… where CD is the effective distance, L is the diameter of the speaker and f is the frequency.
If we define a minimum working distance of 10 metres, this tells us that the diameter of the planar speaker has to be at least 3m to support 1000Hz and at least 6m to support 250Hz. Such speakers would be extremely heavy, expensive and cover a large part of a performance stage, so we don’t see them in real life.
However, all is not lost. In most sound reinforcement situations we actually prefer a wide horizontal dispersion so we can cover a wide audience area. This means we can focus on the vertical dispersion - aiming as much energy as possible for the audience ear height and avoiding it hitting the floor and ceilings. Sadly, this ‘half way’ solution doesn’t give us the 0dB per double distance SPL decay, but instead 3dB. Still, compared to a point source’s 6dB, it’s a big improvement!
This means that the planar loudspeaker could be turned into a line speaker - long in the vertical direction and thin, or ‘regular sized’, in the horizontal direction. Such line speakers could be situated on both sides of a performance stage so the stage is fully visible… easy!
Unfortunately, even in theory, there are no ‘line speakers’ that can produce enough power for sound reinforcement. Instead, we have to use multiple regular loudspeakers lined up as an array - either as separate cabinets or in a combined long and thin speaker enclosure: a line array. To make this work, the individual speakers in the line array have to ‘couple’ - that is to act as a line and radiate their combined acoustic energy as a line-shaped air pressure wave instead of a cone-shaped one. For this to happen, the drivers have to be separated with less than half a wavelength of the lowest supported frequency. For 125 Hz this means about 1.44m, which is no problem - a 15” (about 38cm) speaker can easily do this. The array just needs to be large and heavy, which is the reason they are only used for large systems.
For higher frequencies, however, it's more difficult to achieve coupling. At a frequency of 4000 Hz it requires a distance between the drivers of about 4cm (1.7”), so drivers have to be smaller than that.
This is the challenge of line array speaker design; the trade off between frequency, power, vertical dispersion and effective distance. You can see huge line-arrays with 15” speakers at open air concerts giving spectacular high quality results for audiences of thousands, but smaller line arrays - with speakers down to 1.7” or smaller - are also used for high quality speech and background music in reverberant spaces, such as railway stations and airports.
When loudspeakers in a line array are driven by individual power amplifiers, dedicated digital signal processing DSP can be used to shape one or more ‘beams’ of sound. These are known as ‘beam steering’ loudspeakers, used vertically for software-variable dispersion for sound reinforcement, and horizontally for home audio systems to simulate surround sound with a ‘sound bar’ under the television screen.