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.

(2) Tractrix Horn References

Title: The Edgar Midrange Horn Author: Bruce C. Edgar, 11 pp. Publication: Speaker Builder Magazine, Jan-1986, pg. 11 Abstract: This article is about designing and building a midrange acoustical horn using a tractrix flare. It provides construction details essential to the successful completion of such an undertaking. Title: Acoustical Studies of the Tractrix Horn. I Author: Robert F. Lambert Publication: ASA-J, Vol. 26, No. 6, Pg. 1024-1033, Nov-1954 URL: none Abstract: When predicting and comparing the acoustical properties of horns it is customary practice to formulate the propagation as a one-parameter plane wave front problem. However, when particular attention is paid to the rapid flare near the mouth of a horn structure such as the tractrix, it also seems plausible to formulate the propagation based on a one-parameter spherical wave front theory. Abstract: By visualizing the surfaces of constant phase as spheres of constant radii a and the flow lines as tractrixes having a generating arm of length d, a one-parameter wave equation and Ricatti impedance equation may be derived. Solutions to these equations have been obtained by wave perturbation and by analog computer techniques. Abstract: Axial response and throat impedance measurements are compared with theoretical calculations postulating first a hemispherical and then a plane piston radiation pattern. It appears that the most satisfactory explanation lies somewhere in between these two limiting cases. Title: Acoustical Studies of the Tractrix Horn. II Author: Robert F. Lambert Publication: ASA-J, Vol. 26, No. 6, p.1024-1033, Nov-1954 URL: none Abstract: Experimental investigations have been carried out on the tractrix horn structure to determine its "free-field" radiation characteristics. Axial, off axis, and polar response characteristics, as well as throat impedance data on a single cell horn, are presented for both small and large baffle mounting. Pertinent data on a two-cell structure are also presented. These data show the tractrix performance to be comparable with that of the well known exponential horn. Abstract: A multi-cellular structure, while showing definite improvement in uniformity of angular distribution at high frequencies, exhibits undesirable hand rejection characteristics within the useful frequency range of the horn. Title: A Modeling and Measurement Study of Acoustic Horns Author: Post, John Theodore Publication: Thesis (Ph.D.)--The University Of Texas at Austin, 1994. Source: Dissertation Abstracts International, Vol. 55-06, Sec. B, Pg. 2246, UMI Co. Abstract: Although acoustic horns have been in use for thousands of years, formal horn design only began approximately 80 years ago with the pioneering effort of A. G. Webster. In this dissertation, the improvements to Webster's original horn model are reviewed and the lack of analytical progress since Webster is noted. In an attempt to augment the traditional methods of analysis, a semi-analytical technique presented by Rayleigh is extended. Although Rayleigh's method is not based on one-dimensional wave propagation, it is found not to offer significant improvement over Webster's model. In order to be free of the limitations associated with analytical techniques, a numerical method based on boundary elements has been developed. It is suitable for solving radiation problems that can be modeled as a source in an infinite baffle. The exterior boundary element formulation is exchanged for an interior formulation by placing a hemisphere over the baffled source and using an analytical expansion of the field in the exterior half space. The boundary element method is demonstrated by solving the baffled piston problem, and is then used to obtain the acoustic throat impedance and far field directivity of axisymmetric horns having exponential and tractrix contours. Experiments are performed to measure the throat impedance and the far field directivity of two axisymmetric horns mounted in a rigid baffle. An exponential horn and a tractrix horn with equal throat radius (2.54 cm), length (55.9 cm), and mouth radius (21.1 cm) are critically examined. A modern implementation of the "reaction on the source" method is compared with a new implementation of the two-microphone method for measuring acoustic impedance. The modified two-microphone method is found to be extremely simple and accurate, but the "reaction on the source" method has the advantage of in situ measurements. The far field directivity is measured by a new technique that allows the far field pressure to be calculated from the measured near field pressure. Experimental results compare very well with the numerical predictions obtained by the boundary element method. The annotated bibliography is 34 pages in length and features approximately 200 references that are useful in the general study of acoustic horns.

Excel/VBA UDF Code

1Q)>>An Excel/VBA inverse tractrix function will be available shortly. >??? What is that good for?

1A) For the tractrix curve (flare contour): where

[L] - axial length and [R] - cross-section radius, f(R)= L has an analytic solution f(L) = R, (the inverse function) has no analytic solution; it must solved numerically (programmatically). See below for the code (sans indents eaten by Unix). Use of the latter function for laminar construction should be obvious.

'Title: Horn Function Module 'Author: William H. Geiger 'Date: 4-Feb-2004 'Release: Beta (B1.0) ' 'Return Axial Length of Tractrix Public Function TrctrxL(a As Double, r As Double) As Double 'open function protocol Dim ar As Double 'for avoiding redundant calculation On Error GoTo RtnErr: 'set trap for unexpected errors If a > 0 And r > 0 Or r <= a Then 'valid arguments, so ar = Sqr(a * a - r * r) 'avoid redundant calculation TrctrxL = a * Log((a + ar) / r) - ar 'calculate and return length Exit Function ' return to spreadsheet End If 'invalid argument(s) remain, so RtnErr: 'error trap TrctrxL = CVErr(xlErrValue) 'pass error value Exit Function 'return to Excel End Function 'close function ' 'Return Tractrix Radius (Inverse Function) Public Function _ TrctrxR(a As Double, r As Double, l As Double, p As Integer) As Double '[a] - mouth radius '[r] - section radius estimate, ' set r=a, r=0 or to the value of a previously calculated radius '[l] - length from horn mouth where [rl] is to be calculated '[p] - precision on length match 1-15 significant digits Dim i As Integer 'loop counter Dim rl As Double 'current radius solution Dim r0 As Double 'new radius bound Dim r1 As Double 'radius lower solution bound Dim r2 As Double 'radius upper solution bound Dim aa As Double '=a*a (avoid redundant calculation) Dim ar As Double '=Sqr(aa-r*r) (avoid redundant calculation) Dim rr As Double '=rl/ar (avoid redundant calculation) Dim lr As Double 'current radius position Dim l0 As Double 'length from horn mouth (positive) Dim lu As Double 'length upper bound (tolerance) Dim ll As Double 'length lower bound (tolerance) Dim ld As Double 'length precision variance Const mxi As Integer = 100 'loop limit On Error GoTo RtnErr: 'set trap for unexpected errors If a > 0# And p > 0 And p < 16 Then '[a] and [p] valid If l > 0# Then 'positive length l0 = l 'set positive length ElseIf l < 0# Then 'negative length l0 = -l 'make length positive Else 'l=0, so at mouth, so rl = a 'set [rl] to mouth radius GoTo RtnVal: 'go return radius value End If 'On [l] ld = l0 * 1# / (10 ^ p) 'scale length variance lu = l0 + ld 'set length upper bound ll = l0 - ld 'set length lower bound If r > 0# And r < a Then 'valid radius values rl = r 'set [rl] to argument value Else 'set [rl] arbitrarily rl = a / 2 'radius estimate End If 'on [r] aa = a * a 'avoid redundant calculation r1 = 0 'set lower radius bound r2 = a 'set upper radius bound Else 'a=<0 Or p<1 Or p>15 invalid limit radius or precision parameter GoTo RtnErr: 'go error return End If 'on [a] and [p] (continue for valid arguments) 'Converge on Radius Value Loop (using derivative and bisection) For i = 1 To mxi 'start loop ar = Sqr(aa - rl * rl) \x{2019}avoid redundant calculation rr = rl / ar \x{2018}calculate derivative value lr = a * Log((a + ar) / rl) - ar \x{2018}calculate axial length for [rl] r0 = rl + rr * lr - rr * l0 'calculate new radius boundary If lr < ll Then 'radius too large and length too short r2 = rl 'update upper bound r1 = r0 'update lower bound ElseIf lr > lu Then 'radius too small and length too long r1 = rl 'update lower bound r2 = r0 'update upper bound Else '[lr] within precision bounds, so Exit For 'with radius solution in [lr] End If 'on [lr] rl = (r1 + r2) / 2 'calculate new radius estimate Next i 'loop RtnVal: 'Return Radius Value TrctrxR = rl 'pass radius value Exit Function 'return to Excel RtnErr: 'error trap TrctrxR = CVErr(xlErrValue) 'pass error value Exit Function 'return to Excel End Function 'close function