An exponential midrange horn? You must be kidding! Tractrix is what is used in the midrange, or else you get 'horn sound'!
Why exponential horns in the midrange?
Bruce Edgar has said, and it is common knowledge in the DIY community, that exponential and hypex horns work in the bass range, but tractrix and LeCléa'h horns works better in the midrange. Why is that? Lets look at the differences between the way horns are used in these two different ranges, and the difference between expo/hypex and tractrix/LeCléac'h.
Bass horns are usually shortened, depending on corner placement, wall-floor or wall placement, corner placement beeing the most common, as this gives the shortest horn.
Midrange horns are taken to the full mouth size, as they radiate in free space.
Exponential and hypex horns are usually calculated from the given formulas. These formulas are based on the assumption that the wave fronts in the horns are plane. They are not. The larger the angle between the horn wall and the axis, the more the wavefront curves. But bass horns are shortened at a point where the wavefront curvature is comparably small, and thus the horn contour calculated for a plane wave is not very far from the real contour experienced by the travelling wavefronts. But when it comes to the midrange and full size mouths, the curving of the wavefronts is large, and the difference between the real and theoretical expansion is much greater. So what we really have then is not an exponential horn, but something different.
Tractrix type horns are calculated assuming spherical wavefronts, and considering this, they are an approximation to an exponential expansion of the wavefronts. With full size horns this is a closer approximation to how the horn works in practice, even if it is still not perfect. Real wavefronts in horns are neither plane nor spherical, but something in between ([1],[3],[3]).
Exponential expansion of the wavefront is what we want, and we want the horn to be shaped in a way that allows this.
A test horn
To test this, and also to gain more experience in how horns work and how things are in practice, I started to build a test horn. The horisontal and vertical expansion is shown to the left. This is a simple version of the Reciprocal Flare geometry described by Abraham Cohen [4], using conical expansion in one plane.
Simulations, first version
The horn is designed as a pure plane wave exponential horn matching an Altec 288B compression driver. The first sement of this horn is the part from the 288B phasing plug to the point where the horn is attached, and has a flare rate of 207Hz. From there comes a 200Hz segment extending to a horn mouth with an area of 2354cm2. Total horn length from the phasing plug to the mouth, is 82 cm. The throat impedance for the horn, simulated with spherical wavefronts, is shown to the left. It is not very pretty, but it will be interesting to see how it compares to the measured impedance.
A Hornresp simulation of the on-axis frequency response is also shown. Compare this to the measured frequency response further down the page. The agreement is quite good, even considering that Hornresp simulates axisymmetric horns, and this is a rectangular horn with a wide mouth.
The first prototype
A prototype was buildt from scrap plywood. The drawing of the horn was exported to AutoCAD, and templates for the straight sides were printed out in scale 1:1.
As soon as it was ready, it was hooked up for a listening test. The colorations were obivious, no doubt about that. But there was still something about the sound...
It sounded better in many ways to the short tractrix horns I have now, and with appropriate corrections for the wavefront curving, I believe it will become a good horn.
Measurements
The horn was taken outdoors, and some measurements conducted. Freqency response measurements were taken by ARTA software.
Frequency response
On axis frequency response is fairly smooth from 600hz to 5kHz.
Harmonic distortion vs frequency and level
Distortion is measured at 90, 100 and 110dB SPL @ 1m. The performance is quite good, so at normal listening levels distortion should not be a problem. I'm not yet sure how much of this distortion is from the measuring microphone. The uneven appearance of the distortion curves may be because of the directivity of the horn.
Intermodulation distortion
Intermodulation distortion between 510Hz and 4.4kHz at 100dB SPL @ 1m is shown. This is the level and frequencies used by Klipsch in his measurements, and shows about the same level of distortion.
Throat Impedance
Measuring the acoustic impedance of a horn is not what most amateurs do. Many don't realize the significance of this parameter, but it is vital in many ways. The acoustic impedance at the throat of a horn is the load for the driver. It is in the resistive part of this impedance that the power radiated as sound is dissipated. Variations in throat impedance results in variations in sound pressure, and ripple in this impedance indicates standing waves and transient oscillations in the horn.
As measuring SW for audio is now readily available (see ARTA link above) and computing power is cheap, impulse measurement of throat impedance is now quite simple. It requires some hardware, as can be seen in the picture to the left, but this is simple harware: A tube the same inner diameter as the horn throat, a compression driver, mounting hardware for driver and horn, a microphone placed in a hole drilled in the tube about 2/3 of the tube length from the driver, a microphone preamplifier and a power amplifier.
The measurement is simple. In short the driver sends a pulse through the tube. The pulse is captured by the microphone on it's way to the horn. Since the troat impedance doesn't match the tube impedance at all frequencies, some of the pulse will be reflected, and captured by the mic again. The spectrum of this pulse will contain the frequencies that gets reflected. The impulse response is then stored, the two pulses are separated digitally and FFT'ed, and the spectrum of the second pulse is divided by the spectrum of the first. We then have the reflection coefficient, and from this We can calculate the normalized acoustic impedance. For full description of the method, see [5],[6].
Measured and simulated acoustic impedance at the throat of the horn is shown to the left. They agree quite well, indicating that the software I have written for simulation, geometrical design and measurement of horns works pretty well.
The spikes at 800 and 1120Hz may be because of the abrupt change from circular to rectangular crossection. Response higher than 4500Hz is not shown, because in this range there are higher order modes (cross reflections) in the tube, giving wild, unpredictable variations in the measured throat impedance.
The significance of leaks in the horn was evident when measuring the horn. Here is a measurement of the first part of the horn with leaks, and here is a measurement of the same part without leaks, both compared to the measured values. As can be seen, the difference is quite large, and even if the throat impedance is close to the asymptotic value at higher frequencies, low frequency loading is compromised.
A throat adapter
A throat adapter was build to get a smooth transition from circular to square crossection, but it was not entirely successful. It was build by cutting holes in plywood and laminating it, but the thickness of the plywood was not the same as that used in the calculations of the adapter, so it's flare rate became higher. The first prototype was tested with this adapter, the frequency response is shown to the left. The frequency response is smoother, the bump around 2-4kHz is gone.
Also shown is the simulated and measured throat impedance. Mouth termination is a piston in the end of a long tube.
A Klangfilm mouth
The Klangfim expansion is exponential, but assuming sperical wavefronts of constant radius of one wavelength. It is similar to Tractrix, but not quite, and allows a rolled-back mouth. The flare rate of the klangfilm mouth segment is 200Hz, and the full mouth size (mouth tangent normal to horn axis) is quite a bit bigger than a full normal exponential mouth. The circumference of a circular klangfilm mouth is 1.32 cutoff wavelengths. Mouth termination is simulated as a sphere with the same area as the mouth.
But as can be seen from both frequency response and throat impedance, the low frequency loading is comparable to the other exponential horn. Compared to the other horn, there is also a sligth dip in the mid frequency range.
References
[1] "An investigation of Sound Fields in Regions Restricted by Finite Boundaries"; MS Thesis by William M. Hall; MIT 1928
[2] "Comments on the Theory of Horns; William M. Hall"; JASA 1932 pp. 552-561
[3] "Acoustic Radiation of a Horn Loudspeaker By the Finite Elements Method"; AES Preprint no 1756; N. Kyouno, S. Sakai and S. Morita; 1981
[4] "Wide Angle Dispersion of High-Frequency Sound"; Abraham B. Cohen; Audio Engineering Dec. 1952 pp. 24-25, 57-59
[5] "Measurements of the input impedance of loud speaker horns"; AES Preprint no 1587; Thomas Salava, 1980
[6] "Measurements of the input impedance of loud speaker horns"; Thomas Salava; JAES June 1981 pp. 416-420