Exponential horn

An exponential horn lets its cross-section grow by the same percentage for every unit of length. Starting from the throat area $S_t$, the area at a distance $x$ along the axis is

where $m$ is the flare constant and $f_c$ the cut-off frequency. Below $f_c$ an infinitely long horn of this shape passes no acoustic power at all. Above it the throat sees a resistance that climbs quickly towards $\rho c/S_t$, which is what gives a small diaphragm something substantial to push against.

Length and mouth

Solving the area law for the distance at which the mouth area is reached gives the horn length:

The mouth must be large enough to hand the wave over to the room instead of reflecting it back down the horn. The classic rule for a horn in free space is a mouth circumference of one wavelength at cut-off:

Every boundary the mouth sits against (floor, wall, corner) mirrors it and halves the area needed, which is where the 0.71, 0.50 and 0.35 circumference factors under the slider come from.

Throat impedance

For the infinitely long horn the acoustic impedance at the throat has a closed form:

Fig. 2 plots that ideal as the dashed line and compares it with the horn you have actually specified. The finite horn is sliced into 160 short conical elements, each with an exact transfer matrix, and terminated by the radiation impedance of a piston in a baffle. The smaller the mouth relative to $\lambda_c$, the deeper the ripple just above cut-off.

Air at 20 °C, c = 343.2 m/s. The station table gives round and square sections of equal area; for a rectangular horn keep the area column and choose your own aspect ratio.