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Perhaps the best place to start is a 200 mm drive unit intended for low and mid frequency reproduction. This isn't the biggest drive unit available, so why are larger drive units ever necessary? The answer is to achieve a higher sound level. A 200 mm drive unit only pushes against so much air. Increase the diameter to 300 mm or 375 mm and many more air molecules feel the impact. The next question would be, why are 300 mm or 375 mm drive units not used more often, when space is available? The answer to that is in the behavior of the diaphragm:
200 mm is a good compromise. It will produce enough level at low frequency for the average living room, and it will produce reasonably distortion-free sound up to around 4 kHz or so. When the diaphragm bends, it is called break up, due to the vibration 'breaking up' into a number of different modes. Break up, in this context, doesn't mean severe distortion or anything like that. In fact most low frequency drive units are operated well into the break up region. It is up to the designer to ensure that the distortion created doesn't sound too unpleasant. By the way, it is often thought that a larger drive unit will operate down to lower frequencies. This isn't quite the right way to look at it. Any size of drive unit will operate down to as low a frequency as you like, but you need a big drive unit to shift large quantities of air at low frequency. At high frequency, the drive unit vibrates backwards and forwards rapidly, moving air on each vibration. At low frequencies there are fewer opportunities to move air, therefore the area of the drive unit needs to be greater to achieve the desired level.
The material of the diaphragm has a significant effect on its stiffness. Early moving coil drive units used paper pulp diaphragms, which were not particularly stiff. Modern drive units use plastic diaphragms, or pulp diaphragms that have been doped to stiffen them adequately. Of course, the ultimate in stiffness would be a metal diaphragm. Unfortunately, it would be heavy and the drive unit would be less efficient. Carbon fiber diaphragms have also been used with some success. (It is worth noting that in drive units used for electric guitars, the diaphragm is designed to bend and distort. It is part of the sound of the instrument and a distortion-free sound would not meet a guitarist's requirements).
Moving up the frequency range: as we have said, the diaphragm will bend and produce distortion. Even if it didn't, there would still be the problem that a large sound source will tend to focus sound over a narrow area, becoming narrower as the frequency increases. In fact, this is the characteristic of direct radiator loudspeakers: that their angle of coverage decreases as the frequency gets higher. This is significant in PA, where a single loudspeaker has to cover a large number of people. (It is perhaps counter-intuitive that a large sound source will focus the sound, but it is certainly so. A good acoustics text will supply the explanation).
Because of these two factors, higher frequencies are handled by a smaller drive unit. A smaller diaphragm is more rigid at higher frequencies, and because it is smaller it spreads sound more widely. Often the diaphragm is dome shaped rather than conical. This is part of the designer's art and isn't of direct relevance to the sound engineer, as long as it sounds good.
It might be stating the obvious at this stage, but a low frequency drive unit is commonly known as a woofer, and a high frequency drive unit as a tweeter.
In loudspeakers where a low frequency drive unit greater than 200 mm is used, it will not be possible to use the woofer up to a sufficiently high frequency to hand over directly to the tweeter. Therefore a mid frequency drive unit has to be used (sometimes known as a squawker!). The function of dividing the frequency band among the various drive units is handled by a crossover.