Article: A Horn Consisting of Manifold Exponential Sections

[WMcD]

From time to time there is mention of the concept of a rubber thoat. The following article is usually cited.

The term, rubber throat, seems to be a creation of PWK. In my view, use of the term, particularly with a straight face, causes some people to think there really is some sort of elastomeric material in there.

Actually, the technique is to put a horn with a high flare rate at the driver end of the horn, and then have it feed a horn with a lower flare rate. As a result, the combination acts as if there are two different throat sizes, one bigger than the other. Therefore, in concept, the thoat size stretches. But of course there is no rubber involved at all. Basically, at low frequenies, the initial section with the high Fc is ineffective.

I've marked up one of the figures from the drawing to show what is meant.

I find another matter to be of interest. Figure 5 in the article shows the use of three sections. The horn cross section, to me, starts to look like a conical horn and the throat impedance does too. Therefore, this may give some insights into how the exponential and conical are related.

Olson, in this article, devotes some comments to the virtue of an air chamber. This must be an air chamber between the diphragm and the throat. It is not shown at all. Also, it is a little difficult to see just how much of an improvement is achieved with the manifold section(s). It could be as little as 3 dB but you can make your own estimation. The data here doesn't tell us much of anything about bass horns.

This article may be the first place the equation for calculation of throat impedance below Fc appears. PWK used it to show that there was some finite resistance below Fc, PWKs article must have been to address the notion that K-Horns can possibly work below Fc of the bass horn.

Let me apologize for the problem with the reproduction of the article. Page 517 is missing some text at the left margin. It is possible to read in context.

Best,

WMcD

[WMcD]

The article.

WMcD

A Horn Consisting of Manifold Exponential Sections.pdf

[Groomlakearea51]

The article refers to a M.L.Graham who developed multiple flare horns. Any ideas as to what he did (I don't have a library that can help me find the reference Olson cited). Other question, is the "tractrix" horn a multiple flare horn? Thanks//MWM

[DrWho]

Is the concept of a rubber throat still being used today?

[WMcD]

Sorry, I don't have the Graham article. If I come across it, I'll post it.

The tractrix is a different type of design of course: The exponential horn is described by the exponential equation, of course. It is actually the cross-sectional area which is described by the equation. Further, the equation is actually saying that the growth rate is constant. Or, actually, that the area is going to increase by some percent every X distance interval down the horn.

This exponential growth is like the biology problem where bacteria will double in number every (say) two hours, until it breaks out of the petri dish and takes over the biology building. Similarly, radioactive substances decay with a half-life; the number of particles of the element decay to half of an established amount every hour, or century. In the K-400 horn, the area doubles (or halfs) every 2.75 inches. That distance depends on the Fc of the horn.

The tractrix curve in a pure design actually describes the shape of the wall. So you can make up a horn having a circular cross-section and figure the area. Then you can do some calculations and see how close it comes to mimicing an exponential of an Fc at a given location. But, as you imply, it is a matter of needing slices of exponentials with different Fc to make up the tractrix. (I posted an article by the Klipsch people about the tractrix where they are saying the various sections approximate hyperbolic and conical sections . . . but this really is about the same approach.)

- - - -

Dr. Who. The Jubilee article says it has a rubber throat! I've heard people say the LaScala has a rubber throat, but I don't see it that way.

WMcD

[DrWho]

While the idea for the tractrix equation comes from the shape of the walls, I think it fair to say that Roy is very adament in pointing out that it is still an area expansion equation. In other words, a "perfect" tractrix in the sense of the wall shape would require a spherical horn (like the salad bowls), but the sphere doesn't provide ideal polar patterns. By moving to a rectangular mouth, you no longer have the perfect concept of a tractrix as it pertains to the shape of the wall, but it still has the same area expansion. Roy's "modified tractrix" comes back to address the shape of the horn's walls, while maintaing a perfect tractrix area expansion. Or at least that's my understanding - hopefully Roy will chime in if I've got it all wrong [A]

[WMcD]

I must say I don't disagree with Dr. Who. And what Dr. Who says about Roy's thoughts are not in question.

I did some calculating over the weekend using a pure round tractrix horn. I was surprised. As described below, the tractrix IS an exponential.

The so called wavefront in the tractrix model is a portion of the surface of a sphere of constant radius. It is the radius of the round mouth. That is one calculation to find the area.

(Consider that at the physically flat mouth of the tractrix, we can calculate the flat area as a circle, but the wavefront is a hemisphere. It bulges at the center by an amount equal to the radius of the mouth.)

The other related issue is that the wavefront bulges more and more as we travel from the throat to the mouth. Therefore, the x distance between any given wavefronts going down the horn requires additional calculation.

So then I figured the effective Fc at 1 inch or so slices of the tractrix considering the area of the portion of the sphere and taking into consideration the bulge effects. If you have S1 and S2 and an X, you can find an exponential flare rate.) I expected something a bit odd because the wavefront area is greater because of the sphere effect. . . but the x increment is greater because of the bulge. Just how it was odd was unexpected by me.

The result is that the tractrix has a more or less constant Fc, as an exponential would have (assuming a flat wavefront for the exponential -- and of course Fc is constant). The calculated Fc for the tractrix varies about plus or minus 10 percent. It actually goes lower at the mouth.

I'd like to check some of the older literature on this and recheck my calculations. If I'm correct, it means there is an exponential horn hidden in the classic tractrix. I've never read that. Naturally I'd like to hear from anyone who has tried the calculations. Roy?, Dr.?

WMcD

[JohnA]

Gil: "..... it means there is an exponential horn hidden in the classic tractrix. I've never read that."

That
IS interesting. Just for the mathmatical satisfaction, I ran
through the calculations for a tractrix and designed an elaborate (read
expensive) exponential backward J basshorn that never got built.
The shapes of the 2 horns are quite interesting.

[DrWho]

I have an excel spreadsheet that I created for doing iterations on the

tractrix equation with an arbitrary shape for the expanding surface

area (so circle, square, triangle, whatever...), which would spit out a

graphical representation of each of the cross-sections of the horn (so

you could see the vertical and horizontal flares seperately). I

remember a while back throwing in an exponential horn comparison and

seem to recall that the two were very similar as you were closer to the

throat. However, the closer to the mouth you got, the more they would

diverge...especially right up next to the mouth where the tractrix

explodes into an infinite area and the exponential just stops. So while

it's not exactly number crunching, I think the visual representation is

more striking (at least in my mind). I'll have to see if I can't find

the simulations and post them on the forum.

If I'm understanding correctly, you're trying to force fit the

exponential equation by stretching the change in x inbetween

iterations? Roy mentioned once that he was trying to combine the

exponential and conical area expansions into a sliding morphing thing

until he realized the tractrix accomplished exactly what he was looking

for.

The two equations assume two very different things so I would be

extremely surprised if they ended up with the same solution for optimal

acoustic impedance matching. In other words, I think that by

force-fitting the exponential equation to work, that you're breaking

some of the assumptions that created the exponential model. I think we'd have to look at your specific number crunching to be sure. Can you get your exponential model to create any arbitrary tractrix horn without doing the tractrix first?