Radial vs. Constant Directivity

A radial horn has a shape, when viewed from above as a segment from a circle. The side walls are lines of radius, hence.... Ah you're already there... They have a constant coverage pattern in the horizontal direction but the vertical profile was used to produce an exponential flare rate so the combined directivity is not constant.

Because the directivity is constant, the on axis response is a true reflection (probably a bad technical word) of the power response of drive unit. This requires a boost of about 6dB per octave above the natural roll off point of the driver. This can be between 2KHz and 6KHz depending on the particular drive unit. This characteristic has led to a few arguments about whether CD horns have a falling high frequency response or not.

It depends what you mean by flatter response. An exponential horn will have a flatter on axis response because the increased directivity at higher frequencies compensates for the reduction in power output. Without eq the constant directivity horn will have a falling response at higher frequencies. The power output is the same, but the power is spread over a bigger area

The problem with traditional constant directivity horns is that they depend on difraction slots to achieve the required dispersion. Each time there is an abrupt change in the side wall there is change in acoustic impedance which causes a partial reflection back down the horn. The same thing happens where the horn exits at the mouth. It is these reflections that are considered to cause the characteristic horn honk.

The shape of the mantaray type horn is a result of the difference in horizontal and vertical coverage angles. The usually narrower vertical dispersion requires a horn that is longer than that required for the horizontal dispersion. The characteristic narrow slot and faster flare effectively makes the horn two different lengths. A down side is that this causes the equivalent of astigmatism and the horns only array effectively in one direction.

Radials are not that much better because although the mouth is wide it is not normally very tall causing an imperfect mouth termination.

There is a trend now for horns to have a conical flare with a mouth whose width is not to dissimilar to the height. Line array slots obviously buck that trend. With more robust drivers these days optimum loading is less important. I still have some old JBL drivers rated at 3W RMS.

Imagine the sound starting off as a small bubble at the throat and expanding as it moves down the horn. As long as the bubble is trying to expand faster than the horn and its diameter is smaller than the horn width, the horn will determine the shape, or directivity of the bubble. Higher frequencies produce smaller bubles than lower frequencies. They therfore lose contact with the horn nearer the throat. Their directivity is therefore determined by the angle of the horn walls near the throat. Lower frequencies lose contact further down the horn and their directivity is determined by the angle of the walls near the mouth. With a conical horn, which has straight wall the angle is the same all along the horn which gives a constant directivity. Old style exponential horns have a wall angle that increases along the length which causes the higher frequencies to beam because they only see the first part of the horn.

With a 2" exit compression driver, at frequencies above about 6.5KHz the horn is irrelevant the shape and layout in the compression driver itself determines the response.