# 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 whenÂ  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?

# Spiral Functions for Horns – The Hyperbolic Spiral

It’s been a while since I wrote an article for my blog. As far as I was able to do in time, many other horn profiles have been investigated and programmed with help of my DrBA spread sheet calculator. The blog is therefore clearly behind the current stage of development. Now it is time to reduce the gap somewhat.

When I was working on the True Expansion Tractrix horn, during the research I found something that gave me the base idea of a whole range of new horn profiles. It was a reference to the hyperbolic spiral, which was something like this: if you unroll a hyperbolic spiral on a line and place a virtual pen in the center of the spiral, you get a Tractrix curve. What? A Tractrix curve? If the Tractrix curve is known to be suitable as a horn profile, is this also the case for the hyperbolic spiral?