Edgar

Again, we are talking about two different things. You are talking about the assumptions made to derive the expressions that describe the acoustical impedance, while I am talking about the actual shape of the wave front as it propagates down the horn. In the real world, neither assumption is exactly correct. Greg

Hmmmm ... I think that an infinitely long tractrix would have a point source, just like an infinitely long exponential, hyperbolic, etc. Exactly. It comes back to those assumptions again. Ah, but amplifier distortion is typically orders of magnitude smaller than speaker distortion. Again, it's about what you want to optimize. There is a whole school of speaker design that says that the sound should be omnidirectional, which would maximize the room reflections that you note. There is yet another school of speaker design that says that the sound should be tightly controlled, which would minimize the room reflections. The fact that superb-sounding speakers are available from both camps tells me that neither assumption is 100% correct (or incorrect). I am not disagreeing with you ... just playing Devil's advocate.

Roy and Edgar are in full agreement that, "how each distributes the area is related to how each thinks the wave is propagating down the horn and how each thinks the loading should be. in all cases, assumptions are made but real world measurments can reveal some errors in our thinking." I'm not familiar with Post's thesis, but then I have to admit that horns are a side interest for me and I'm always learning new stuff about them. All that aside; even in a round tractrix horn the wave front is not perfectly spherical. The only horn that supports a truly spherical wave front is conical with a circular cross section. Earl Geddes published papers on this sort of thing. It really comes down to what you want to optimize. Nowadays, when Watts are super-cheap, do you really care about maximum power transfer? In the pro audio world, pattern control is most important -- they want everybody in the audience to hear essentially the same thing. In your living room you don't care much about pattern control -- you just want the best sound quality possible at a single listening location. People have been moving away from exponential horns and toward tractrix horns because tractrix horns sound better. Since the frequency response and loading characteristics of similarly sized exponential and tractrix horns differ only by small amounts, the differences in their sonic quality must be due to something else. Again, Geddes enters the picture and claims that the sonic differences are due to "higher-order-modes" -- resonances within the horn body due entirely to the shape of the horn itself. Maybe there is something to that. I honestly don't know. Edgar (Greg)

I have argued that the tractrix curve describes a radius expansion that is translated into an area expansion, but it's a minor point that does not affect the current discussion. Exactly. I see now that I misunderstood your statements. We are in complete agreement here. Was that the recent JAES paper (within the past few years) where they analyzed the errors in the plane-wave assumptions behind Webster's equation? Again, we are in better agreement than I originally thought. See my earlier comment, in which I said, "I just apply a small fudge-factor to the horn design ... and otherwise don't worry about it." Thank you and have a safe and prosperous new year. Greg

Respectfully, Roy, I don't see how you can start with the defining differential equation for the tractrix curve (see the PDF in my previous message), and reach your conclusions 1, 2, and 3. Greg

Okay, I'm going to attach this as a PDF, but I don't know how well it will work because it's four pages long. Tractrix Horn Path Length.pdf

No, my tractrix calculator is in C, and my drawings are in TurboCAD. And the equations are in Word, using Equation Editor.

There is. Anybody want to see it? I'm open to suggestions as to how to post math equations to the forum.

I'll answer in reverse order. A tractrix horn definitely can be interpreted as a specific area expansion. That's just simple geometry. But, since acoustical waves in the horn don't propagate as plane waves, the actual area of the curved wave front will always be somewhat larger than the calculated area of the circular cross section of the horn. For non-round horns the effect of this curvature can range from insignificant to profound. For a horn of square cross section, that expands as a tractrix in two dimensions, the area error between the square bubble and the square plane is probably about the same as the area error between the round bubble and the round plane, so a square horn should behave very much like an "equivalent" round horn. For a horn of rectangular cross section, that is tractrix in one dimension and conical (or constant) in the other, the bubble error difference gets larger and is also dependent upon the aspect ratio of the rectangular cross section. But my point is that this is all of only academic value. As an engineer I find it interesting to analyze, but the fact is that horns are very tolerant of modest discrepancies in the contour. Play around with Hornresp and you'll find that, as long as you don't get too extreme with the contours, all horns of the same length, throat, and mouth dimensions perform very similarly. (Bruce Edgar himself -- no relation -- has even stated this explicitly.) So to answer your first question: I don't worry about it too much. Sometimes, if I'm feeling particularly anal-retentive, I'll try to approximate the bubble shapes in my calculations, but mostly I just apply a small fudge-factor to the horn design (like most people do -- if you want to use a horn to 100 Hz, design a 70 Hz horn, for example), and otherwise don't worry about it. Edgar (Greg)

You can certainly make a horn of any shape have the same cross sectional area as the circular cross section of the tractrix horn. But my point is that, even if you do THAT perfectly, you still haven't correctly translated the area expansion of the actual wave front because the wave front doesn't propagate as a plane. When I design tractrix horns I use my own tractrix calculator, because that way I know exactly what assumptions it uses.

Yes and no. Remember, the tractrix curve is a line in two dimensions. The tractrix horn has a circular cross section, and the radius of the cross section at any point along its length is equal to the distance from the tractrix curve to the centerline at that same length. Most tractrix calculators compute the area of the tractrix horn at any point along its length as the area of the circular cross section at that length. While easy to compute, it does not accurately represent how the wave front propagates down the horn. The wave front is curved outward -- Roy has described this as a "bubble". The bubble is NOT spherical, but it's pretty close. So a better approximation of the area expansion can be obtained by computing the area of a spherical section at each point along the length of the circular tractrix horn. However, when translating the area expansion of the circular tractrix horn into any other cross section (square, rectangular, elliptical, etc.), the shape of the "bubble" appropriate for THAT cross section also needs to be taken into account. For a rectangular tractrix horn that only expands in one dimension (the blue figure in my Image2), the bubble will be approximately cylindrical. For other shapes, it will be different and generally much more complex. In fact, the best way to determine the actual bubble shape is finite element analysis. I guess the point of all of this is to tell you not to get too worried about matching the dimensions exactly. Even if you do, it'll still be wrong. So all that you can do is design it, build it, and see if it works. Edgar (Greg)

It is all an approximation. The original tractrix horn design had a circular cross section. When designing circular tractrix horns, I approximate the wave front with a section of a sphere. It's not perfect, but it's a lot closer than using a planar wave front. For non-circular horns, it gets even more complicated. Greg

And, finally, the fifth image shows what happens when you lay the original flat tractrix curve on top of the resulting solid. They definitely do not match. That is because the tractrix curve designates the centerline of the horn, not the top and bottom.

The fourth image shows both the original flat tractrix curve and the resulting solid.

This next one shows what it looks like after cutting the top and bottom angles.

The next shows what happens if you really did cut it out on a band saw.

You are right. This is a bit intricate, but I have made some drawings that I will include in the next several messages. The first shows a tractrix curve laid-out on top of a square block, as if you were going to cut it out on a band saw.

-i*ln[2i+sqrt(-3)]

I think that James Bongiorno still does GAS repairs. GASworks used to be the place to go, but apparently they are no longer with us. Greg

Familiar discussion between engineering and marketing: "You told the customer that our product can do things that it can't." "Well, then you have to make the product do those things, because we can't sell the product if it doesn't." "Why do you keep making promises that we can't keep? Why don't you just sell what we make?" "Well, why don't you make what we can sell?

When describing a sine wave, the general form is "sin(2*PI*f*t)", where PI is the familiar "3.14159265358979...", "f" is the frequency of the sine wave (the thing you're asking about in your post), and "t" is time, in seconds. So saying "just sin(x)" doesn't describe a sine wave; it describes the value of the sine function at a particular angle "x" radians.

There must be something special about that $148K figure -- Steinway. "While most speakers are sealed boxes, Steinway's engineers left these open, replicating the resonance of live instruments." "And unlike other systems, which convert the audio signal from digital to analog in the amp many sound-distorting circuits and wires away from the speakers the C is digital from source to driver."

With the Wilsons it isn't even bling -- they look like industrial robots, or maybe Daleks on steroids.

Circum-aural or supra-aural? Certainly not in-ear!

Many years ago, probably around 2000, I heard Wilson's then top-of-the-line loudspeaker in a demo at the Audio Research facility near Minneapolis. Frankly, I thought that my triamplified Legacy Focus sounded better.