# Horn Math

kindly provided by W.H.Geiger.

## 1) Front Cavity

At low frequencies the body of air in the front chamber behaves as an incompressible fluid and moves into an out of the horn neck as a unit mass. At higher frequencies, less air movement occurs as its compliance becomes dominant. The front chamber air volume is typically minimized to increase horn bandwidth. In fact, it can be chosen to extend horn response to cover a 4+-octive span. A phase plug, that follows diaphragm contour, is used to reduce the volume of the front chamber.

## 2) Horn Throat Impedance [Zat]

Analysis of horn throat impedance, to be tractable, requires evaluation of the non-dissipative case where no back-wave propagation is present because a horn of infinite extent is presumed. For Salmon horns, the following evaluation is provided: [Zat] = [Rat] + [i]*[Xat] [1] Note that the characteristic impedance of air, [p0]*[c] = 407 N*s/(m^3). [2] a) Throat Acoustical Resistance (N*s)/(m^5) [Rat] = [A]*([p0]*[c])/[St] [3] The frequency dependence of acoustical throat resistance may be characterized by the coefficient: [A] = {[u]*(([u]^2 - 1)^(1/2))}/{([u]^2) + ([T]^2) - 1} [4] For [T] => 2^(-1/2), max([A]) <= 1 for [u] -> [oo] For [T] < 2^(-1/2), max([A]) > 1 for [u] -> 1 b) Throat Reactance (N*s)/(m^5) [Xat] = [B] * ([p0]*[c])/[St] [5] The frequency dependence of acoustical throat reactance may be characterized by the coefficient: [B] = {[T]*[u]}/{([u]^2)+([T]^2)-1}. [6] For [T] > 1 max([B]) < 1 (Bessel horn) For [T] = 1 max([B]) =1 ([u] = 1) (exponential horn) For [T] < 1 max([B]) < 1 ([u] -> 1) (hyperbolic horn, reactance annulling value) 3) Throat impedance may be modeled by an equivalent mechanical or acoustical circuit consisting of resistance c) or d) and shunt inductance a) or b) where

a) Throat Air Load Mechanical Mass (kg)

[Mmt] = [p0]*[c]*[St]/(2*[pi]*[fc]*[T]) [7] b) Throat Air Load Acoustical Mass (kg/(m^4)) [Mat] = [Mmt]/([St]^2) [8] c) Throat Mechanical Resistance (N*s/m) [Rmt] = [p0]*[c]*[u]*[St]/{(([u]^2) \x{2013}1)^(1/2)} [9] d) Throat Acoustical Resistance [Rat] = [Rmt]/([St]^2) [10] 4) For reactance annulling, the following equality is satisfied: 2*[pi]*[fc]*[Mat] = ([p0]*[c])/([T]*[St]) = [p0]*([c]^2){[Vas]*[Vab])/([Vas] + [Vab])} [11] Thus [T] = {(2*[pi]*[fc])/([c]*[St])}*{([Vas]*[Vab])/([Vas]+[Vab])} [12]

### Legend:

[i] = (-1)^(1/2) [pi] = 3.14159 [T] - Salmons Horn Shape Factor [u] = [f]/[fc] (Frequency Ratio) [f] - Frequency of Interest (Hz) [fc] - Horn Cut-Off Frequency (Hz) [Vas] - Driver Equivalent Air Volume Compliance (m^3) [Vab] - Back-Box Effective Air Volume (m^3) [St] - Throat Cross-Section Area (m^2) [p0] - Air Density (kg/(m^3)) [c] - Sound Velocity (m/s) [Qts] = [Qms]*[Qes]/([Qms]+[Qes]) - Total Driver Damping [Qms] = 1/[Rms]*([Mms]/[Cms])^(1/2) - Mechanical Damping Component [Qes] = {[Re]/([B]*[l])^2}*([Mms]/[Cms])^(1/2) - Electrical Damping Component [Mms] = [Mmd] + 2* [Mm1] - Mechanical Mass of Moving Elements, Plus Air Load Upon Them (kg) [Cms] = [Vas]/{[p0]*([c]^2)*([Sd])^2)} - Mechanical Compliance of the Moving Element Suspension (m/N) [fs] = 1/{2*[pi]*([Mms]*[Cms])^(1/2)} - Resonant Frequency as a Direct Result of [Mms] and [Cms] (Hz) [Mm1] = 8*([a]^3)*[p0] - The Air Load on One Side of Driver Diaphragm (Infinite Baffle Assumed) (kg) Note: The enclosure used will change this for at least one side of the driver diaphragn. For a horn, both sides. [Sd] = [pi]*([a]^2) - Effective Radiating Area of Moving Elements (m^2) [p0] = 2.18 kg/(m^3) - Density of Air [c] = 345 m/s - Velocity of Sound in Air (m/s) Note theses are under "standard conditions". For measurements in situ, adjust these constants according to local conditions of ambient temperature, barometric pressure and relative humidity.

### Legend

[Vas] - Volume of Air Exhibiting (when compressed) an Elasticity Equivalent to that of the Driver Suspension) (m^3) [a] - Effective Piston Radius of the Moving Elements (m)

## Horn math Q&A

Q1) Who can tell me what the difference between Tractrix- and Exponential horns in regards to acoustic performance is? People seem to be biased for one or the other.

A1a) The bias comes from blind men describing an elephant while fondling its appendages. Neither horn shape is intrinsically superior. In fact, other issues are far more important in horn design than the arbitrary selection of a horn shape. For example, these include, use of a phase plug, folding the horn path, driver selection, and use of room corners.

A1b) Typical comparisons of the two horn flares are unfortunate as they include the assertion that a shorter horn may be achieved for the tractrix profile that is "equivalent" to a longer exponential counterpart. This assertion is patently false because the horns are not acoustically equivalent and the exponential flare is only one of many alternatives that should be included is such an evaluation. In fact, the shorter tractrix design produces a sub-optimal variant to what is possible. The benefit of the tractrix flare is that the mouth perimeter may be seamlessly joined to a flat baffle. Even in the case where such a baffle is not used, an abrupt mouth termination is avoided. This leads to reduced mouth reflectance, and commensurate reduction in the amplitude of back-waves returning to impinge on the driver diaphragm. The detraction is, that horn flare parameters are set by mouth radius only and typically determined by setting the product

[kc]*[Rm]=1. First, [kc]*[Rm]>1 is preferred. Second, and of equal importance, when set, it fixes of tangent angle of the flare at the throat aperture. In cases where a compression driver is used, the flare angle should match that of the driver exit. In this case, mouth size is then fixed by this match as demonstrated by the flare derivative: d[Rs]/d[Ls] = -tan (ts) = -[Rs]/{([Rm]^2-[Rs]^2)^((1/2)} rearranging we get, [Rm] = [Rs]*csc(ts) setting [Rs]=[Rt] for driver throat radius [ts]=[tt] for driver flare tangent angle at [Rt] then [Rm] = [Rt]*csc(tt) Bottom line: the tractrix flare is useful for designing the bells of mid and high frequency horn mouths. For the design of horn necks as well as entire low frequency horns, its use is contraindicated. Alternatively, use of a Salmon family horn flare (that includes exponential flare) is preferred. By setting, tangent angles equal at the junction of the Salmon horn neck and tractrix bell, a near ideal mid- or high-frequency horn design may be achieved provided other design issues have been properly and successfully addressed.

Note that wave number

[kc] = (2*[pi]*[fc])/[c] where [fc] - Horn (Mouth) Cut-Off Frequency [c] - Sound Velocity [Rm] = Mouth Radius

Q2) What are the differences between front loaded and back loaded horns, shouldn\x{2019}t a front loaded one be better due to just one source emitting sound (ideally)?

A2) Back loaded horns do not have a back-box. Typically, they are used to extend the low frequency response of a not so "full-range", "full-range" driver. Note that a front loaded horn, like all horns is a band-pass device. The neighborhood of one decade to 4-octives is the horn frequency limit.

Q3) How would I go about front AND back loaded horn designs?
A3) To start, see Olson (1) for details.

Q4) What is preferable, back loaded horns of half a wavelength with mouth opening to the front or back?

A4) For a low frequency horn, mouth designed to work out of a room corner is preferred. Higher frequency horns, the mouth should be far away from the corners. If placed there, line adjoining walls with an acoustical material that suppresses the "early" reflections.

Q5) What about mouth pointing sideways, how does that work out in regards to directivity?

A5) Directivity is fundamentally determined by horn neck geometry including and the aspect ratio of its section and the projected size of the driver diaphragm (as seen through the phase plug aperture).

## Bibliography

### (1) Olson Reference

Title: A Compound Horn Loudspeaker Author: Harry F. Olson Author: Frank Massa Publication: ASA-J, Vol. 8, No. , p. 48-52, (Jul-1936) Abstract: A new type of loudspeaker is described in which a single mechanism is coupled to two horns: a straight axis high frequency horn and a folded low frequency horn. A theoretical analysis of the combined system is given and experimental data are shown which indicate smooth uniform response from 50 to 9000 cycles, and an efficiency of the order 50 percent over a large portion of this range.