Constant-directivity mouth

A constant-directivity horn uses straight, conical walls to fix the coverage angle. The walls can only do that while the mouth is large enough in wavelengths. D. B. Keele's 1975 study of such horns gave a simple relation between mouth dimension $x$, coverage angle $\theta$ and the lowest frequency at which that angle is still held:

Apply it separately to width and height. A 90° horn 600 mm wide holds its horizontal pattern down to about 470 Hz; if it is only 300 mm tall and aims for 40° vertically, vertical control is lost below about 2.1 kHz. The narrower the angle, the larger the mouth has to be. Narrow vertical coverage is the expensive one.

Below the limit

Under $f$ the beamwidth is set by diffraction at the mouth instead of the walls and widens in inverse proportion to frequency, as in Fig. 1. Real horns also show some narrowing just above the limit, which the asymptotes ignore.

Wall depth

Straight walls from a throat of width $d_t$ reach the mouth dimension $x$ at an axial depth

The two planes generally need different depths, as Fig. 2 shows. Practical designs start the narrow-angle walls first and let the wide-angle walls open later, or use a diffraction slot, and finish with a wider flare near the mouth to tame the mid-range narrowing.

The price

Spreading high frequencies over a constant angle instead of letting them beam means the on-axis response falls at about 6 dB per octave above the driver's mass break point. That is corrected electrically; see the CD-horn equaliser sheet.

After D. B. Keele Jr., “What's So Sacred About Exponential Horns?”, AES preprint 1038, 1975.