steep cutoff characteristics at both ends of the frequency range
problems with impulse response, diffraction, and smooth dispersion


Contours/flare rate

The horn contour is the expansion (flare) rate of a horn. There are few curves appropriate for audio application; these are conical, exponential, hyperbolic, tractrix contour and permutations between them dependant of designer's needs. Parabolic horns are so inefficient that they are really of no practical use.
Of these, the conical is the easiest one to calculate and stuff into a box, but it's also the least efficient. Conical contours are never employed for bass horns, because of the poor response and the impossibly long horns that result.

The exponential is the most commonly used, and is easy to calculate. It is probably the best flare trade-off high efficiency vs. low distortion. This is why a very large number of commercially produced horns feature exponential contours.

The hyperbolic contour is a variety of the exponential, and is the most efficient type. A hyperbolic horn by comparison is pretty much like a trumpet stretched out, a tube that flares very little until it gets to the end where it flares suddenly. The trade-off is more distortion in the deep bass region. The problem with such a tight flare is that as sound pressures increase, the restricted passage for the air causes it to begin to compress. This causes distortion - not good. You don't want to compress the air, cause it gets hot, etc, etc - and the sound starts to sound really bad. Hyperbolic horns are also somewhat longer than exponential horns.

The tractrix is sometimes called the tractory or equitangential curve. It is a curve well-known in the world of mechanics. The tractrix was first studied by Huygens in 1692, who gave it the name ``tractrix.''. Later, Leibniz, Johann Bernoulli and others studied the curve. P.G.A.H. Voigt "reinvented" and applied to horn speaker acoustics in 1926. The tractrix contour has characteristics similar to the exponential, but has the advantage of being shorter (the curve expands faster). The disadvantage is that it's somewhat awkward to calculate, since you can't directly calculate the area A(x) at a distance x from the throat. This should not be problem today, since computer software is accessible through net (http://www.melhuish.org/audio/).

Flair rate (T) - from conical to hyperbolic curve:
the gain of a horn depends upon its shape



Mouth size

The mouth size of a horn determines the lowest frequency at which there still is any significant reinforcement of the sound.
Graph shows output of two horn with same throat size and contour, but with different lenghts, therefore different mouth size



In the end the Final Rolloff of the Horn will be 18db/Octave as cancellation and horncutoff combine. This together defines the absolute available low-end cutoff and it is obvious from what is said, that a small horn cannot go very low.


Throat size

The throat of the horn can be as big as the transducer cone or smaller. Maximum efficiency is achieved when they are the same size, but the bandwidth is decreased. To increase the bandwidth, mouth must be smaller. The ratio of the cone to the throat is known as Sd/St. Of course, as you increase Sd/St (or reduce the throat size) the efficiency drops. The improved efficiency of the horn is gained at the loss of bandwidth. So, it may be louder but it will drop off in frequency at the upper end sooner.

Front loaded horns

Front horns are primarly used for high, mid or higher bass freq. regions. They can be used as bass enforcement especially at PA loudspeakers in multi-way configurations, where one unit covers only limited frequency range.
They don't sufffer from combo effect of rear horns (two different sources (a driver and a mouth) which are missaligned in phase) and are great partner for full range drive units, which tend to loose efficiency around 1kHz and below.

The front horn will (again depending upon it's profile) give a certain reinforcement to the lower midrange, so ideally this horn is designed to raise the lower midrange to the level of the "bump" in the drivers frequency response.

At higher frequencies the coupling of the Driver to the Horn will become "looser" and the direct radiation of the Driver will dominate the Frequency Response. Again skillfull blending of these two will result in a very flat frequency response.


Rear loaded horns

A rear loaded horn radiates acoustic energy from two separate sources; from a driver fireing directly and horn's mouth. Regardless of horn's path lenght it radiates 180 degrees opposite in phase with a driver. Horn's lenght adds additional phase difference between two sources.
Because of this dipole action there is a frequency below which the driver's rear and front action will cancel. This frequency is dependent upon the length of the horn. The longer the horn, the lower the frequency at which the the cancellation set's in.


Real life tricks

A horn-loaded cabinet has to be large to have smooth frequency response. This is of course dependent on the lowest frequency response the cabinet needs to reproduce. One of the old rules of thumb said that you need a horn with an opening of 10 metres in diameter to reproduce frequencies down to about 30 Hz with smooth response - a touch or two larger than your average vented cabinet. In fact, those huge bass bins at the rock venue are usually useless below about 50 Hz or so - even though it feels like the sound pressure will crush your body.
There are only a bunch of people who can afford monstrous no compromise bass horns and not be thrown out by their living partners. Because physical dimensions of unfolded bass horn are so nonpractical, horn contour must be folded. This building condition makes actual horn response unpredictable. Instead of smooth, uniform contour, we have an array of resonating chambers. Functioning of this relatively weak resonators can be noted as resonance peaks and dips in impedance and frequency response plot. This could be rendered to certain degree with an aid of computers, thou it would not be easily covered mathematically.

Even when size of a horn is reduced with foldings, they are still pretty big. There are few tricks which can be used to reduce horn size without loosing to much low bass performance.

Hornmouth is placed near the floor, so reduced radiation by the floor allows reduction of hornmouth to 1/2 the orginal, with a theoretical 100Hz hornmouth effectively offering a response down to 60Hz.
Hornmouth is placed into a corner. Now it radiates only in 1/8 of full space and increases the accoustical output of the hornmouth. This is the case with classic fullrange horns a'la Klipsch and Voight Horn (1936) or majority of conteporary horns.