Query re. Conical/Exponential hybrid

[Dave Zan]

If you have a squarish cross section horn flare with both sides linear you end up with area expansion that is not linear but quadratic aka conical.
area a(x) = a(0)x**2 (except trivially when it's a tube)

Both sides exponential is still exponential area expansion
a(x) = a(0)e**mx
These two contours are well studied

One side flat and one exponential
a(x) = a(0)xe**mx
This seems less studied -any one seen any information on this?
TIA

[Steve Schell]

Hi Dave,

Just today I ran across the very interesting U.S. Patent #2,135,610 issued to Edward Wente of Bell Labs. He designed a horn for use in public address in auditoriums, where the pattern control of a conical horn was desirable as was the loading of an exponential. He accomplished this with cigar shaped vertical vanes, which allowed the outer horn shape to be a plane sided (conical) horn.

By saying "both sides" do you mean both pairs of sides? If you were to take a curved wall exponential horn and simply flatten two of opposing sides, you would change both the flare expansion and pattern of the horn. I suppose it would still be exponential, though with a different flare frequency. Or would it? I need to think about this. If you increase the curvature of the other sides to compensate, then you would still have an exponential with the same flare rate. All exponentials are a bit messed up in their calculations anyway, as they are based on the assumption of a plane wave propagation. To the extent that the wavefront bulges into a spherical shape, the exponential horn actually increases its flare frequency toward the mouth.

One thing Tom Danley once wrote that has stuck with me is something to the effect that the many different horn flare types are all points along a continuum, each with their stronger and weaker aspects. I guess that a hybrid horn would form another point along that continuum.

[whgeiger]

If you have a squarish cross section horn flare with both sides linear you end up with area expansion that is not linear but quadratic aka conical.
area a(x) = a(0)x**2 (except trivially when it's a tube)

Both sides exponential is still exponential area expansion
a(x) = a(0)e**mx
These two contours are well studied

One side flat and one exponential
a(x) = a(0)xe**mx
This seems less studied -any one seen any information on this?
TIA

DZ,

Used (z) for horn axis instead of (x) so the (x) and (y) may represent the ordinates of section a(z).

So,

a(z) = a(0)*e^(m*z) [1]

and assuming symmetry about the horn axis, then

a(z) = 4*x(z)*y(z) [2]

a(0) = 4*x(0)*y(0) [3]

Thus from [1], [2] & [3],

y(z)*x(z)=y(0)*x(0)*e^(m*z) [4]

Note that either y(z) or x(z) does not have to be an exponential function to deliver an exponential horn; e.g.,

if x(z) is some arbitrary function of (z) (for a “flat” side or otherwise), then

y(z) = {x(0)*y(0)/x(z)}*e^(m*x) [5]

and area expansion conforms to equation [1].

Note that particularly for H.F. horns with ‘fast’ flair rates, error intrinsic to the plane wave assumption is considerable; thus, the area a(z) should be taken over some assumed curvilinear surface that is normal to horn boundaries and 'stream lines'. This makes the math a bit more complicated, the effect of which, is to reduce the magnitude of the Cartesian coordinates x(z) and y(x) that define horn boundries.

Regards,

WHG

[Dave Zan]

Hi Dave,
By saying "both sides" do you mean both pairs of sides?

Yes.


If you were to take a curved wall exponential horn and simply flatten two of opposing sides, you would change both the flare expansion and pattern of the horn. I suppose it would still be exponential, though with a different flare frequency. Or would it?

If you take a square cross section exponential (area flare) horn then the curves of the walls are (coincidentally) exponential. If, as you say, you simply flatten two opposite sides (i.e you replace one pair of sides with flat, flared sides and don't alter the other pair of sides) the result is _not_ exponential but xe**mx - kind of half way between exponential and conical. That's exactly the point.


If you increase the curvature of the other sides to compensate, then you would still have an exponential with the same flare rate.

As the other post spells out explicitly, it is very simple to alter the profile of one pair of sides to compensate for variation in the other pair - but that's not my question.


All exponentials are a bit messed up in their calculations anyway, as they are based on the assumption of a plane wave propagation. To the extent that the wavefront bulges into a spherical shape, the exponential horn actually increases its flare frequency toward the mouth.

Yes - a reasonably complete solution isn't easy. I've battled several articles on the topic in the Journal of the AES. In one case I recall an errata in a subsequent issue. It seems even the author hadn't completely subdued the maths http://audioheritage.org/vbulletin/images/smilies/smile.gif My study of this subject is more or less what prompted the inquiry.

[JAN-813]

Hi all

I know Wilson had a method for compensating for plane wavefronts,
but it is very cumbersome, being applied after the initial design of the horn.
Does a mathemathical model for curved wavefronts exist?

cheers

[Dave Zan]

Hi all

I know Wilson had a method for compensating for plane wavefronts,
but it is very cumbersome, being applied after the initial design of the horn.
Does a mathemathical model for curved wavefronts exist?

cheers

Yes. The best work I have read is in the JAES. Lots of reasonably unpleasant maths, partly because of the unfamiliarity of co-ordinate systems that don't seem to be used much anywhere else. It's easy to find details of the relevant journal issues at the AES website. As I noted, there's at least one erratum - I strained my brain on one article and then read the next issue to find much of my effort was wasted! I won't name the culprit except to say it was Earl:) Actually he seems to be the main man for this topic, so one can't complain - he's better than anyone else and he's honest.

[JAN-813]

Thanks, Dave

cheers ;)

[Steve Schell]

I have designed and built radial bass horn flares, using what I believe to be at least an approximately correct method. Beginning at an imaginary point where the two straight sides would meet, I plot arcs every few inches out with a compass, using this point to swing the compass from. This way the wavefront remains perpendicular to the straight walls and the spherical expansion is accounted for, at least in one plane. The length of the arc at a given point in the horn can be calculated by finding the circumference of a circle with the given radius, and calculating its length based on the angle between the side walls (portion of the circle). Then, knowing the width of the flare at this point, you divide the desired cross sectional area at this point in the horn to find the height. In this way the curved wall dimensions can be determined and plotted. Clear as mud? Hope not.

The funny thing about radial flares is that usually the curved walls initially become closer together, proceeding out from the throat, before they begin expanding apart. This is because the straight walls flare so rapidly from the throat that the dimension between the curved side needs to decrease to maintain the cross section. This creates the distinctive look of most radial horns, such as the RCA MI-9462 (Ubangi) bass horn.

To fully account for the spherical wave expansion of a horn with two straight and two curved walls, I suppose it would be necessary to calculate the arcs in both planes so that they intersect the side walls at 90 degrees as the flare progresses. At least this strikes me intuitively as the way to proceed. There might be trouble though, if the straight side walls flare rapidly enough to require the aforementioned narrowing in the throat between the other two walls, as this would require an arc in the wrong direction!

WHG, I have heard for years that John Volkmann wrote a paper explaining the design of radial horns, which were his invention. Would you happen to know where this paper is in the literature?

[whgeiger]

)snip( WHG, I have heard for years that John Volkmann wrote a paper explaining the design of radial horns, which were his invention. Would you happen to know where this paper is in the literature?

SS,

As of 1957, Olson in his book [1] referenced an unpublished report by Volkmann for the horn design to which you refer. Based on a recent search, no U.S. patents were issued to Volkmann for his horn design. A related presentation [2] was made by May, Caldwell and Volkmann during an ASA meeting held in 1953. At this time, have no additional information concerning Volkmann’s work on acoustic horns.

Regards,

WHG

P.S.: Most sectoral horns have an abrupt transition from circular to ‘bow-tie’ section. If this transition is accomplished over some distance, then a smooth transition may be implemented in a manner that permits matching the slope of the horn entrance to that of the driver exit. Under these conditions the impedance discontinuity disappears and frequency response is commensurately improved. At the mouth, use of a tractrix bell, also reduces mouth reflectance. It avoids the abrupt discontinuity presented by the mouths of horn members from Salmon’s family. At the end of the horn neck, neck and bell slopes may be set equal. Once set, the tractrix mouth radius [a] is determined. Note that for a sectoral horn, the surface area is that over a toroidal patch formed by differing horizontal and vertical tractrix flares. In this setting the horn response characteristics are set by neck geometry, and the [m] and [T] parameters used to determine area expansion.


References:

[1]
Title: Acoustical Engineering
Author: Harry F. Olson
Publication (1): D. Van Nostrand Co., Inc. (1957)
Publication (2): Professional Audio Journals, Inc. (1991)
URL: http://www.audioxpress.com/bksprods/books/bkpa1.htm
Abstract: A comprehensive, but dated, text on the subject of acoustical engineering. It’s reprinting at this late date, serves as a tribute to the value and significance of Olsen’s work. For those involved in this discipline, a copy of this book should be considered a necessary addition to their reference library.

[2]
Title (1): Program of The Forty-Fifth Meeting of the Acoustical Society of America, May 7, 8 & 9, 1953
Title (2): Session A, Sound Reproduction
Title (3): Session A6, Distribution Characteristics of Cylindrical Radiators
Author (1): A. J. May
Author (2): S. A. Caldwell
Author (3): J. E. Volkmann
Affiliation: RCA Victor Division, Camden, New Jersey
Publication: ASA-J, Vol. 25, No. 4, p. 820, Jul-1953
Abstract (1): Experimental data on the directional characteristics of a number of quasi-cylindrical sector radiators vibrating radially are given. The variables considered are those of sector angle, height, and frequency. The exact pressure distributions over equiphase surfaces in the vicinity of several of the horns are shown. These data are useful in predicting the directional characteristics of other similar radiators having different height dimension.
Abstract (2): The accuracy of this application is also shown by measurements made on a similar radiator.