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:
<![CDATA[
[Zat] = [Rat] + [i]*[Xat] [1]
]]>
Note that the characteristic impedance of air,
<![CDATA[
[p0]*[c] = 407 N*s/(m^3). [2]
]]>
a) Throat Acoustical Resistance (N*s)/(m^5)
<![CDATA[
[Rat] = [A]*([p0]*[c])/[St] [3]
]]>
The frequency dependence of acoustical throat resistance may be characterized by the coefficient:
<![CDATA[
[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)
<![CDATA[
[Xat] = [B] * ([p0]*[c])/[St] [5]
]]>
The frequency dependence of acoustical throat reactance may be characterized by the coefficient:
<![CDATA[
[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)
<![CDATA[
[Mmt] = [p0]*[c]*[St]/(2*[pi]*[fc]*[T]) [7]
]]>
b) Throat Air Load Acoustical Mass (kg/(m^4))
<![CDATA[
[Mat] = [Mmt]/([St]^2) [8]
]]>
c) Throat Mechanical Resistance (N*s/m)
<![CDATA[
[Rmt] = [p0]*[c]*[u]*[St]/{(([u]^2) \x{2013}1)^(1/2)} [9]
]]>
d) Throat Acoustical Resistance
<![CDATA[
[Rat] = [Rmt]/([St]^2) [10]
]]>
4) For reactance annulling, the following equality is satisfied:
<![CDATA[
2*[pi]*[fc]*[Mat] = ([p0]*[c])/([T]*[St]) = [p0]*([c]^2){[Vas]*[Vab])/([Vas] + [Vab])} [11]
]]>
Thus
<![CDATA[
[T] = {(2*[pi]*[fc])/([c]*[St])}*{([Vas]*[Vab])/([Vas]+[Vab])} [12]
]]>
Legend:
<![CDATA[
[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.
<![CDATA[
[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
<![CDATA[
[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
<![CDATA[
[kc]*[Rm]=1.
]]>
First,
<![CDATA[
[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:
<![CDATA[
d[Rs]/d[Ls] = -tan (ts) = -[Rs]/{([Rm]^2-[Rs]^2)^((1/2)}
]]>
rearranging we get,
<![CDATA[
[Rm] = [Rs]*csc(ts)
]]>
setting
<![CDATA[
[Rs]=[Rt] for driver throat radius
[ts]=[tt] for driver flare tangent angle at [Rt]
]]>
then
<![CDATA[
[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
<![CDATA[
[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
<![CDATA[
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
<![CDATA[
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
<![CDATA[
[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.
<![CDATA[
'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
]]>