JMLC Inspired Horn Calculator

I find it difficult to formulate the appropriate introductory words for a person like Jean-Michel Le Cléac’h (JMLC). Unfortunately, I didn’t have the opportunity to discuss my work with him, although theoretically it would have been possible but my interest in horns arose just a few years ago. I would have been honoured to have received feedback on my work from Jean-Michel.

Recently, I realized what meaning as human being Jean-Michel Le Cléac’h must have had for other people when I recognized that Bjørn Kolbrek and Thomas Dunker dedicated their excellent book about “High-Quality Horn Loudspeakers Systems” to him.

Here are two links to diyaudio.com for those who are not so familiar with his work and his life: Link1Link2.

Interestingly, JMLC emphasized that we should understand his work on horns more as a method to calculate horn profiles than rather a new expansion. This is exactly how I understand it and this post will describe my implementation of JMLC’s method.

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Progressive Expansion T-Factor Horns

In this post I would like to describe a method that I have been using since some time to get more flexibility with regard to different horn profile slopes. The fact that slow expansion profiles like the exponential or spherical wave horn provide very good loading but suffer somewhat with respect to directivity control inspired me to look for an algorithm that influences the given specific horn expansion function. I ended up with a modification of the well known wave front surface area expansion formula for hyperbolic (hypex) horns which is given by:

\tag{1a}S_z = S_0\cdot \left( cosh(\frac{m}{2}\cdot z) + T \cdot sinh(\frac{m}{2}\cdot z) \right)^2
\tag{1b}m=\frac{4\pi \cdot f_c}{c_s}

I will not go into more detail but with T=1 we get exactly the well known exponential horn for a two dimensional surface area expansion. This formula can also be used for the spherical wave horn with it’s assumed spherical wave fronts and will put out the same SWH profile already described on this site. Equation (1) is extremely flexible because when  T \lt 1 an hyperbolic profile is the result and when T \to \infty the horn profile becomes conical. This is the reason why I switched to this formula as it gives me the flexibility to produce horn shapes with different slopes only by changing one parameter. Up to here nothing new but what if we make T a variable function that depends on a reference point somewhere on the horn axis z_{off} and the distance from or to this offset?

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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}

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