Why Horns??

So why would anyone want to use horn speakers today, when we can have 100W amps at low costs? As I hinted in the previous paragraph, horns DO have other special qualities than just high efficiency. Okay, it's necessary to understand a few basic things about HOW HORNS WORK in order to say WHY they do some things better than other speakers. (You'd be amazed at how many people who simply refuse to believe that horns can be used for serious music listening because somebody told them so or because they think horns all sound like megaphones.)
A horn can be viewed as an acoustic impedance transformer. Turning mechanical motion (a vibrating diaphragm) into sound waves in air is in many ways a difficult thing to do. The most fundamental problem, which has a lot to do with the issue of efficiency, is that the difference in DENSITY between a paper or metal diaphragm and AIR is huge. There is a tremendous impedance mismatch. This fact explains that sound travels very far through denser media like metal, water or rock. In a speaker-air situation, the speaker diaphragm can be seen as a high impedance source (solid material) and the air being a low impedance medium (the air does not easily load down solid moving objects). There is a reason why humans can't fly by waving their arms!
What the horn does is to help the transducer couple its radiated energy into sound waves in air by means of an impedance transformation. What this means is that it creates a higher acoustic impedance for the transducer to work into, which means that more power is transfered. (Analogous to putting an antenna on a radio transmitter, which seems like an obvious thing to do!) Basically, a horn is a tube or conduit with increasing cross-section along its axis. The narrow end (where the driver sits) is called the horn throat, and the large end (which opens into the room) is called the horn mouth.
Sound pressure is defined as pressure change per unit area. In a horn, the wave front is restricted by the inner walls of the horn, and the area across the horn increases as the wave front approaches the horn mouth. So what happens here is that at the throat we have a small area and high pressure with small amplitudes, efficiently loading the diaphragm. As the wave fronts travel towards the horn mouth, the pressure drops, while the amplitude and the area increases.
A horn also has the property of directing the sound into a narrower beam, which increases the on-axis sensitivity (SPL/1W/1m). Increased directivity combined with high electric-acoustic conversion efficiency means that horn speakers are very easy to power, even with very small amplifiers.
What does all this really mean, then? In what ways does the horn 'help' the driver/transducer. And how does all this make horn drivers a bit different from direct radiators? I will try to sum this up in a few points:

-Improved energy conversion means that for a given SPL, a horn loaded diaphragm will have to move less than a direct radiating diaphragm of equal size. For any electromechanical transducer, the distortion generated by the driver itself will be proportional to diaphragm excursion magnitude. Thus, for any given SPL, the horn loaded speaker will have lower distortion than the same size direct radiator.

-A smaller diaphragm on a horn can be used to generate the same SPL as a larger direct radiating diaphragm for the same excursion amplitude. This means that you have a smaller mass to accelerate for the same acoustic output when you horn load a driver. This helps the transient response of the speaker regardless of what Fourier said. Subjectively, horns will be noted for their effortless, snappy handling of transients.

-The smaller diaphragm excursions allow the use of short, underhung voice coils (reduced mass again) taking full advantage of the flux in the pole piece gap. This increases the efficiency of the transducer, allowing the amplifier to work with more headroom and greater ease. Horn drivers need to have powerful magnets and tight magnetic coupling because of the high pressure they are asked to produce when sitting in a horn throat.

-Because the amp has more headroom, and because the driver handles signal peaks and high outputs more ideally, horns will be able to produce much higher SPLs than comparable direct radiators before distortion becomes objectionable. In short, there will be room for more dynamics, at lower distortion, with better transient response, with less stress on the amp.

Since this is an enthusiastic pro-horn text, I have not emphasized the problematic aspects of horns. One thing I haven't mentioned is that the lower the frequency one wants to reproduce through a horn, the larger the horn must be. The size of a horn quickly multiplies when you go down a few octaves. Bass horns can be next to impossible to fit into a normal home. This really shouldn't be a problem to a true enthusiast, but even I have had to postpone any dream of a bass horn until I get a bigger place to live.
Some people say that horns have 'horn sound'. I'm not sure but I think what they mean is a sort of megaphone-like quality to the sound. A good horn should not have any of this. Unfortunately, many bad horn designs have led people to think that this is how horns 'are supposed to sound'. To the true horn fan, 'horn sound' will be a compliment that means, clean, dynamic, 'fast', physical, detailed and present sound.
Admittedly, there are a few things that horns don't always handle quite as well as your typical small direct radiator speakers. Particularly the much hyped concepts of 'imaging', 'neutrality' and 'transparency'. Horns will often lean towards a more 'solid' (as opposed to 'transparent') presentation with more 'body'. (Less ghost-like if you will!!) And horns will definitely not sound laid back! The music will jump at you rather than shyly hiding some place far behind the speakers.
Anyhow, horns have their problems like any kind of speaker, and can be extremely sensitive to the room and the rest of the system. The enthusiast would say that the advantages outweigh the disadvantages and that the problems are challenges.