# 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.

# Spiral Functions for Horns – The Sici Spiral

The last post in this series deals with the Sici spiral (Link1). It is similar to the Nielson spiral. I already mentioned that this spiral is my personal favourite of the three spiral functions presented on my page. On the one hand this is because the curve reminds me of JMLC horns and on the other hand because of the simple relationship of the tangent with respect to the basic rotation angle. In addition, we will see that the tangent vector, when the parameters are selected appropriately, results in a nearly constant length over large areas of the horn curve and only expands towards the horn mouth.

The cartesian parametrization look simple on the first view

\tag{1a}x =-a \cdot Ci({\phi})

\tag{1b}y = a \cdot \left( \frac{\pi}{2}-Si({\phi})\right)

but Ci(\phi) and Si(\phi) are the cosine and sine integrals (Link2). These integrals need to be solved but again as for the Cornu spiral these integrals can be developed as a series expansion. With proper offsets defined the Sici spiral becomes usable as horn profile function.

\tag{2}x_0 =-a \cdot Ci({\phi_s})

\tag{3a}y_0 = y(\phi_s) =  a \cdot \left( \frac{\pi}{2}-Si({\phi_s})\right)

\tag{3b}y_0 = y(\phi_0) =  a \cdot \frac{\pi}{2}

# Spiral Functions for Horns – The Cornu Spiral

The next spiral function in this series is the so called Cornu spiral (Link1). Other known names are Euler or Fresnel spiral or Clotoide. The most fascinating thing is that the curvature of the Cornu spiral is proportional to it’s arc length. Therefore, it is often used as base function for car lanes or railway traces. The mathematical treatment bear some challenges as the so called Fresnel integrals have to be solved. Fortunately, these integrals can be developed as a series expansion of quite simple terms. Several examples can be found in the web an there is nothing really new about this. The biggest failure for a series expansion happens if it diverges but for the range of parameters needed for a horn it has been found that the used algorithm converges properly throughout.