Horn profiles

Spherical-wave horn

The Klangfilm Kugelwellen contour: exponential growth of spherical wave-fronts, out to a 180° mouth.

Sheet: 05 / 25
Units: Metric
Air: 20 °C

Fig. 1Axial section, to scale

Length $L$: 32.9cm
Mouth $d_m$: 40.9cm
Wave-front radius $r_0$: 36.4cm
Wall angle at mouth: 73.2°
Full 180° mouth: 48.2cm
Air volume: 8.10L

Fig. 2Throat impedance of this finite horn

ResistanceReactancedrag on plot to read

Plane-wave estimate, normalised to ρc/St. The contour assumes curved wave-fronts, so treat the curve as a guide to where loading begins.

Table 1Build stations

Stations

#	x (mm)	⌀ round (mm)	□ side (mm)	area (cm²)
0	0	50.00	44.31	19.63
1	27.41	58.21	51.59	26.61
2	54.81	67.80	60.08	36.10
3	82.22	79.03	70.04	49.05
4	109.6	92.21	81.72	66.78
5	137.0	107.7	95.48	91.17
6	164.4	126.1	111.8	125.0
7	191.8	148.1	131.3	172.3
8	219.3	174.7	154.8	239.7
9	246.7	207.5	183.9	338.0
10	274.1	249.3	220.9	488.0
11	301.5	306.6	271.8	738.5
12	328.9	409.5	362.9	1317

How it works

The spherical-wave horn, or Kugelwellenhorn, came from Klangfilm in the late 1940s. Like the tractrix it assumes spherical wave-fronts of constant radius, but it asks for the area of those curved fronts to grow exponentially. It is often confused with the tractrix; the two contours are different.

Construction

Choose a cut-off frequency and a throat radius $y_0$ . The wave-front radius, flare constant and the height of the spherical cap sitting in the throat are

r_0 = \frac{c}{\pi f_c}, \qquad m = \frac{4\pi f_c}{c}, \qquad h_0 = r_0 - \sqrt{r_0^{2}-y_0^{2}}

A cap of height $h$ on a sphere of radius $r_0$ has area $2\pi r_0 h$ , so exponential area growth means exponential growth of $h$ . With $x$ the distance travelled by the top of the wave-front:

h = h_0\,e^{mx}, \qquad y = \sqrt{2 r_0 h - h^{2}}, \qquad x_h = x - h + h_0

The pair $(x_h,\,y)$ is a point on the wall: $y$ is the horn radius and $x_h$ its axial position.

The mouth

The wall turns perpendicular to the axis when $mh = 1$ , that is at $h = r_0/4$ , giving a mouth radius of about $0.66\,r_0$ . That is roughly a third wider than the tractrix mouth for the same cut-off. Continue past that point and the contour curls back towards the throat, so the slider stops at 100 %. Most builders stop earlier.

Formulae after B. Kolbrek, “Horn Theory: An Introduction”, audioXpress, 2008.

Loading sheet…

x (mm)

⌀ round (mm)

□ side (mm)

area (cm²)

50.00

44.31

19.63

27.41

58.21

51.59

26.61

54.81

67.80

60.08

36.10

82.22

79.03

70.04

49.05

109.6

92.21

81.72

66.78

137.0

107.7

95.48

91.17

164.4

126.1

111.8

125.0

191.8

148.1

131.3

172.3

219.3

174.7

154.8

239.7

246.7

207.5

183.9

338.0

274.1

249.3

220.9

488.0

301.5

306.6

271.8

738.5

328.9

409.5

362.9

1317

How it works

Construction

Choose a cut-off frequency and a throat radius $y_0$ . The wave-front radius, flare constant and the height of the spherical cap sitting in the throat are

r_0 = \frac{c}{\pi f_c}, \qquad m = \frac{4\pi f_c}{c}, \qquad h_0 = r_0 - \sqrt{r_0^{2}-y_0^{2}}

A cap of height $h$ on a sphere of radius $r_0$ has area $2\pi r_0 h$ , so exponential area growth means exponential growth of $h$ . With $x$ the distance travelled by the top of the wave-front:

h = h_0\,e^{mx}, \qquad y = \sqrt{2 r_0 h - h^{2}}, \qquad x_h = x - h + h_0

The pair $(x_h,\,y)$ is a point on the wall: $y$ is the horn radius and $x_h$ its axial position.

The mouth

Formulae after B. Kolbrek, “Horn Theory: An Introduction”, audioXpress, 2008.

Spherical-wave horn

Inputs

How it works

Construction

The mouth

Inputs

How it works

Construction

The mouth

§ How it works

Construction

The mouth

§ How it works

Construction

The mouth

How it works

How it works