In the previous article I already presented some basic information about the semi cubic parabola (Neile parabola) and why I got interested in this mathematical function (Link1). The initial intention was indeed to calculate curved fins with equal path lengths (arc length). But first, a verification of this mathematical function for it’s use as a horn profile should show acceptable results because the two outer Neile parabolas define the outer horn contour. Within most of my previous horn calculators two different functions in two orthogonal planes were sufficient to generate a point cloud by blending the two functions together to form an ellipse at each iteration step along the horn axis. But I decided against this procedure because in the meantime I could make some experiences how to generate a point cloud for radial like horns which can quite easily be made out of wood with a CNC milling machine. The reader should be familiar with the basic blueprint of a radial horn. For the use of the Neile parabola to this type of horns some things are a little bit different as we have no exact radial expansion from a pre-defined point of origin of the profile. In this article I will describe the basic steps how to create a William Neile horn and show some initial BEM simulations as proof of concept for this type of horns.

# A Tribute to William Neile

For some time now I have been investigating for the so-called fin horns. And for over a year now I have been able to generate complete point clouds for such horns. I myself own a replica of the TAD TH4001 horn and I think these horns sound very good and load the driver perfectly down to the lowest octave of the usable range of the driver. A passive crossover can therefore be implemented without significant problems.

What has always surprised or bothered me about these fin horns is the straight construction of the fins, which start on a section of a circle near the throat. My understanding is that a point sound source at the intersection of the two side walls is assumed. But what if the arriving wave front is not curved like the fin start circular arrangement? And what is happening in the small “pre-chamber” before the fins? My initial idea for a new approach was to use a mathematical function that starts directly at throat and which is curved and an analytical expression of the functions arc length is known. The resulting fins should start at throat, they should be be curved and they should end at the same path length along the curved trajectory. At this time I came across with a function discovered already in the 17th century that is named after William Neile (Link1) as Neile parabola. Another well known name is semi cubical parabola (Link2). It was the first algebraic curve to have its arc length computed (Link3). The semi cubic parabola bears an interesting property that it is an isochronous curve of Leibnitz (Link4) . All in all a very attractive looking function and worth to try it as a horn function. This article deals with my way to use the Neile parabola as horn function.

# PETF Applied to JMLC Horns – Simulation Results

In two previous posts I presented my PETF algorithm and JMLC inspired horn calculator. Assuming that many of my readers are familiar with the native JMLC horn performance like loading or radiation polar it should be a common acceptable consensus that JMLC horns do not belong to the so called constant directivity (CD) category. Towards higher frequencies they tend to slightly beam which is because of the curved horn walls. This might not be an issue for some applications or some people might even like this behaviour but as general rule of thumb the lower the horn cut-off value the longer the horn profile will be and the smaller the initial opening angle both causing an increasing tendency to narrow the dispersion of higher frequencies. A more focused dispersion of higher frequencies might be an advantage in small environments if it is fairly constant or if the intention was to compensate the natural roll-off of most compression drivers but generally a design goal of wider dispersion is one of my personal preferences. More precisely, one of my main goals is to find a good compromise between good horn loading and good directivity control.

There can be found some BEM simulation examples for round JMLC horns in the web that clearly show the increasingly more narrowed dispersion towards higher frequencies especially for the lower loading versions with 350Hz or lower cut-off. On the opposite JMLC horns shine if the target design intention was mainly a nearly perfect horn loading down to the desired cut-off frequency or by looking at the smoothness of radiated wave fronts when the formalism inherited roll-back is present.

I already presented that the PETF algorithm produces a faster opening of a horn profile while straightening the horn walls. In this article I will investigate what horn properties we can expect by applying PETF to a given profile.