# 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?

The following procedure is inspired by a patent filed 1957 by Manfred Harsdorff DE1182301 “Schallführung für elektroakustische Wandler”. Harsdorff intended to shorten an exponential horn without negative effects with respect  to loading or simply speak to reduce the footprint of the large cinema horns used these days. He presented an algorithm that increases the expansion coefficient near mouth as a function of the horn axis and a target mouth point. The result was a shorter horn with a profile that expands much faster in the mouth region. Well done!

This patent inspired me to implement a progressive expansion T-factor by making  T a function of the horn axis with different possible offsets. One offset is simply the horn throat.

\tag{2a}T_z = T_0 + T_{add} \cdot f(z)
\tag{2b}f(z) = 1 - e^{-m \cdot z \cdot f_{mult}}

If we start for example with T_0  = 1 and want to add T_{add}=0.7 the multiplication factor f_{mult} influences how fast the addition takes place. After some initial tests I found that it is a good choice to use the expansion factor m as base for the slope of the transition function. With proper values we could even start with an hyperbolic horn at throat and gently change to conical after some horn length. I will present examples with two different offsets. The first is for an offset directly at throat which means that the T-factor increase is occurring just from throat on and for the second example T-factor increases after an offset on the horn axis. But first we need a reference to compare to and this is the well known spherical wave horn with cut-off 400 Hz, throat radius 17.5mm and T = 1:

With an offset at throat the profile is more steep and less curved. The horn length decreases, so the trade-off would be slightly less LF loading. Unchanged by this procedure is the horn mouth radius which is a very nice feature:

With the less curved profile directivity properties could be improved but this comes always with the disadvantage of less loading capabilities. When the offset is somewhere along the horn axis we get the following profile:

The sudden increase of T produces a knee with this set of parameters but it can be smoothed with a multiplication factor of 0.5.

Another would be to use a different transition function to mooth the knee. The last example is to show that it is possible to make a profile that starts exponential and becomes conical for a certain section until the roll-back occurs that is typical for the spherical wave horn. The parameters are T_0  = 0.7, T_{add}=3, f_{mult}=1 while cut-off parameter is still 400 Hz.

Just by decreasing the cut-off parameter value to 350 Hz we get back some more horn length but this also increases the mouth radius:

It is fascinating that the base for all these shapes was the spherical wave horn formalism. If someone looks at the last two examples without knowing this it would be hard to recognize.

Many combinations of different parameters are possible and will provide much flexibility. Of course, other transition functions for equation 2b could be used and I have tested also a \tanh function especially when the offset is in the target mouth region. This procedure in now included in most of my calculators that rely on a construction wave front that follows a pre-defined expansion rule. An example horn that uses this algorithm investigated with BEM simulations can be found here (Link_to_diyaudio.com).

This method can also be adopted to other horn types that take a pre-defined wave front surface area expansion into account. I will call these PETF horns.