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Sonic Design, Sweden:
"The Tractrix horn is designed to produce a hemispherical wave front
and to have less coloration than other horns. In October 5, 1927
P.G.A. Voight got the British patent 278,078 for a horn based on
the tractrix curve.
The following is an excerpt from his patent application:
"In a horn constructed according to this invention, the taper at any
point is as nearly as possible such that the sound is expanding as if
the source was at a fixed distance which is at least a quarter wavelength,
of the lowest frequency at which full efficiency is required, away".
And: "Since according to this invention, the rate of expansion shall be
as if the source was a fixed distance away, it follows that the length
of the tangent is equal to this distance. The property of this curve
is therefore that the length of tangents is constant.
At the point where the tangent is at right angles to the centre line,
it becomes the radius. I call this point the "mouth" and prefer either
to terminate the curve at this point, or to continue the plane (baffle)
at right angle to the centre line. The curve whose tangent to the centre
is of constant length is called the "Tractrix", and may be drawn by
drawing a small portion of one tangent after the other.
In the case of a square horn, the tangent to the corners is longer than
that to the middle of the sides and a compromise is unavoidable.
I prefer to make the section correspond to the tractrix.
The shortest tangent is then correct, but the area is 4/pi=1.27 times
that of the corresponding tractrix. If the area is made equal to the
corresponding tractrix, the tangent at the sides will be short,
a defect which is partly compensated for by the excess length of the
tangent to the corners."
"If the horn is assumed to have a horizontal partition along the centre
which is preferably continued for some distance in front of the mouth,
the sound will still expand perfectly in each half.
When the sound is not required to expand downwards, as for example in
a table gramophone [or a horn placed on the floor -- Sonic Design],
the lower half of the horn can be omitted and considerable space will
3. Designing tractrix horns with HornCalc.
HornCalc is based on this formula:
x = a * ln((a + sqrt(a^2 - r^2)) / r) - sqrt(a^2 - r^2)
x is the distance from the mouth of the horn,
a is the radius at the mouth, and
r is the radius at distance x from the mouth.
Different authors on loudspeaker horns have proposed different ideas
about approximating a square or rectangular horn shape to the tractrix
contour. P.G.A. Voight preferred the section height to equal the diameter
of the (round) tractrix. The area of a square horn is then 1.27 times
larger than the tractrix contour, and the circumference, to which some
authors pay great attention, will be even larger compared to the circular
circumference. To calculate from the circumference, on the other hand,
would yield a smaller area from square horns than the circular tractrix,
and since the square and rectangular shapes are compromises anyway,
it seems safe not to complicate things more than necessary.
Consequently HornCalc calculates the sectional area of square and
rectangular horns based on equal area as a circular horn of the selected
The cut-off frequency is determined by the size of the horn, and the
circumference of the mouth should be at least one wavelength of the
lowest reproduced frequency for a free field. The horn length should
be at least 1/4 wavelength of this frequency.
If you have input values that make the length shorter than 1/4 wavelength,
the calculated table will print a warning. But if you are to add it to
another horn, you may ignore the warning.
The cut off is calculated according to this and the formula;
where "c" is the speed of sound, 340m/s, and "r" is the radius of the
horn mouth for the fully expanded tractrix. This is in line with Bruce
Edgar's articles. You may note, however that P.G.A. Voight was of the
opinion that the length of the tangent from the horn flare to the centre
line should be a quarter wave of the lowest frequency at which full
efficiency isrequired. This tangent is a constant throughout the horn
and finally defines the opening radius. Thus, P.G.A. Voight can be claimed
to calculate: Fc=c/(4*r). His remark "at full efficiency" may suggest a
somewhat different definition of cut off than -3dB, and this perhaps could
explain the difference.
A ratio of at least one to five between the throat and the mouth area
should be achieved. If this and the length are considered, the mouth
area can be reduced to 1/2 for placement in one wall (baffle), to 1/4
for placement close to the floor and one wall, and to 1/8 if the opening
is placed in a corner. If the room walls are not heavy enough, their
supporting effect may be lost and a larger mouth may be necessary.
One cannot simply choose a smaller mouth to get the reduction of size,
since the contour would not become right in that case.
A full-size horn is the base for the calculation of the reduced horn,
and the reduction is done by HornCalc by dividing the full tractrix horn
section into half, a quarter or an eighth, and, as Voight said about the
horn for half space, "The sound will still expand perfectly in each half".
For midrange and tweeter horns, many of the tractrix horn advantages are
lost if the horn curve is terminated before the full, right-angle turn is
reached. Unwanted edge diffractions would be created by the sharp edges
of such a mouth. Thus no provision is made for the reduction of circular
The produced table of the calculations for circular horns has an
interesting feature in the angle column. If you would like to add a
tractrix mouth to an existing horn or compression driver with a known
or measured opening angle, it is possible to make the start angle fit.
Set the throat diameter to fit the preceeding horn, and try different
cut-off frequencies until you get the desired start angle.
For rectangular horns it is possible to select two flat sides if you would
like the horn to be easier to build. If you make the distance between
sides 1A and 1B equal at both the throat and the mouth, these sides will
become parallel -- this can be useful when designing folded horns.
The two flat sides may be used to design a horn for constant directivity
with the desired projection angle in one plane.
The produced table, when calculating rectangular horns with two flat sides,
has a column with values for the distance along the horn measured on the
flat side. This will greatly simplify the construction of the horn sides." Zitat Ende