Horn Theory: an Introduction, Part 1 by Bjørn Kolbrek

Tube, Solid State, Loudspeaker Technology Article prepared for www.audioXpress.com Horn Theory: An Introduction, Part 1 By Bjørn Kolbrek This author presents a two-part introduction to horns—their definition, features, types, and functions. ρ 3 his article deals with the theory of acous- 0: Density of air, 1.205 kg/m . there will be a considerable mismatch tical horns, as it applies to loudspeakers. f: Frequency, Hz. between the source and the load. The It reviews the basic assumptions behind ω: Angular frequency, radians/s, ω = 2πf. result is that most of the energy put into Tclassical horn theory as it stands, presents k: Wave number or spatial frequency, a direct radiating loudspeaker will not the different types of horns, and discusses their ω2πf reach the air, but will be converted to = properties. Directivity control, wave-front shapes, radians/m, k = cc. heat in the voice coil and mechanical and distortion are also discussed. resistances in the unit. The problem is In this article, I try to keep the math S: Area. worse at low frequencies, where the size simple, and, where it is required, I explain p: Pressure. of the source will be small compared to or illustrate the meaning of the equations. ZA: Acoustical impedance. a wavelength and the source will merely The focus is on understanding what is j: Imaginary operator, j = √-1. push the medium away. At higher fre- going on in a horn. The practical aspects quencies, the radiation from the source of horn design are not treated here. THE PURPOSE OF A HORN will be in the form of plane waves that It can be useful to look at the purpose do not spread out. The load, as seen from TERMINOLOGY of the horn before looking at the theory. the driver, is at its highest, and the system The article includes the following termi- Where are horns used, and for what? is as efficient as it can be. nology: Throughout the history of electroa- If you could make the driver radiate Impedance: Quantity impeding or re- coustics, there have been two important plane waves in its entire operating range, ducing flow of energy. Can be electrical, aspects: efficient operation would be secured at all mechanical, or acoustical. • Loading of the driver frequencies. The driver would work into Acoustical Impedance: The ratio of • Directivity control a constant load, and if this load could be sound pressure to volume velocity of air. You would also think that increasing the made to match the impedances of the In a horn, the acoustical impedance will output would be one aspect of horns, but driver, resonances would be suppressed. increase when the cross-section of the this is included in both. Increasing the This is because the driver is a mechani- horn decreases, as a decrease in cross sec- loading of the driver over that of free air cal filter, which needs to be terminated tion will limit the flow of air at a certain increases efficiency and hence the out- in its characteristic impedance, ideally a pressure. put, and concentrating the sound into a pure resistance. If the driver is allowed Volume Velocity: Flow of air through a certain solid angle increases the output to radiate plane waves, resistive loading surface in m3/s, equals particle velocity further. is secured. times area. Loading of the Driver. The loudspeaker, The easiest way to make the driver Throat: The small end of the horn, which is a generator of pressure, has an radiate plane waves is to connect it to a where the driver is attached. internal source impedance and drives an long, uniform tube. But the end of the Mouth: The far end of the horn, which external load impedance. The air is the tube will still be small compared to a radiates into the air. ultimate load, and the impedance of air is wavelength at low frequencies. To avoid Driver: Loudspeaker unit used for driv- low, because of its low density. reflections, the cross section of the tube ing the horn. The source impedance of any loud- must be large compared to a wavelength, c: The speed of sound, 344m/s at 20° C. speaker, on the other hand, is high, so but, at the same time, it must also be audioXpress 2008 1 small to fit the driver and present the Initially, it is a three-dimensional prob- is insignificant. required load. To solve this dilemma, you lem, but solving the wave equation in 3D 2. The medium is considered to be a need to taper the tube. is very complicated in all but the most uniform fluid. This is not the case with When you do this, you can take radia- elementary cases. The wave equation for air, but is permissible at the levels (see tion from the driver in the form of plane three dimensions (in Cartesian coordi- 1) and frequencies involved. waves and transform the high pressure, nates) looks like this2 3. Viscosity and friction are neglected. low velocity vibrations at the throat into ∂φ22 ∂φ∂φ∂φ22 The equations involving these quanti- −++=c2 0 2222 low pressure, high velocity vibrations that ∂∂∂∂txyz(1) ties are not easily solved in the case of can efficiently be radiated into the air. horns. Depending on how the tube flares, it is This equation describes how sound waves 4. No external forces, such as gravity, act possible to present a load to the driver that of very small (infinitesimal) amplitudes on the medium. is constant over a large frequency range. behave in a three-dimensional medium. 5. The motion is assumed irrotational. Directivity Control. The directivity of I will not discuss this equation, but only 6. The walls of the horn are perfectly a cone or dome diaphragm is largely un- note that it is not easily solved in the case rigid and smooth. controlled, dictated by the dimensions of horns. 7. The pressure is uniform over the of the diaphragm, and heavily depen- In 1919, Webster3 presented a solu- wave-front. Webster originally consid- dent on frequency, becoming sharper tion to the problem by simplifying equa- ered tubes of infinitesimal cross-section, and sharper as frequency increases. You tion 1 from a three-dimensional to a and in this case propagation is by plane can solve this problem by using multiple one-dimensional problem. He did this waves. The horn equation does not re- driving units and digital signal process- by assuming that the sound energy was quire plane waves, as is often assumed. ing, but a far simpler and cheaper way to uniformly distributed over a plane wave- But it requires the wave-front to be a achieve predictable directivity control is front perpendicular to the horn axis, and function of x alone. This, in turn, means to use a horn. The walls of the horn will considering only motion in the axial di- that the center of curvature of the wave- restrict the spreading of the sound waves, rection. The result of these simplications fronts must not change. If this is the so that sound can be focused into the is the so-called “Webster’s Horn Equa- case, the horn is said to admit one-pa- areas where it is needed, and kept out of tion,” which can be solved for a large rameter (1P) waves6, and according to areas where it is not. number of cases: Putland7, the only horns that admit 1P Directivity control is most important d2φ d lnS dφ waves are the uniform tube, the para- + −k2 φ=0 in sound reinforcement systems, where a dx dx dx (2) bolic horn with cylindrical wave-fronts, large audience should have the same dis- and the conical horn. For other horns, tribution of low and high frequencies, and where you need the horn radius to be small where reverberation and reflections can be 2πf compared to the wavelength. a problem. In a studio or home environ- k = c , the wave number or spatial fre- ment, this is not as big a problem. quency (radians per meter), Because the horn equation is not able to As the art and science of electroacous- φ is the velocity potential (see appendix predict the interior and exterior sound tics has developed, the focus has changed for explanation), and S is the cross-sec- field for horns other than true 1P horns, from loading to directivity control. Most tional area of the horn as a function of x. it has been much criticized. It has, how- modern horns offer directivity control at ever, been shown8, 9 that the approxima- the expense of driver loading, often pre- The derivation of equation 2 is given tion is not as bad as you might think in senting the driver with a load full of reso- in the appendix. You can use this equa- the first instance. Holland10 has shown nances and reflections. Figure 1 compares tion to predict what is going on inside that you can predict the performance of the throat impedance of a typical constant a horn, neglecting higher order effects, horns of arbitrary shape by considering directivity horn (dashed line) with the but it can’t say anything about what is the wave-front area expansion instead of throat impedance of a tractrix horn (solid going on outside the horn, so it can’t the physical cross-section of the horn. I line)1. The irregularities above 8kHz come predict directivity. Here are the assump- have also developed software based on from higher order modes. tions equation 2 is based on4,5: the same principles, and have been able 1.

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