Horn profiles
Tractrix horn
Voigt’s constant-tangent contour with its 90° mouth. Full or truncated, with wave-fronts drawn normal to the walls.
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Horn profiles
Voigt’s constant-tangent contour with its 90° mouth. Full or truncated, with wave-fronts drawn normal to the walls.
Loading sheet…
Plane-wave estimate, normalised to ρc/St. The tractrix assumes curved wave-fronts, so read this as a guide to where loading begins rather than an exact prediction.
| # | x (mm) | ⌀ round (mm) | □ side (mm) | area (cm²) |
|---|---|---|---|---|
| 0 | 0 | 50.00 | 44.31 | 19.63 |
| 1 | 25.54 | 57.62 | 51.07 | 26.08 |
| 2 | 51.09 | 66.44 | 58.88 | 34.67 |
| 3 | 76.63 | 76.66 | 67.93 | 46.15 |
| 4 | 102.2 | 88.53 | 78.46 | 61.56 |
| 5 | 127.7 | 102.4 | 90.73 | 82.33 |
| 6 | 153.3 | 118.6 | 105.1 | 110.5 |
| 7 | 178.8 | 137.8 | 122.1 | 149.1 |
| 8 | 204.3 | 160.7 | 142.4 | 202.8 |
| 9 | 229.9 | 188.6 | 167.1 | 279.2 |
| 10 | 255.4 | 223.5 | 198.1 | 392.4 |
| 11 | 281.0 | 270.5 | 239.7 | 574.8 |
| 12 | 306.5 | 364.2 | 322.7 | 1042 |
Paul Voigt patented the tractrix horn in 1927. The curve is the path of a weight dragged on a string of fixed length: at every point of the wall, the tangent drawn back to the axis has the same length . Measured from the mouth plane back towards the throat, the axial position of radius is
The full mouth radius makes the mouth circumference exactly one wavelength at cut-off, and the wall finishes perpendicular to the axis.
Voigt assumed the wave travels as spherical caps of constant radius that always meet the wall at right angles, becoming a full hemisphere at the mouth. Fig. 1 draws wave-fronts on that basis: flat at the throat, increasingly curved as the wall opens. The result is a horn noticeably shorter than an exponential of the same cut-off, with a smooth hand-over to the room.
The final few centimetres of the curve add a lot of width for very little length. Cutting the mouth at 70 to 85 % of the full radius is common practice; the wall-angle read-out shows how far the flare has opened where you stop. Many builders also add a generous round-over to the cut edge.
The contour is defined for round sections. The station table lists equal-area squares for convenience, but a square tractrix is an approximation.