Tractrix horn flare

An email conversation with a friend regarding the myth of the tractrix horn flare

### My initial email 24 June 2007

I remember we recently discussed the merits of a tractrix horn and its relationship to an expanding spherical wavefront.  I think if you assume a spherical wavefront (which is not a correct assumption, but we’ll leave that for the moment), you need a conical horn.

Straight sided waveguides are fundamentally different from traditional flared horn contours. Imagine the layers (skins) of an onion represent the isophase acoustic wave fronts radiated from a tiny source at the onion centre. Imagine poking a long thin needle directly towards the centre, it will be perpendicular to each layer at the point of piercing.

Now imagine using a sharp knife to cut the onion in half, directly though the centre. Each of the layers comes to the plane of the cut at exactly 90°. If the plane of the knife cut does not pass through the centre, the layers will not intersect the cut at 90°. Imagine the needle edges are sharp enough to carve out a cone (with the apex fixed at the centre of the onion), you would also see that the onion layers remained perpendicular to the cut. Actually the needle could carve out a pyramid, or oval, or any other 3D shape into the onion, and as long as the tip of the needle stayed at the centre, the cut would always be 90° to the onion layers.

For spherically expanding wave fronts, (or any 1P wave), the guiding boundaries (horn walls) which are perpendicular to the wave front act as acoustic mirrors, and create reflected phantom images of the wave front (continuing the same curvature beyond the wall), which effectively cause the 1P wave to continue to propagate exactly as if it were in full 360° (3D) space.

As you can see, only a straight sided horn remains perpendicular to an expanding spherical wavefront. And it doesn’t have to have a circular cross section: any shape of straight sided horn is perfect. The only proviso is that the point of convergence of the horn needs to be the point source of the wave front.

But if you tried to draw a tractrix through the onion layers, using a “specially curved” needle, it would not remain perpendicular to each layer.

Fortunately, it is not critical anyway, because compression drivers are designed to emit a planar wavefront at the throat of the horn, not spherical. And a tractrix horn is no better suited for a planar wave than it is for a spherical wave.

The only reason to pick a tractrix horn is if you are committed to first order crossovers for some reason, which means you will have loads of output below the mouth frequency, in which case the wide mouth with its 180 degree flare will produce less backpressure down the horn at those low frequencies. And that is the Edgar demo where he speaks with a male voice that goes down to 80Hz into a 400Hz horn, which produces major problems with an exponential 400Hz horn, and less severe in the tractrix.

I prefer to use a horn within its design bandwidth, with a sharp crossover controlling the out-of-bandwidth problems.

(my friend replied with an article published in July 1997 Hi Fi News and Record Review about Avantgarde horns, with a sidebar about the tractrix flare, which is ironic because Avantgarde refer to their products as spherical horns, which are not tractrix! -- see here for a visual comparison of most horn flares including tractrix and spherical)

### second email 25 june 2007

Aha, Avantgarde horns! I have always admired them for making a serious attempt to customise horns to the home environment. It was the layout of their Trio that inspired me towards trying something similar, but once I noticed how audible was the huge distance separating the drive units, I realised I needed something better.

If a tractrix horn is used to take a plane wavefront (which compression drivers are designed to produce) at the throat and ‘convert’ it to spherical at the mouth, there will be reflections along the way: the claims in the HFNRR ‘advertorial’ that they do not occur is flat-out wrong. The truth is, any change in the slope of the walls of a horn will cause reflections along the horn walls. That is an incontrovertible acoustical law. Any curved wall (tractrix, exponential, etc) is constantly changing the slope of its walls, so every millimetre produces more and more reflections. Any articles claiming the opposite are w-r-o-n-g, including the HFN/RR advertorial you sent me.

The *only* way not to create reflections is to have straight walls: conical flare. That is why the conical horn has always been known to produce the least distortion of all horn profiles. Measurement and theory agree on this. Of course, discontinuities at the mouth and (usually) throat of the conical horn produce their own problems, but at least the flare itself does not produce reflections.

P.S. There is so much misinformed information about horns. It is a worry. An old email from Thomas Dunker (who I had many emails with and admire greatly for several reasons) says “The important concept that led directly to Voigt's "invention" of the tractrix was that he not only assumed spherical wave fronts, but also that the wave fronts would be of a constant radius (the radius of the whole sphere) at all points along the length of the horn. Further, assuming that the wave fronts would have to be perpendicular to the inner surface of the horn wall, the curve simply had to be a tractrix.”

Well, spot the error! A spherical wavefront *expands* as it moves down the horn, so its radius changes: it is not of a constant radius. The only way to have a constant radius would be if the source of the wave, the diaphragm, raced along the axis of the horn at the same speed as the wavefront – which is impossible.

(my friend replied seeking proof that curved walls must produce reflections, and claiming that the wavefront in a tractrix horn starts off flat and curves to a hemisphere at the mouth)

### third and final email 26 June 2007

My, this discussion is fun.  I could say “prove me wrong”, but since I put the words down in the first place, let me try to substantiate them:-

First of all, we seem to be talking a lot about the shape of the wavefront as it travels along the horn, for example, flat at the throat and spherical at the mouth. There is nothing natural about that. Wavefronts don’t do that by themselves. In Dinsdale’s words, “the composite wavefront resulting from the original wavefront and its reflection will itself be normal to the walls”. In other words, every horn forces the composite wavefront to always be normal to its walls, and the way it does that is by reflections off the walls.

If a tractrix starts with a flat wavefront and finishes with a spherical one, it forces that to occur by reflections off its walls.

So what is the natural form of sound waves? If we consider a sound source originating in free space, let’s say a small pulsating sphere, the natural wavefront is an expanding sphere, starting with the diameter of the pulsating source, and expanding uniformly in every direction, with the centre of the expanding spherical wavefronts remaining in the centre of the source of the sound.

If we now place an imaginary shape in this soundfield, a shape which is always perpendicular to the expanding wavefront (so that it never causes any reflections off its walls), we have a cone. See my “onion layers” email. It doesn’t have to be circular, but it does have to be conical (straight sided).
The conical horn does not distort the natural wavefront, whereas all others do, hence lowest distortion. It has less amplitude distortion, phase distortion, and harmonic distortion.
QED.

By the way, I’m not saying anything new. It is well known that a conical horn produces lowest distortion of all horns. All I tried to do above is demonstrate it. It is also well known that changing the slope of a horn wall will cause reflections. But the math is above my head, so I can’t provide you ‘proof’. You might find it in Geddes’ book on waveguides, but he warned me it is very mathematical, so I reckon it would be incomprehensible to me.

P.S. You asked “What is the radius of a flat wave at the throat?”. Not sure if you were asking rhetorically, but the radius is infinity, or curvature is zero.

P.P.S. if Dunker is wrong, as you claim, about constant radius of the wavefront in a tractrix flare, so is Voigt. Dinsdale wrote “Voigt had commenced his analysis on the assumption that wavefronts within the horn will be spherical and of the same radius throughout their progression through the horn. This assumption leads to a tractrix curve for the horn contour”. Voigt states this assumption in his 1927 patent.