Loading & acoustics

Flare-rate finder

Measure an existing horn at two stations and recover its flare constant and cut-off frequency.

Sheet: 11 / 25
Units: Metric
Air: 20 °C

Fig. 1Equivalent round section between the two stations

Flare cut-off $f_c$: 139Hz
Flare constant $m$: 5.10m⁻¹
Area doubles every: 13.6cm
Growth per 10 cm: 66.5%
Area 1 $S_1$: 100cm²
Area 2 $S_2$: 768cm²

Table 1Areas an exponential flare would have in between

Stations

#	x (mm)	⌀ round (mm)	□ side (mm)	area (cm²)
0	0	112.8	100.0	100.0
1	33.33	122.8	108.9	118.5
2	66.67	133.7	118.5	140.5
3	100.0	145.6	129.0	166.5
4	133.3	158.5	140.5	197.3
5	166.7	172.5	152.9	233.8
6	200.0	187.8	166.5	277.1
7	233.3	204.5	181.2	328.4
8	266.7	222.6	197.3	389.3
9	300.0	242.4	214.8	461.3
10	333.3	263.9	233.8	546.8
11	366.7	287.2	254.6	648.0
12	400.0	312.7	277.1	768.0

How it works

This sheet works backwards. Measure the inside width and height of an existing horn at two places, note how far apart they are along the axis, and it returns the flare rate and the cut-off frequency of the exponential that passes through both.

m = \frac{1}{d}\,\ln\frac{S_2}{S_1}, \qquad f_c = \frac{m\,c}{4\pi}, \qquad \ell_2 = \frac{\ln 2}{m}

Here $S_1$ and $S_2$ are the two cross-sectional areas, $d$ their spacing and $\ell_2$ the distance over which the area doubles. Only areas matter, so the same formula serves round, square and rectangular horns.

Getting a trustworthy number

Measure inside dimensions and stay clear of the throat adaptor and of any lip or round-over at the mouth.
Use the longest spacing the horn allows. A one-millimetre error matters less over 40 cm than over 5 cm.
Repeat with a different pair of stations. If the cut-off comes out much the same, the horn is exponential. If it drifts lower towards the mouth the flare is closer to a cone; if it drifts higher, it is a hypex, or a tractrix opening out at its mouth.
For a folded horn, measure the distance along the centre line of the air path, not through the panels.

Table 1 lists the areas a true exponential would have between your stations, handy for checking a third measurement.

Loading sheet…

x (mm)

⌀ round (mm)

□ side (mm)

area (cm²)

112.8

100.0

33.33

122.8

108.9

118.5

66.67

133.7

118.5

140.5

100.0

145.6

129.0

166.5

133.3

158.5

140.5

197.3

166.7

172.5

152.9

233.8

200.0

187.8

166.5

277.1

233.3

204.5

181.2

328.4

266.7

222.6

197.3

389.3

300.0

242.4

214.8

461.3

333.3

263.9

233.8

546.8

366.7

287.2

254.6

648.0

400.0

312.7

277.1

768.0

How it works

m = \frac{1}{d}\,\ln\frac{S_2}{S_1}, \qquad f_c = \frac{m\,c}{4\pi}, \qquad \ell_2 = \frac{\ln 2}{m}

Getting a trustworthy number

Measure inside dimensions and stay clear of the throat adaptor and of any lip or round-over at the mouth.
Use the longest spacing the horn allows. A one-millimetre error matters less over 40 cm than over 5 cm.
Repeat with a different pair of stations. If the cut-off comes out much the same, the horn is exponential. If it drifts lower towards the mouth the flare is closer to a cone; if it drifts higher, it is a hypex, or a tractrix opening out at its mouth.
For a folded horn, measure the distance along the centre line of the air path, not through the panels.

Table 1 lists the areas a true exponential would have between your stations, handy for checking a third measurement.

Flare-rate finder

Inputs

Station 1, nearer the throat

Station 2, nearer the mouth

Spacing

How it works

Getting a trustworthy number

Inputs

Station 1, nearer the throat

Station 2, nearer the mouth

Spacing

How it works

Getting a trustworthy number

Station 1, nearer the throat

Station 2, nearer the mouth

Spacing

§ How it works

Getting a trustworthy number

Station 1, nearer the throat

Station 2, nearer the mouth

Spacing

§ How it works

Getting a trustworthy number

How it works

How it works