Logarithmic Horns

Horns are used in acoustic and antenna applications. Typically it is optimal for the horn to taper outward not linearly, but as an exponential function of the horn's length. A logarithmic horn having a rectangular cross section everywhere is not a developable surface: it is not possible to make one by cutting and then bending or folding a sheet of paper. It is, however, possible to make the horn out of four sheets of paper.

This will generate a ready-to-print PDF template for any kind of rectangular logarithmic horn, given its dimensions. Just cut out the four pieces, and join them along their edges.

All dimensions are in mm.

The `shape factor' goes from zero to infinity; it determines how nonlinear the horn's taper is. One is a good value to start. In the limit, a value of zero corresponds to a linear horn.

The actual expression for the exponential curve is

            $MinLength +
                (($MaxLength - $MinLength)/
                    (exp(($ShapeLength/$MaxHeight)*$MaxHeight)-1)) *
                (exp(($ShapeLength/$MaxHeight)*$height) - 1)
    

This expression describes the taper of the assembled horn. It does not (necessarily) describe any of the curves on the paper templates; the taper of the horn gets distorted when it is `flattened out.'

Nov 2005, Cambridge