I assume you're talking about "coolers" as in CPU-coolers or other cooling systems in computers. The noise-spectrum from such a fan is not white strictly speaking, you can see it measured on this page. The spectrum is fairly level if you look at small parts though and ignore the tonal components.
Anyway, by your definition, two uncorrelated white noise sources add without interference. The "peaks and troughs" (sound pressures) don't add up coherently (in ideal sources of course :), but they don't cancel each other perfectly either. Therefore the sound power is doubled. By definition the RMS-amplitude (proportional to the sound pressure) is then multiplied by $\sqrt 2$. The sound intensity is proportional to the sound power and hence is doubled as well.
I actually simulated this in Matlab, because the terminology with sound power, sound energy, sound levels, sound pressures etc is daunting, misleading and confusing.. I'm specifically avoiding talking about decibels.
>> w1 = wgn(8192, 1, 10, 'real'); % generates real white noise, 8192 samples
>> w2 = wgn(8192, 1, 10, 'real');
>> mean(([w1 w2]).^2) % show mean powers of both noise signals
ans =
10.0410 10.0832
>> mean(([(w1+w1) (w1+w2)]).^2) % add one of them to itself, add both
ans =
40.1640 20.6295
So yes you do get a doubling of power when adding uncorrelated white-noise sources.. and when perfectly adding a single source, you quadruple the power. By definition the RMS amplitude is the root of the above.
It can be done, but there's some trade offs. Larger speakers are better at moving longer wavelength (low frequency) waves. When you try to combine a bunch of small surfaces in different locations to recreate a single wave you end up with a some random interference where the wave is stronger or weaker (in 3d-space) (see phased-array antenna for some examples). Whereas with one large speaker you get a more uniform reproduction of the wave in all directions.
In addition the size of the speaker can also aid in the how well it transmits the wave. Larger speakers better match the frequency of the low sounds so they are broadcast better and with more force than it would be on 3 smaller speakers with the equivalent surface area.
Edit: Typically woofers are built different than midrange or tweeters because they need to move a lot more air to create the powerful low frequency sounds you bounce to at the clubs. Midrange/tweeter speakers are not meant to be driven hard at low frequencies so you have to be careful with the amplification when driving these speakers outside of their optimized frequency range.
The design is optimized for low frequency response and allows the sound to resonate better (more efficiently). Likewise they don't handle the higher frequencies as well so this is why a filter called a cross over network is used to send the high notes to the tweeter and low to the bass.
Part of the design that is important is getting the box to also resonate with the bass because this will help project a stronger and cleaner sound and keep the backside of the low frequency wave from canceling the front side. This need to isolate sound waves between the front and back of an open-back driver does not exist at all frequencies. It occurs only for lower frequencies whose wavelength is relatively large when compared to the diameter of the cone. Because the diaphragm becomes directional as the wavelengths become shorter and so the sound will naturally not mix between the front and back even when the driver is in open air. This is why a driver in open air lacks bass or sounds "thin". It produces mid and high frequencies without the rear sound waves canceling the front ones. But the low frequencies are diminished or canceled.
Hopefully this helps you understand the differences in performance due to design. You can drive small speakers fine at low frequencies, look at headphones for example. However if you want to bounce the house with a powerful bass, you need a big diaphragm. And if you want to go cheap with a bunch of smaller ones, you have to be careful not to over drive them at low frequencies and you'll have to play with the box some to get the sound right. In the end the results will probably be better with one bass speaker instead of trying to wire in and adjust 9 others.
And naturally there is a limit to how well you can build something so that all their sounds combine to equal the surface area of one good speaker. As in your extreme example the 100 speakers are going to have a very weak low frequency sound and they are going to be spread out over a very big area. Getting that sound to combine right and not cancel each other out is probably impossible. You'll have 100 speakers all operating at different phases mixing and canceling each other around the room.
Best Answer
No, it has nothing to do with sound quality. In fact, the grid or covering is carefully chosen to interfere with the sound as little is possible.
Speaker cones must be light weight, so are made from paper or other thin and delicate material. The grill is to physically protect the delicate speaker cone from getting dinged, a curious cat, or some moron with a poky finger.
The tweeter in your picture doesn't have a covering because it is significantly recessed and behind a rigid and narrow barrier. Poky fingers can't fit in there to hurt the tweeter. Also, the high frequencies that the tweeter produces are more susceptible to attenuation by a cover, and a cover would cause diffusion and alter the radiation pattern. From the horn shape, it looks like the sound is intended to be directed in a somewhat narrow beam straight out.