Cup a hand behind a phone’s speaker and a voice gets louder while the bass stays as thin as before. A horn does the same job with a calculated shape. How low it keeps working depends on two things you can work out before cutting any material: how fast the horn opens and how big its mouth is.
What the cutoff frequency is
Webster’s horn equation treats a horn as a duct whose cross-section changes along its length. It assumes the same pressure across each cross-section and walls that do not move. Within those assumptions a horn behaves like a high-pass filter.
In an exponential horn the area grows by the same factor over every equal step along the axis:
S(x) = St · emx
Here St is the throat area, x the distance from the throat and m the flare constant. The cutoff frequency follows from m and the speed of sound c:
fc = m · c / (4π)
Below fc an infinitely long exponential horn presents the driver with a purely reactive load, and no power travels down it. Above fc the resistive part of the load climbs towards ρc/St, the load of an endless straight pipe with the throat’s area. As a fraction of that value it equals √(1 − (fc/f)²): 0.60 at 1.25 times the cutoff and 0.87 at twice the cutoff. A real, finite horn still lets some sound out below fc. The cutoff frequency marks where the horn stops helping the driver.
Turned around, the formula gives the flare that a target cutoff needs: m = 4π fc/c. Halve the cutoff and the horn must open half as fast. The area doubles every ln 2/m along the axis.
Numbers for a horn that has to fit beside a phone
The table works through three target cutoff frequencies for a phone-sized source. It takes c = 343 m/s, the speed of sound in dry air at 20 °C. The throat is 20 mm², the area of a slot 2 mm by 10 mm, and the mouth circumference is one wavelength at the cutoff. These are round numbers for the arithmetic, not Braeburn dimensions.
| Target cutoff | Flare constant m | Area doubles every | Mouth diameter | Length, throat to mouth |
|---|---|---|---|---|
| 500 Hz | 18.3 m⁻¹ | 37.8 mm | 218 mm | 411 mm |
| 1 kHz | 36.6 m⁻¹ | 18.9 mm | 109 mm | 168 mm |
| 2 kHz | 73.3 m⁻¹ | 9.5 mm | 55 mm | 65 mm |
| Formula | 4π fc/c | ln 2/m | c/(π fc) | ln(Smouth/St)/m |
The 500 Hz horn comes out 41 cm long with a mouth 22 cm across. That is a gramophone horn. At 2 kHz the same rules give 65 mm of length and a 55 mm mouth, which fits inside a desk cradle. The table explains why compact passive amplifiers lift the upper midrange and treble and leave the bass where it was.
The arithmetic also runs backwards. A round mouth 50 mm across has a circumference of 157 mm, one wavelength at about 2.2 kHz. Folding can fit the 168 mm path of the 1 kHz horn into a shorter body, but it leaves the 109 mm mouth as wide as before. The page on the Auricle waveguide explains why a desk cradle folds its horn. Each bend adds reflections of its own.
Why the mouth sets the floor
When a long wave leaves a narrow duct, the pressure at the opening falls and part of the wave travels back down the horn. A shorter wave meets the same opening as a relatively larger one and reflects less.
The usual guideline asks for a mouth circumference of at least one wavelength at the cutoff, written krm ≥ 1. Here k = 2πfc/c is the wavenumber at the cutoff and rm is the mouth radius. D. B. Keele modelled exponential horns in an AES convention paper and placed the optimum slightly below krm = 1, assuming plane waves. With spherical wavefronts, Kolbrek reports, there is no obvious optimum. Kolbrek’s conclusion: a mouth can hardly be too large but easily too small, and midrange and treble horns benefit from krm ≥ 1. Keele also showed that the size needed depends on the solid angle the mouth radiates into. A cradle mouth at desk level radiates into roughly half-space, so the free-space guideline errs on the large side.
The mouth also limits pattern control. For a horn built to hold a constant angle, Kolbrek gives the lowest controlled frequency as fI = 25 × 10⁶ / (x · θ). Here x is the mouth width in millimetres and θ the angle in degrees. With a 55 mm mouth, even such a horn holds a 90° pattern only down to about 5.1 kHz.
Three flare profiles with the same ends
The first figure draws the three horns to scale for the 2 kHz case.
Three horns for a 2 kHz cutoff with the same throat and mouth. Upper half of the section, drawn to scale.
- Throat: a circle with the area of the 2 × 10 mm slot, radius 2.5 mm
- Exponential wall: area doubles every 9.5 mm
- Conical wall: a straight line between the same ends
- Tractrix wall: ends at right angles to the axis
- Mouth radius 27.3 mm: circumference one wavelength at 2 kHz
- Axis; the lower half mirrors the upper
- Exponential and conical length: 65 mm
- Tractrix length: 56.7 mm
Exponential
The exponential horn opens slowly near the throat and quickly near the mouth. Its throat resistance rises steeply just above the cutoff, so it loads the driver well from close to fc upwards. An infinitely long horn with the same 2 kHz flare reaches 80% of its final value at about 3.3 kHz. The 65 mm horn from the table is less than half a wavelength long at 2 kHz, so mouth reflections shape its response near the cutoff frequency.
Conical
A cone has straight walls, so its area grows with the square of the distance from its apex. It has no sharp cutoff. The throat impedance of an infinitely long cone equals the radiation impedance of a pulsating sphere with radius x0, the distance from the apex to the throat. The resistive part is (kx0)²/(1 + (kx0)²).
Give the cone the same throat, mouth and length as the 2 kHz exponential horn, and its apex sits 6.6 mm behind the throat. An infinitely long cone with that apex distance reaches 80% only at about 16.5 kHz. Cones are simple to mill and print, and Kolbrek notes their virtues in directivity control: Keele’s constant-directivity horns use conical sections near the mouth. To load lower down, a cone must open slowly, which makes it long.
Throat resistance as a fraction of ρc/St, frequency on a log scale. Infinitely long horns, so the mouth adds no ripple.
- Cutoff of the exponential horn: 2 kHz
- Exponential: zero below the cutoff, 0.87 at twice it
- Cone with the same ends: no cutoff, slow rise
- Full load ρc/St of an endless pipe with the throat’s area
- At 4 kHz: 0.87 against 0.19
Tractrix
P. G. A. H. Voigt patented the tractrix horn. Its mouth radius is normally c/(2π fc), which makes the mouth circumference one wavelength at the cutoff, so the profile meets the mouth guideline by construction. It opens faster than the exponential near the mouth and ends at right angles to the axis. With the same ends it is shorter: 56.7 mm instead of 65 mm.
Hyperbolic and catenoidal
Salmon’s family of horns has the area S(x) = St (cosh(x/x0) + T sinh(x/x0))², with x0 = c/(2π fc). T = 1 gives the exponential horn, T = 0 the catenoidal one and a large T a cone. Below T = 1 the throat resistance rises towards its final value faster.
What the formulas leave out
Webster’s equation is exact only for shapes that carry true one-parameter waves: the uniform tube, the cone and a parabolic horn with two parallel sides. Other shapes must stay narrow compared with the wavelength, or higher-order modes appear. At 10 kHz the wavelength is 34 mm, less than the 55 mm mouth of the 2 kHz example. The top octave of a desk horn lies outside the simple theory.
The theory also assumes rigid, smooth walls and contains no term for a bend. Whether a printed or wooden wall is rigid enough is the subject of the guide to phone amplifier materials. The overview of passive phone amplifiers compares a horn with a resonant tube and a box with a slot. The formulas give a starting geometry; the fit at the throat and the bends decide much of what a finished cradle measures.
Common questions
How big would a horn have to be to reach bass?
At 100 Hz the wavelength is 3.43 m. A mouth with that circumference is about 1.09 m across. An exponential horn from a 20 mm² phone slot to that mouth would be about 2.9 m long. No desk cradle comes near that, so a passive amplifier leaves a phone’s bass essentially unchanged.
Is a straight cone good enough for a 3D-printed horn?
Only for the top octave, and even there it loads the driver less. In the second figure the cone with the same ends reaches 0.60 at 10 kHz, where the exponential horn is at 0.98. If the horn has to help between 2 and 5 kHz, the exponential profile is worth the extra drawing work.
What happens to the sound below the cutoff frequency?
An infinitely long exponential horn passes no power below its cutoff. A finite horn still lets some sound through, but it no longer works as an acoustic transformer and only adds a mass reactance. Near the cutoff, reflections from the mouth add ripples; a larger mouth keeps them smaller.
Sources
- W. M. Leach Jr., “A two-port analogous circuit and SPICE model for Salmon’s family of acoustic horns”, J. Acoust. Soc. Am., vol. 99, no. 3, pp. 1459–1464. Webster’s equation, Salmon’s family, cutoff.
- B. Kolbrek, “Horn Theory: An Introduction, Part 1”, audioXpress. Throat impedance, mouth guideline, tractrix, assumptions.
- B. Kolbrek, “Horn Theory: An Introduction, Part 2”, audioXpress. Higher-order modes, pattern control, constant-directivity horns.
- D. B. Keele Jr., “Optimum Horn Mouth Size”, AES 46th Convention, preprint 933.
- U.S. Standard Atmosphere, NASA. Speed of sound c = √(γRT/M): γ = 1.40, R = 8.314 J/(mol·K), M = 28.96 g/mol, T = 293 K give 343 m/s.