This is the part where the thought meets some slightly awkward physics. Before any wood gets cut, I need to work out the shape of the horn: how small the throat should be where the driver sits, how big the mouth has to be, how quickly the walls should flare out, and how long the whole path needs to be. Get those wrong and the horn simply won't do what I'm hoping for.
I'll be honest — an exact solution to the wave equation for the sound inside a horn is basically off the table for a home builder like me. It gets messy fast. So the trick is to lean on sensible simplifications, and there are several well-known ways to go about it. The good news is that I don't have to grind through the partial differential equations by hand.
The expansion formula
The shape of an exponential horn is described by a wonderfully compact equation. The cross-sectional area grows exponentially as you move away from the throat:
- \(S(x)\) — the cross-sectional area at a distance \(x\) along the horn
- \(S_0\) — the throat area, right at the driver end
- \(m\) — the flare constant, i.e. how fast the horn opens up
- \(x\) — the distance measured from the throat
That single constant \(m\) is where all the magic (and all the compromise) lives. Make the horn open up slowly — a small \(m\) — and it will load down to a low frequency, but it needs to be long. Open it up quickly and it stays compact, but the deep bass slips away.
The flare constant is tied directly to the lowest frequency the horn can support, its cutoff frequency \(f_c\):
- \(f_c\) — the cutoff frequency (below this the horn stops "horn-loading")
- \(c\) — the speed of sound, about 343 m/s in air
Plug in my target of 20 Hz and you get a very gentle flare — which is exactly why the horn ends up so long, and why the space under the floor matters so much.
What I'm actually solving for
Beyond that flare, dimensioning a horn comes down to juggling four numbers against each other, and against the space I have under the floor. Push one and the others move. The art is finding a combination that fits and still hits my targets.
| Parameter | What it does |
|---|---|
| Throat area \(S_0\) | The small end where the driver loads into the horn — sets how hard the driver is working. |
| Mouth area | The big end. Too small and the deep bass just rolls off and reflects back. |
| Flare rate \(m\) | How fast the cross-section grows. This is what really sets the low-frequency cutoff. |
| Path length | Throat to mouth. Long horns go deep — which is exactly the problem in a normal house. |
The constraint that hurts
Here's the catch, and it's a big one: to properly load down to 20 Hz you need a genuinely enormous mouth. The rule of thumb is that the mouth area has to be a decent fraction of the wavelength squared at the cutoff frequency — and at 20 Hz the wavelength is roughly 17 metres. That's why serious bass horns are measured in cubic metres, and why the space under the floor suddenly looks very attractive as free volume.
An exponential flare is the classic compromise here: it goes deep, behaves well, and it's realistic to build. Purely conical horns are gentle but lose the deep bass; hyperbolic horns dig deepest but bring more distortion along for the ride. Exponential sits nicely in the middle, which is why it's the one I keep coming back to.
Where this is heading
Right now I'm in the fun phase — feeding numbers into Hornresp, watching the response curve wiggle, and slowly learning what actually matters versus what just looks good on screen. As the design firms up I'll post the real dimensions, the simulated curves, and hopefully some sketches of how it all folds into the space.