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See also: The Use of a Horn, Amplification, The Folded Horn  
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Horn ShapesAuthor: N.H. Crowhurst A horn thus serves two purposes: it confines the sound into a narrower angle, but, more important than this, it improves acoustic efficiency by preventing the losses that occur at sudden transitions. This improvement in efficiency is sometimes the only reason that a horn is used, since some horns are designed to distribute sound in all directions.
The megaphone is a straightsided cone, because this happens to be a convenient way to make it. Although it does improve acoustic efficiency, other shapes will do so even better. The purpose of a horn is to "stretch'* the developing wave gradually. (The wave should not expand too rapidly suddenly, and making the expansion too gradual will make the horn unduly long.) In the conical horn, the rate of stretch varies. Near the throat the area doubles in only a short distance, whereas it expands more slowly near the mouth. A horn is effective in producing a smooth transition only if it takes not less than l/18th of the wavelength corresponding to the lowest frequency required to double its area. (This relationship has a mathematical basis too advanced to give here, although it has been verified experimentally.) Suppose that the lowest frequency for a horn is to be 400 cycles per second (this is not very low, but is often used for practical equipment, because it results in a useable size): the wavelength for 400 cycles at 1100 feet per second is about 33 inches; so the horn should double its area in not less than about 2 inches (15/6 inches to be exact). In addition, the diameter at the mouth of a horn should not be less than about half a wavelength for the lowest frequency that is to be radiated. For a 400cycle horn, the diameter at the mouth should not be less than about 16% inches. To get the best acoustic energy match into the throat of the horn, a diaphragm about 1*4 inches in diameter works into a hole about 4 inch *n diameter. The horn thus has to stretch the y^inch diameter to about 16^ inches. Doubling the diameter will quadruple the area, and this should never happen in less than 4 inches of horn length. If it takes 4 inches to build up from 3/4 to 1 1/2 inches, it will require 4x(16 1/2  3/4)/(1 1/2  3/4) or 84 inches to reach 16 1/2 inches with a conical horn. This is a horn 7 feet long, to handle frequencies only down to 400 cycles per second. But it is only this long to get the correct rate of stretch near the throat. If we make a horn of a group of cones so that the diameter doubles every 4 inches, it will be: Distance from throat (inches) 0 4 8 12 16 20 Diameter (inches) % l*/2 3 6 12 24 The required diameter is reached between 16 and 20 inches from the throat. Even this shape is not ideal, because the rate of stretch is not quite uniform  it changes suddenly where the conical sections join. The best shape smooths out these joints, so that the stretch is uniform. This shape is called exponential.
If a horn is required only for frequencies above 800 cycles the rate of flare could be increased. It could double its diameter every 2 inches of length instead of every 4 inches. In addition, the mouth could be half the size.
From this we can see a pattern emerging: low frequencies need a big thing to radiate them successfully, but high frequencies can come from quite small objects or sources. Which explains why small birds chirp at higher frequencies, and only larger animals, like lions, can produce a deepthroated roar.


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