Mouth reflections (reposted from Thomas Dunker)

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Posted by Thomas Dunker on June 07, 2004 at 04:34:20
In Reply to: Re: Mouth Reflections etc... posted by Paul Eizik on June 01, 2004 at 16:40:53:

Hi Paul, everybody...
Could be that Jean-Michel is on vacation (I haven't had much time for e-mail the past few days, so haven't read the JoeNet mail either).
If I am to understand Jean-Michel's theory (and the resulting horns)
the way he expects it to be understood, it would be that the "design" of the horn mouth makes "all the difference" when it comes to reflections, and consequently if reflections at the mouth can be *eliminated* as a matter of design, there would be no requirement for
the axial length of the horn in terms of preventing reflections making it back to the throat, thus permitting a shorter horn. This all sounds
very nice in theory, but I wonder...
I know that Jean-Michel has devoted a lot of time to the study of
wavefront expansion and propagation in the horn. This is a very difficult subject (at least to me...) in terms of judging a horn design. Except in very rare cases, we don't get to *know* precisely how the waves actually propagate towards the mouth and where/how they
"let go" of the horn, and how this might differ greatly with the wavelength of the sound. For instance, there is no evidence that a round tractrix horn is capable of radiating hemispherical waves from the plane of the mouth, although the assumption of "spherical wave" radiation being unique to, and only possible with tractrix horns now seems to be something of a "proclaimed truth", and a very questionable one at that...
I am thinking of the papers by Newell, Holland and Fahey ("prediction and measurement of the one-parameter behavior of horns", and "round the horn"), where, in accordance with other experimental evidence, it is found that the wavefronts just outside the horn mouth typically assume the shape of "flattened spherical caps". This can be even
when the wavefronts have a considerable (greater) curvature just before "letting go" of the horn. Voigt assumed the wavefronts to have a constant radius of curvature from throat to mouth, and wavefronts being perpendicular to the inner horn walls, and this produced the tractrix curve. I've been listening to tractrix horns myself for ten years, and I'm not saying they "don't work", just that they are based on some assumptions that aren't necessarily all correct.
One of these assumptions appear to be that the wave "tears loose" at
the extreme end of the physical horn, i.e. in the plane defined by the
physical mouth perimeter, AND that the curvature of the wavefront at this point is defined by the "opening angle" of the mouth. If this were to be the case, and for all frequencies transmitted, very wide and constant (with frequency) dispersion from horns just wouldn't be a problem, and it would all simply depend on the horn mouth geometry.
This clearly is not the case. One reason for this, as Jean-Michel points out, is that the "mouth" is not actually a well defined point
along the horn axis. It would seem that for frequency independent constant dispersion, a starting point would have to be establishing
conditions for the "wavefront" having a constant curvature at the point where it is radiated into free space, and for this to happen at a well defined point near the horn mouth - for all frequencies/wavelengths in question. A further requirement would be that once the wave has "let go" of the horn it is allowed to propagate
with the same curvature as that defined by the horn, again, for all
frequencies/wavelengths within some predefined frequency band. These particular requirements would seem to be met in modern "wave guides" and constant directivity/constant dispersion horns, which are frequently less than ideal "horn loads" for the driver except at higher frequencies, more than anything often having properties similar to those of conical horns, typically restricting their use to higher frequencies (which is where wide dispersion is the most difficult to achieve).
Recently I have spent a great deal of time examining the design of the early wide range horns from Western Electric, a generation of horns that was made "obsolete" 70 years ago and clearly not very well understood except by their creators. Their exceptionally wide range response is primarily due to the use of a small compression driver being used down to unusually low frequencies (and well below the driver's resonant frequency as well). However, having very large mouths, how could these horns have reasonably good dispersion
as much as 5-6 octaves above cutoff? These horns would seem to embody
very careful design meeting tough challenges at both extremes of the
frequency band they were designed to cover. To avoid excessive diaphragm excursions at the lowest frequencies, the horn would have to have a very high resistive throat impedance, and low reactive impedance right down to the "minimum frequency". There is evidence that these horns were refered to as exponential horns, BUT that Edward Wente specified for the area expansion of the assumed *curved*
wavefronts to follow an exponential law, not merely that the horn would have an exponentially increasing cross section (assuming plane waves). Therefore, horns such as the 15A, 13A etc. must have a "modified exponential" expansion taking the wavefront curvature into account. They would therefore appear to expand more slowly than a "plane wave exponential horn", since the increasing curvature of the propagating wave contributes a term of area increase in addition to that provided by the increasing cross section of the horn.
Therefore, a "true exponential horn" is something other than a horn with exponentially increasing *cross section*. Its cross sectional expansion would seem to follow something more in the direction of a "hypex", which would also serve to maximise throat resistance and minimise throat reactance at extreme low frequencies approaching the
cutoff frequency. Considering that these horns were used as full range
speakers, the axial length did not have to be considered a big
problem, rather, I think the length was as "necessary" as every other
aspect of these horns. The most obvious problem with a long horn with a very slow initial expansion is harmonic distortion due to the nonlinear compression characteristic of air. But this distortion is
of a simple and predictable nature, unlike most of the distortion produced in the driver. Conceivably, the air compression induced harmonic distortion could be canceled out to a great extent by making it complement the distortion of a single ended triode output stage in the amplifier powering the driver, since the nonlinear characteristic of a triode is quite similar to that of air's nonlinear compression/rarefaction characteristic.
In this picture, dynamic nonlinearities and mechanisms responsible for intermodulation distortion would compromise the performance of the
driver/horn speaker much more than the simple and pure even harmonic
distortion introduced by the long throat, which in itself has a far higher subjective audibility threshold than odd harmonics and IMD.
The latter, odd harmonics and IMD primarily relates to nonlinear mechanisms in the driver, and increases with increasing diaphragm
excursions. Odd harmonics are symptomatic of symmetrical nonlinearities, such as that of the diaphragm suspension and to a varying degree, the magnetic field in the gap, which depending on pole piece geometry may be more or less symmetric. At any rate, an underhung voice coil has no business moving out of the gap and into
nonlinear fringes of the field. If this happens, the coil is too long and/or the excursions are too great.
Intermodulation distortion would seem to be among the worst problems
in a wide range single driver horn speaker, since all forms of IMD
increase with proportion to the upward bandwidth.
The most significant source of IMD in a compression driver is probably the variation of the air volume between diaphragm and phase plug. For LF applications this poses a challenge, that was elegantly solved in the WE 555 driver by making the diaphragm to phase plug clearance increase from the center of the diaphragm to the edge, making the change in volume smaller at large excursions than if the
clearance had been constant across the whole surface.
Well, I am digressing, but "everything relates to everything else"...
How then, about the length of these horns, in terms of desired wave propagation/dispersion AND avoiding severe throat impedance fluctuations at the lowest frequencies...? There is fortunately some
data available on the mouth dimensions and length of some of these
horns. The mouth circumference is typically 1.2-1.6 times the length of the horn. If we assume that the "mouth cutoff" is placed some way below the "useful cutoff" of the horn (where we'd place the high pass crossover freq. in a multiway system), we see that for this frequency, the horn is about as long, or slightly longer than the wavelength
of the lowest frequency transmitted.
An "oddity" with these horns is also the "flare reversal" at the point where the curled and therefore suitably flattened horn straightens out and the transition to the "mouth" is made. Here, there is a quite sudden increase in vertical wave expansion, whereas the gentle horizontal expansion continues without abrupt change. This causes wider dispersion in the vertical plane due to diffraction of the wave emanating from the narrow, flattened "throat". The exact
same principle has been used for decades in constant directivity horns, and before that, in "reverse flare" horns of various makes.
As with the CD horns, this feature makes it tempting to divide the horn into a "throat section" and a "mouth section".
In the WE horns, most of the horn's length is in the flattened "throat" part, and it would seem that even this length of the horn is made nearly equal to a wavelength at the lowest frequency (somewhere around 3-3.5m or close to a wavelength at 100Hz). Clearly, this constitutes a kind of wide band wave guide that prevents even the high frequency wave components from "leaving the horn" prematurely, and assures that all frequency components (wavelengths) undergo the same slow, controlled area expansion and give them a reasonably fixed
degree of curvature at the point where the vertical expansion begins to increase.
There's more to be considered, not least the effects of curving the horn upon the wave propagation, as the wavefronts are tilted due to unequal inner and outer horn wall lengths. If this is also taken into consideration when computing the horn for "true exponential wavefront area expansion" (it would have been!), and at the same time minimizing the effects of the bends upon the HF response, it is seen just how much work must have gone into designing these horns - in the pre-computer era!
Yes, these horns were unusually long and had a low initial flare rate due to their low cutoff frequency (and possibly a hyperbolic term in the expansion profile), large mouth and small throat, but it is quite easily seen that everything from bass horns to tweeters today quite typically are considerably shorter than the cutoff wavelength. Tractrix horns don't really have a "flare constant", they're computed from the mouth backwards, and the throat area is defined by the choice of driver, and for a tractrix horn with a given mouth size, the length
of the horn is therefore given by the throat dimensions. And in most other cases it is always made a big priority to keep horns as short as possible! Regardless of the cutoff frequency/wavelength?!?!?!?!
I am sure "horn sound" can be eliminated by removing "most of the horn", but that does give you less of an impedance transformer
and more of a direct radiator. Why did we we want horns in the first place?
All the classical horn theory is very clear on this: There is the theoretical infinite horn, in which no reflections take place and the
impedances taper off without any ripple. The infinite horn is infinitely long and has an infinitely large mouth, we learn.
Maybe it's only Leo Beranek who really "takes the bull by the horns"
in stating that:
"If the horn is a number of wavelengths long and if the mouth circumference is larger than the wavelength, we may call the horn "infinite" in length."
(L.L. Beranek, "Acoustics", 1954, page 269)
This says something about the *combination* of horn length and mouth size, but it is left for the the reader to interpret the meaning of this. I'm still thinking...
Thomas

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The Cable Cooker

Topic - Mouth reflections (reposted from Thomas Dunker) - JMLC 00:10:16 06/08/04 (5)

Re: Mouth reflections (reposted from Thomas Dunker) - JMLC 09:34:37 06/08/04 (3)

Re: Mouth reflections (reposted from Thomas Dunker) - Steve Schell 00:42:31 06/10/04 (2)

Re: Mouth reflections (reposted from Thomas Dunker) - JMLC 09:49:38 06/10/04 (1)

Re: Mouth reflections (reposted from Thomas Dunker) - Thomas Dunker 14:10:29 06/13/04 (0)

Re: Mouth reflections (reposted from Thomas Dunker) - Magnetar 07:08:09 06/08/04 (0)

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